Theorem hhsssh 9139

Description: The predicate "H is a subspace of Hilbert space."

Hypotheses

Ref	Expression
hhsst.1	⊢ U = ⟨⟨ +h , ·h ⟩, normh⟩
hhsst.2	⊢ W = ⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩

Assertion

Ref	Expression
hhsssh	⊢ (H ∈ Sℋ ↔ (W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ))

Proof of Theorem hhsssh

Step	Hyp	Ref	Expression
1		hhsst.1	. . . 4 ⊢ U = ⟨⟨ +h , ·h ⟩, normh⟩
2		hhsst.2	. . . 4 ⊢ W = ⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩
3	1, 2	hhsst 9136	. . 3 ⊢ (H ∈ Sℋ → W ∈ (SubSp ‘U))
4		shss 9079	. . 3 ⊢ (H ∈ Sℋ → H ⊆ ℋ )
5	3, 4	jca 288	. 2 ⊢ (H ∈ Sℋ → (W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ))
6		eleq1 1534	. . 3 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → (H ∈ Sℋ ↔ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) ∈ Sℋ ))
7		eqid 1475	. . . 4 ⊢ ⟨⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩, (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))⟩ = ⟨⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩, (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))⟩
8		xpeq1 3200	. . . . . . . . . . . . 13 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → (H × H) = ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × H))
9		xpeq2 3201	. . . . . . . . . . . . 13 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × H) = ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))
10	8, 9	eqtrd 1507	. . . . . . . . . . . 12 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → (H × H) = ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))
11		reseq2 3369	. . . . . . . . . . . 12 ⊢ ((H × H) = ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )) → ( +h ↾ (H × H)) = ( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))))
12	10, 11	syl 10	. . . . . . . . . . 11 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ( +h ↾ (H × H)) = ( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))))
13		xpeq2 3201	. . . . . . . . . . . 12 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → (ℂ × H) = (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))
14		reseq2 3369	. . . . . . . . . . . 12 ⊢ ((ℂ × H) = (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )) → ( ·h ↾ (ℂ × H)) = ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))))
15	13, 14	syl 10	. . . . . . . . . . 11 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ( ·h ↾ (ℂ × H)) = ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))))
16	12, 15	opeq12d 2495	. . . . . . . . . 10 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩ = ⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩)
17		reseq2 3369	. . . . . . . . . 10 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → (normh ↾ H) = (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))
18	16, 17	opeq12d 2495	. . . . . . . . 9 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩ = ⟨⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩, (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))⟩)
19	18, 2	syl5eq 1519	. . . . . . . 8 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → W = ⟨⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩, (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))⟩)
20	19	eleq1d 1540	. . . . . . 7 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → (W ∈ (SubSp ‘U) ↔ ⟨⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩, (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))⟩ ∈ (SubSp ‘U)))
21		sseq1 2082	. . . . . . 7 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → (H ⊆ ℋ ↔ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) ⊆ ℋ ))
22	20, 21	anbi12d 628	. . . . . 6 ⊢ (H = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ) ↔ (⟨⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩, (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))⟩ ∈ (SubSp ‘U) ⋀ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) ⊆ ℋ )))
23		xpeq1 3200	. . . . . . . . . . . 12 ⊢ ( ℋ = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ( ℋ × ℋ ) = ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × ℋ ))
24		xpeq2 3201	. . . . . . . . . . . 12 ⊢ ( ℋ = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × ℋ ) = ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))
25	23, 24	eqtrd 1507	. . . . . . . . . . 11 ⊢ ( ℋ = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ( ℋ × ℋ ) = ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))
26		reseq2 3369	. . . . . . . . . . 11 ⊢ (( ℋ × ℋ ) = ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )) → ( +h ↾ ( ℋ × ℋ )) = ( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))))
27	25, 26	syl 10	. . . . . . . . . 10 ⊢ ( ℋ = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ( +h ↾ ( ℋ × ℋ )) = ( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))))
