Theorem hhsssh2 9095

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

Hypothesis

Ref	Expression
hhsssh2.1	⊢ W = ⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩

Assertion

Ref	Expression
hhsssh2	⊢ (H ∈ Sℋ ↔ (W ∈ NrmCVec ⋀ H ⊆ ℋ ))

Proof of Theorem hhsssh2

Step	Hyp	Ref	Expression
1		eqid 1475	. . 3 ⊢ ⟨⟨ +h , ·h ⟩, normh⟩ = ⟨⟨ +h , ·h ⟩, normh⟩
2		hhsssh2.1	. . 3 ⊢ W = ⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩
3	1, 2	hhsssh 9094	. 2 ⊢ (H ∈ Sℋ ↔ (W ∈ (SubSp ‘⟨⟨ +h , ·h ⟩, normh⟩) ⋀ H ⊆ ℋ ))
4	1	hhnv 8987	. . . . 5 ⊢ ⟨⟨ +h , ·h ⟩, normh⟩ ∈ NrmCVec
5	1	hhva 8988	. . . . . 6 ⊢ +h = ( +v ‘⟨⟨ +h , ·h ⟩, normh⟩)
6	2	fveq2i 3725	. . . . . . 7 ⊢ ( +v ‘W) = ( +v ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)
7		eqid 1475	. . . . . . . . 9 ⊢ ( +v ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = ( +v ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)
8	7	vafval 8179	. . . . . . . 8 ⊢ ( +v ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = (1st ‘(1st ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩))
9		opex 2780	. . . . . . . . . . 11 ⊢ ⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩ ∈ V
10	9	op1st 4083	. . . . . . . . . 10 ⊢ (1st ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = ⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩
11	10	fveq2i 3725	. . . . . . . . 9 ⊢ (1st ‘(1st ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)) = (1st ‘⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩)
12		hilabl 8982	. . . . . . . . . . 11 ⊢ +h ∈ Abel
13		resexg 3392	. . . . . . . . . . 11 ⊢ ( +h ∈ Abel → ( +h ↾ (H × H)) ∈ V)
14	12, 13	ax-mp 7	. . . . . . . . . 10 ⊢ ( +h ↾ (H × H)) ∈ V
15	14	op1st 4083	. . . . . . . . 9 ⊢ (1st ‘⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩) = ( +h ↾ (H × H))
16	11, 15	eqtr 1494	. . . . . . . 8 ⊢ (1st ‘(1st ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)) = ( +h ↾ (H × H))
17	8, 16	eqtr 1494	. . . . . . 7 ⊢ ( +v ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = ( +h ↾ (H × H))
18	6, 17	eqtr2 1495	. . . . . 6 ⊢ ( +h ↾ (H × H)) = ( +v ‘W)
19	1	hhsm 8991	. . . . . 6 ⊢ ·h = ( ·s ‘⟨⟨ +h , ·h ⟩, normh⟩)
20	2	fveq2i 3725	. . . . . . 7 ⊢ ( ·s ‘W) = ( ·s ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)
21		eqid 1475	. . . . . . . . 9 ⊢ ( ·s ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = ( ·s ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)
22	21	smfval 8181	. . . . . . . 8 ⊢ ( ·s ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = (2nd ‘(1st ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩))
23	10	fveq2i 3725	. . . . . . . . 9 ⊢ (2nd ‘(1st ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)) = (2nd ‘⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩)
