Theorem hhshsslem1 9122

Description: Lemma for hhsssh 9124.

Hypotheses

Ref	Expression
hhsst.1	⊢ U = ⟨⟨ +h , ·h ⟩, normh⟩
hhsst.2	⊢ W = ⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩
hhssp3.3	⊢ W ∈ (SubSp ‘U)
hhssp3.4	⊢ H ⊆ ℋ

Assertion

Ref	Expression
hhshsslem1	⊢ H = (Base ‘W)

Proof of Theorem hhshsslem1

Step	Hyp	Ref	Expression
1		eqid 1475	. . . 4 ⊢ (Base ‘W) = (Base ‘W)
2		eqid 1475	. . . 4 ⊢ ( +v ‘W) = ( +v ‘W)
3	1, 2	bafval 8208	. . 3 ⊢ (Base ‘W) = ran ( +v ‘W)
4		hhsst.1	. . . . . . . 8 ⊢ U = ⟨⟨ +h , ·h ⟩, normh⟩
5	4	hhnv 9017	. . . . . . 7 ⊢ U ∈ NrmCVec
6		hhssp3.3	. . . . . . 7 ⊢ W ∈ (SubSp ‘U)
7		eqid 1475	. . . . . . . 8 ⊢ (SubSp ‘U) = (SubSp ‘U)
8	7	sspnv 8370	. . . . . . 7 ⊢ ((U ∈ NrmCVec ⋀ W ∈ (SubSp ‘U)) → W ∈ NrmCVec)
9	5, 6, 8	mp2an 697	. . . . . 6 ⊢ W ∈ NrmCVec
10	2	nvgrp 8221	. . . . . 6 ⊢ (W ∈ NrmCVec → ( +v ‘W) ∈ Grp)
11	9, 10	ax-mp 7	. . . . 5 ⊢ ( +v ‘W) ∈ Grp
12		grprndm 8039	. . . . 5 ⊢ (( +v ‘W) ∈ Grp → ran ( +v ‘W) = dom dom ( +v ‘W))
13	11, 12	ax-mp 7	. . . 4 ⊢ ran ( +v ‘W) = dom dom ( +v ‘W)
14		hhsst.2	. . . . . . . . 9 ⊢ W = ⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩
15	14	fveq2i 3727	. . . . . . . 8 ⊢ ( +v ‘W) = ( +v ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)
16		eqid 1475	. . . . . . . . . 10 ⊢ ( +v ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = ( +v ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)
17	16	vafval 8207	. . . . . . . . 9 ⊢ ( +v ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = (1st ‘(1st ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩))
18		opex 2782	. . . . . . . . . . . 12 ⊢ ⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩ ∈ V
19	18	op1st 4085	. . . . . . . . . . 11 ⊢ (1st ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = ⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩
20	19	fveq2i 3727	. . . . . . . . . 10 ⊢ (1st ‘(1st ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)) = (1st ‘⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩)
21		hilabl 9012	. . . . . . . . . . . 12 ⊢ +h ∈ Abel
22		resexg 3394	. . . . . . . . . . . 12 ⊢ ( +h ∈ Abel → ( +h ↾ (H × H)) ∈ V)
23	21, 22	ax-mp 7	. . . . . . . . . . 11 ⊢ ( +h ↾ (H × H)) ∈ V
24	23	op1st 4085	. . . . . . . . . 10 ⊢ (1st ‘⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩) = ( +h ↾ (H × H))
25	20, 24	eqtr 1495	. . . . . . . . 9 ⊢ (1st ‘(1st ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩)) = ( +h ↾ (H × H))
26	17, 25	eqtr 1495	. . . . . . . 8 ⊢ ( +v ‘⟨⟨( +h ↾ (H × H)), ( ·h ↾ (ℂ × H))⟩, (normh ↾ H)⟩) = ( +h ↾ (H × H))
27	15, 26	eqtr 1495	. . . . . . 7 ⊢ ( +v ‘W) = ( +h ↾ (H × H))
28	27	dmeqi 3312	. . . . . 6 ⊢ dom ( +v ‘W) = dom ( +h ↾ (H × H))
29		hhssp3.4	. . . . . . . . 9 ⊢ H ⊆ ℋ
30		ssxp 3256	. . . . . . . . 9 ⊢ ((H ⊆ ℋ ⋀ H ⊆ ℋ ) → (H × H) ⊆ ( ℋ × ℋ ))
31	29, 29, 30	mp2an 697	. . . . . . . 8 ⊢ (H × H) ⊆ ( ℋ × ℋ )
32		ax-hfvadd 8855	. . . . . . . . 9 ⊢ +h :( ℋ × ℋ )–→ ℋ
33	32	fdmi 3632	. . . . . . . 8 ⊢ dom +h = ( ℋ × ℋ )
34	31, 33	sseqtr4 2094	. . . . . . 7 ⊢ (H × H) ⊆ dom +h
35		ssdmres 3381	. . . . . . 7 ⊢ ((H × H) ⊆ dom +h ↔ dom ( +h ↾ (H × H)) = (H × H))
36	34, 35	mpbi 189	. . . . . 6 ⊢ dom ( +h ↾ (H × H)) = (H × H)
37	28, 36	eqtr 1495	. . . . 5 ⊢ dom ( +v ‘W) = (H × H)
38	37	dmeqi 3312	. . . 4 ⊢ dom dom ( +v ‘W) = dom ( H × H)
39		dmxpid 3333	. . . 4 ⊢ dom ( H × H) = H
40	13, 38, 39	3eqtr 1499	. . 3 ⊢ ran ( +v ‘W) = H
41	3, 40	eqtr 1495	. 2 ⊢ (Base ‘W) = H
42	41	eqcomi 1479	1 ⊢ H = (Base ‘W)