28		xpeq2 3201	. . . . . . . . . . 11 ⊢ ( ℋ = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → (ℂ × ℋ ) = (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))
29		reseq2 3369	. . . . . . . . . . 11 ⊢ ((ℂ × ℋ ) = (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )) → ( ·h ↾ (ℂ × ℋ )) = ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))))
30	28, 29	syl 10	. . . . . . . . . 10 ⊢ ( ℋ = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ( ·h ↾ (ℂ × ℋ )) = ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))))
31	27, 30	opeq12d 2495	. . . . . . . . 9 ⊢ ( ℋ = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ⟨( +h ↾ ( ℋ × ℋ )), ( ·h ↾ (ℂ × ℋ ))⟩ = ⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩)
32		reseq2 3369	. . . . . . . . 9 ⊢ ( ℋ = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → (normh ↾ ℋ ) = (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))
33	31, 32	opeq12d 2495	. . . . . . . 8 ⊢ ( ℋ = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ⟨⟨( +h ↾ ( ℋ × ℋ )), ( ·h ↾ (ℂ × ℋ ))⟩, (normh ↾ ℋ )⟩ = ⟨⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩, (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))⟩)
34	33	eleq1d 1540	. . . . . . 7 ⊢ ( ℋ = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → (⟨⟨( +h ↾ ( ℋ × ℋ )), ( ·h ↾ (ℂ × ℋ ))⟩, (normh ↾ ℋ )⟩ ∈ (SubSp ‘U) ↔ ⟨⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩, (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))⟩ ∈ (SubSp ‘U)))
35		sseq1 2082	. . . . . . 7 ⊢ ( ℋ = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ( ℋ ⊆ ℋ ↔ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) ⊆ ℋ ))
36	34, 35	anbi12d 628	. . . . . 6 ⊢ ( ℋ = if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) → ((⟨⟨( +h ↾ ( ℋ × ℋ )), ( ·h ↾ (ℂ × ℋ ))⟩, (normh ↾ ℋ )⟩ ∈ (SubSp ‘U) ⋀ ℋ ⊆ ℋ ) ↔ (⟨⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩, (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))⟩ ∈ (SubSp ‘U) ⋀ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) ⊆ ℋ )))
37		ax-hfvadd 8870	. . . . . . . . . . . . 13 ⊢ +h :( ℋ × ℋ )–→ ℋ
38		ffn 3627	. . . . . . . . . . . . 13 ⊢ ( +h :( ℋ × ℋ )–→ ℋ → +h Fn ( ℋ × ℋ ))
39	37, 38	ax-mp 7	. . . . . . . . . . . 12 ⊢ +h Fn ( ℋ × ℋ )
40		fnresdm 3596	. . . . . . . . . . . 12 ⊢ ( +h Fn ( ℋ × ℋ ) → ( +h ↾ ( ℋ × ℋ )) = +h )
41	39, 40	ax-mp 7	. . . . . . . . . . 11 ⊢ ( +h ↾ ( ℋ × ℋ )) = +h
42		ax-hfvmul 8875	. . . . . . . . . . . . 13 ⊢ ·h :(ℂ × ℋ )–→ ℋ
43		ffn 3627	. . . . . . . . . . . . 13 ⊢ ( ·h :(ℂ × ℋ )–→ ℋ → ·h Fn (ℂ × ℋ ))
44	42, 43	ax-mp 7	. . . . . . . . . . . 12 ⊢ ·h Fn (ℂ × ℋ )
45		fnresdm 3596	. . . . . . . . . . . 12 ⊢ ( ·h Fn (ℂ × ℋ ) → ( ·h ↾ (ℂ × ℋ )) = ·h )
46	44, 45	ax-mp 7	. . . . . . . . . . 11 ⊢ ( ·h ↾ (ℂ × ℋ )) = ·h
47	41, 46	opeq12i 2492	. . . . . . . . . 10 ⊢ ⟨( +h ↾ ( ℋ × ℋ )), ( ·h ↾ (ℂ × ℋ ))⟩ = ⟨ +h , ·h ⟩
48		normf 8989	. . . . . . . . . . . 12 ⊢ normh: ℋ –→ℝ
49		ffn 3627	. . . . . . . . . . . 12 ⊢ (normh: ℋ –→ℝ → normh Fn ℋ )
50	48, 49	ax-mp 7	. . . . . . . . . . 11 ⊢ normh Fn ℋ
51		fnresdm 3596	. . . . . . . . . . 11 ⊢ (normh Fn ℋ → (normh ↾ ℋ ) = normh)
52	50, 51	ax-mp 7	. . . . . . . . . 10 ⊢ (normh ↾ ℋ ) = normh
53	47, 52	opeq12i 2492	. . . . . . . . 9 ⊢ ⟨⟨( +h ↾ ( ℋ × ℋ )), ( ·h ↾ (ℂ × ℋ ))⟩, (normh ↾ ℋ )⟩ = ⟨⟨ +h , ·h ⟩, normh⟩
54	53, 1	eqtr4 1498	. . . . . . . 8 ⊢ ⟨⟨( +h ↾ ( ℋ × ℋ )), ( ·h ↾ (ℂ × ℋ ))⟩, (normh ↾ ℋ )⟩ = U
55	1	hhnv 9032	. . . . . . . . 9 ⊢ U ∈ NrmCVec
56		eqid 1475	. . . . . . . . . 10 ⊢ (SubSp ‘U) = (SubSp ‘U)
57	56	sspid 8384	. . . . . . . . 9 ⊢ (U ∈ NrmCVec → U ∈ (SubSp ‘U))
58	55, 57	ax-mp 7	. . . . . . . 8 ⊢ U ∈ (SubSp ‘U)
59	54, 58	eqeltr 1544	. . . . . . 7 ⊢ ⟨⟨( +h ↾ ( ℋ × ℋ )), ( ·h ↾ (ℂ × ℋ ))⟩, (normh ↾ ℋ )⟩ ∈ (SubSp ‘U)
60		ssid 2080	. . . . . . 7 ⊢ ℋ ⊆ ℋ
61	59, 60	pm3.2i 285	. . . . . 6 ⊢ (⟨⟨( +h ↾ ( ℋ × ℋ )), ( ·h ↾ (ℂ × ℋ ))⟩, (normh ↾ ℋ )⟩ ∈ (SubSp ‘U) ⋀ ℋ ⊆ ℋ )
62	22, 36, 61	elimhyp 2390	. . . . 5 ⊢ (⟨⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩, (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))⟩ ∈ (SubSp ‘U) ⋀ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) ⊆ ℋ )
63	62	pm3.26i 320	. . . 4 ⊢ ⟨⟨( +h ↾ ( if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))), ( ·h ↾ (ℂ × if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ )))⟩, (normh ↾ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ))⟩ ∈ (SubSp ‘U)
64	62	pm3.27i 324	. . . 4 ⊢ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) ⊆ ℋ
65	1, 7, 63, 64	hhshsslem2 9138	. . 3 ⊢ if((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ), H, ℋ ) ∈ Sℋ
66	6, 65	dedth 2383	. 2 ⊢ ((W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ) → H ∈ Sℋ )
67	5, 66	impbi 157	1 ⊢ (H ∈ Sℋ ↔ (W ∈ (SubSp ‘U) ⋀ H ⊆ ℋ ))