24		hvmulex 8836	. . . . . . . . . . 11 ⊢ ·h ∈ V
25		resexg 3392	. . . . . . . . . . 11 ⊢ ( ·h ∈ V → ( ·h ↾ (ℂ × H)) ∈ V)
26	24, 25	ax-mp 7	. . . . . . . . . 10 ⊢ ( ·h ↾ (ℂ × H)) ∈ V
27	14, 26	op2nd 4084	. . . . . . . . 9 ⊢ (2nd ‘⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩) = ( ·h ↾ (ℂ × H))
28	23, 27	eqtr 1494	. . . . . . . 8 ⊢ (2nd ‘(1st ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)) = ( ·h ↾ (ℂ × H))
29	22, 28	eqtr 1494	. . . . . . 7 ⊢ ( ·s ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = ( ·h ↾ (ℂ × H))
30	20, 29	eqtr2 1495	. . . . . 6 ⊢ ( ·h ↾ (ℂ × H)) = ( ·s ‘W)
31	1	hhnm 8993	. . . . . 6 ⊢ normh = (norm ‘⟨⟨ +h , ·h ⟩, normh⟩)
32	2	fveq2i 3725	. . . . . . 7 ⊢ (norm ‘W) = (norm ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)
33		eqid 1475	. . . . . . . . 9 ⊢ (norm ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = (norm ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)
34	33	nmfval 8183	. . . . . . . 8 ⊢ (norm ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = (2nd ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)
35		normf 8944	. . . . . . . . . . 11 ⊢ normh: ℋ –→ℝ
36		ax-hilex 8824	. . . . . . . . . . 11 ⊢ ℋ ∈ V
37		fex 3650	. . . . . . . . . . 11 ⊢ ((normh: ℋ –→ℝ ⋀ ℋ ∈ V) → normh ∈ V)
38	35, 36, 37	mp2an 697	. . . . . . . . . 10 ⊢ normh ∈ V
39		resexg 3392	. . . . . . . . . 10 ⊢ (normh ∈ V → (normh ↾ H) ∈ V)
40	38, 39	ax-mp 7	. . . . . . . . 9 ⊢ (normh ↾ H) ∈ V
41	9, 40	op2nd 4084	. . . . . . . 8 ⊢ (2nd ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = (normh ↾ H)
42	34, 41	eqtr 1494	. . . . . . 7 ⊢ (norm ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = (normh ↾ H)
43	32, 42	eqtr2 1495	. . . . . 6 ⊢ (normh ↾ H) = (norm ‘W)
44		eqid 1475	. . . . . 6 ⊢ (SubSp ‘⟨⟨ +h , ·h ⟩, normh⟩) = (SubSp ‘⟨⟨ +h , ·h ⟩, normh⟩)
45	5, 18, 19, 30, 31, 43, 44	isssp 8340	. . . . 5 ⊢ (⟨⟨ +h , ·h ⟩, normh⟩ ∈ NrmCVec → (W ∈ (SubSp ‘⟨⟨ +h , ·h ⟩, normh⟩) ↔ (W ∈ NrmCVec ⋀ (( +h ↾ (H × H)) ⊆ +h ⋀ ( ·h ↾ (ℂ × H)) ⊆ ·h ⋀ (normh ↾ H) ⊆ normh))))
46	4, 45	ax-mp 7	. . . 4 ⊢ (W ∈ (SubSp ‘⟨⟨ +h , ·h ⟩, normh⟩) ↔ (W ∈ NrmCVec ⋀ (( +h ↾ (H × H)) ⊆ +h ⋀ ( ·h ↾ (ℂ × H)) ⊆ ·h ⋀ (normh ↾ H) ⊆ normh)))
47		resss 3381	. . . . 5 ⊢ ( +h ↾ (H × H)) ⊆ +h
48		resss 3381	. . . . 5 ⊢ ( ·h ↾ (ℂ × H)) ⊆ ·h
49		resss 3381	. . . . 5 ⊢ (normh ↾ H) ⊆ normh
50	47, 48, 49	3pm3.2i 818	. . . 4 ⊢ (( +h ↾ (H × H)) ⊆ +h ⋀ ( ·h ↾ (ℂ × H)) ⊆ ·h ⋀ (normh ↾ H) ⊆ normh)
51	46, 50	mpbiran2 729	. . 3 ⊢ (W ∈ (SubSp ‘⟨⟨ +h , ·h ⟩, normh⟩) ↔ W ∈ NrmCVec)
52	51	anbi1i 481	. 2 ⊢ ((W ∈ (SubSp ‘⟨⟨ +h , ·h ⟩, normh⟩) ⋀ H ⊆ ℋ ) ↔ (W ∈ NrmCVec ⋀ H ⊆ ℋ ))
53	3, 52	bitr 173	1 ⊢ (H ∈ Sℋ ↔ (W ∈ NrmCVec ⋀ H ⊆ ℋ ))