Syntax hints:   = wceq 956   ∈ wcel 958  Vcvv 1811   ⊆ wss 2047  ⟨cop 2411   × cxp 3168  dom cdm 3170  ran crn 3171   ↾ cres 3172   ‘cfv 3182  1^st c1st 4077  ℂcc 5224  Grpcgr 8018  Abelcabl 8084  NrmCVeccnv 8188   +_v cpv 8189  Basecba 8190  SubSpcss 8365   ℋ chil 8773   +_h cva 8774   ·_h csm 8775  norm_hcno 8779

This theorem is referenced by:  hhshsslem2 9123  hhssba 9126

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 4617  ax-hilex 8854  ax-hfvadd 8855  ax-hvcom 8856  ax-hvass 8857  ax-hv0cl 8858  ax-hvaddid 8859  ax-hfvmul 8860  ax-hvmulid 8861  ax-hvmulass 8862  ax-hvdistr1 8863  ax-hvdistr2 8864  ax-hvmul0 8865  ax-hfi 8931  ax-his1 8934  ax-his2 8935  ax-his3 8936  ax-his4 8937

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 4566  df-ni 4992  df-pli 4993  df-mi 4994  df-lti 4995  df-plpq 5027  df-mpq 5028  df-enq 5029  df-nq 5030  df-plq 5031  df-mq 5032  df-rq 5033  df-ltq 5034  df-1q 5035  df-np 5078  df-1p 5079  df-plp 5080  df-mp 5081  df-ltp 5082  df-plpr 5156  df-mpr 5157  df-enr 5158  df-nr 5159  df-plr 5160  df-mr 5161  df-ltr 5162  df-0r 5163  df-1r 5164  df-m1r 5165  df-c 5232  df-0 5233  df-1 5234  df-i 5235  df-r 5236  df-plus 5237  df-mul 5238  df-lt 5239  df-sub 5348  df-neg 5350  df-pnf 5479  df-mnf 5480  df-xr 5481  df-ltxr 5482  df-le 5483  df-div 5691  df-n 5913  df-2 5958  df-3 5959  df-4 5960  df-n0 6088  df-z 6124  df-seq1 6294  df-exp 6555  df-sqr 6656  df-re 6737  df-im 6738  df-cj 6739  df-abs 6740  df-grp 8022  df-gid 8023  df-ginv 8024  df-abl 8085  df-vc 8150  df-nv 8196  df-va 8199  df-ba 8200  df-sm 8201  df-0v 8202  df-nm 8204  df-ssp 8366  df-hnorm 8822  df-hvsub 8825

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