Syntax hints:   ↔ wb 146   ⋀ wa 223   = wceq 956   ∈ wcel 958   ⊆ wss 2047   ifcif 2361  ⟨cop 2411   × cxp 3168   ↾ cres 3172   Fn wfn 3177  –→wf 3178   ‘cfv 3182  ℂcc 5232  ℝcr 5233  NrmCVeccnv 8203  SubSpcss 8380   ℋ chil 8788   +_h cva 8789   ·_h csm 8790  norm_hcno 8794   S_ℋ csh 8797

This theorem is referenced by:  hhsssh2 9140

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625  ax-hilex 8869  ax-hfvadd 8870  ax-hvcom 8871  ax-hvass 8872  ax-hv0cl 8873  ax-hvaddid 8874  ax-hfvmul 8875  ax-hvmulid 8876  ax-hvmulass 8877  ax-hvdistr1 8878  ax-hvdistr2 8879  ax-hvmul0 8880  ax-hfi 8946  ax-his1 8949  ax-his2 8950  ax-his3 8951  ax-his4 8952

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-sup 4574  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-n 5925  df-2 5970  df-3 5971  df-4 5972  df-n0 6100  df-z 6136  df-seq1 6308  df-exp 6569  df-sqr 6670  df-re 6751  df-im 6752  df-cj 6753  df-abs 6754  df-grp 8037  df-gid 8038  df-ginv 8039  df-gdiv 8040  df-abl 8100  df-subg 8115  df-vc 8165  df-nv 8211  df-va 8214  df-ba 8215  df-sm 8216  df-0v 8217  df-vs 8218  df-nm 8219  df-ssp 8381  df-hnorm 8837  df-hvsub 8840  df-hlim 8841  df-sh 9076  df-ch 9092  df-ch0 9125

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