Syntax hints:   ↔ wb 146   ⋀ wa 223   ⋀ w3a 775   = wceq 956   ∈ wcel 958  Vcvv 1809   ⊆ wss 2045  ⟨cop 2409   × cxp 3166   ↾ cres 3170  –→wf 3176   ‘cfv 3180  1^st c1st 4075  2^nd c2nd 4076  ℂcc 5220  ℝcr 5221  Abelcabl 8056  NrmCVeccnv 8160   +_v cpv 8161   ·_s cns 8163  normcnm 8166  SubSpcss 8337   ℋ chil 8743   +_h cva 8744   ·_h csm 8745  norm_hcno 8749   S_ℋ csh 8752

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 2691  ax-sep 2701  ax-nul 2708  ax-pow 2740  ax-pr 2777  ax-un 2864  ax-inf2 4613  ax-hilex 8824  ax-hfvadd 8825  ax-hvcom 8826  ax-hvass 8827  ax-hv0cl 8828  ax-hvaddid 8829  ax-hfvmul 8830  ax-hvmulid 8831  ax-hvmulass 8832  ax-hvdistr1 8833  ax-hvdistr2 8834  ax-hvmul0 8835  ax-hfi 8901  ax-his1 8904  ax-his2 8905  ax-his3 8906  ax-his4 8907

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 1586  df-nel 1587  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2502  df-int 2532  df-iun 2566  df-br 2618  df-opab 2665  df-tr 2679  df-eprel 2830  df-id 2833  df-po 2838  df-so 2848  df-fr 2915  df-we 2932  df-ord 2949  df-on 2950  df-lim 2951  df-suc 2952  df-om 3130  df-xp 3182  df-rel 3183  df-cnv 3184  df-co 3185  df-dm 3186  df-rn 3187  df-res 3188  df-ima 3189  df-fun 3190  df-fn 3191  df-f 3192  df-f1 3193  df-fo 3194  df-f1o 3195  df-fv 3196  df-rdg 3930  df-opr 3963  df-oprab 3964  df-1st 4077  df-2nd 4078  df-1o 4131  df-oadd 4133  df-omul 4134  df-er 4259  df-ec 4261  df-qs 4264  df-en 4365  df-dom 4366  df-sdom 4367  df-sup 4562  df-ni 4988  df-pli 4989  df-mi 4990  df-lti 4991  df-plpq 5023  df-mpq 5024  df-enq 5025  df-nq 5026  df-plq 5027  df-mq 5028  df-rq 5029  df-ltq 5030  df-1q 5031  df-np 5074  df-1p 5075  df-plp 5076  df-mp 5077  df-ltp 5078  df-plpr 5152  df-mpr 5153  df-enr 5154  df-nr 5155  df-plr 5156  df-mr 5157  df-ltr 5158  df-0r 5159  df-1r 5160  df-m1r 5161  df-c 5228  df-0 5229  df-1 5230  df-i 5231  df-r 5232  df-plus 5233  df-mul 5234  df-lt 5235  df-sub 5344  df-neg 5346  df-pnf 5475  df-mnf 5476  df-xr 5477  df-ltxr 5478  df-le 5479  df-div 5686  df-n 5893  df-2 5938  df-3 5939  df-4 5940  df-n0 6068  df-z 6104  df-seq1 6272  df-exp 6529  df-sqr 6630  df-re 6711  df-im 6712  df-cj 6713  df-abs 6714  df-grp 7994  df-gid 7995  df-ginv 7996  df-gdiv 7997  df-abl 8057  df-subg 8072  df-vc 8122  df-nv 8168  df-va 8171  df-ba 8172  df-sm 8173  df-0v 8174  df-vs 8175  df-nm 8176  df-ssp 8338  df-hnorm 8792  df-hvsub 8795  df-hlim 8796  df-sh 9031  df-ch 9047  df-ch0 9080

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