hhshsslem2 - Hilbert Space Explorer

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Theorem hhshsslem2 30785

Description: Lemma for hhsssh 30786. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)

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

**Assertion**
Ref	Expression
hhshsslem2	⊢ 𝐻 ∈ S_ℋ

Proof of Theorem hhshsslem2 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hhssp3.4	. . 3 ⊢ 𝐻 ⊆ ℋ
2		hhsst.1	. . . . . 6 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
3	2	hhnv 30682	. . . . 5 ⊢ 𝑈 ∈ NrmCVec
4		hhssp3.3	. . . . 5 ⊢ 𝑊 ∈ (SubSp‘𝑈)
5	2	hh0v 30685	. . . . . 6 ⊢ 0_ℎ = (0_vec‘𝑈)
6		eqid 2731	. . . . . 6 ⊢ (0_vec‘𝑊) = (0_vec‘𝑊)
7		eqid 2731	. . . . . 6 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈)
8	5, 6, 7	sspz 30252	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → (0_vec‘𝑊) = 0_ℎ)
9	3, 4, 8	mp2an 689	. . . 4 ⊢ (0_vec‘𝑊) = 0_ℎ
10	7	sspnv 30243	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑊 ∈ NrmCVec)
11	3, 4, 10	mp2an 689	. . . . . 6 ⊢ 𝑊 ∈ NrmCVec
12		eqid 2731	. . . . . . 7 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
13	12, 6	nvzcl 30151	. . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (0_vec‘𝑊) ∈ (BaseSet‘𝑊))
14	11, 13	ax-mp 5	. . . . 5 ⊢ (0_vec‘𝑊) ∈ (BaseSet‘𝑊)
15		hhsst.2	. . . . . 6 ⊢ 𝑊 = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩
16	2, 15, 4, 1	hhshsslem1 30784	. . . . 5 ⊢ 𝐻 = (BaseSet‘𝑊)
17	14, 16	eleqtrri 2831	. . . 4 ⊢ (0_vec‘𝑊) ∈ 𝐻
18	9, 17	eqeltrri 2829	. . 3 ⊢ 0_ℎ ∈ 𝐻
19	1, 18	pm3.2i 470	. 2 ⊢ (𝐻 ⊆ ℋ ∧ 0_ℎ ∈ 𝐻)
20	2	hhva 30683	. . . . . . 7 ⊢ +_ℎ = ( +_𝑣 ‘𝑈)
21		eqid 2731	. . . . . . 7 ⊢ ( +_𝑣 ‘𝑊) = ( +_𝑣 ‘𝑊)
22	16, 20, 21, 7	sspgval 30246	. . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥( +_𝑣 ‘𝑊)𝑦) = (𝑥 +_ℎ 𝑦))
23	3, 4, 22	mpanl12 699	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +_𝑣 ‘𝑊)𝑦) = (𝑥 +_ℎ 𝑦))
24	16, 21	nvgcl 30137	. . . . . 6 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +_𝑣 ‘𝑊)𝑦) ∈ 𝐻)
25	11, 24	mp3an1 1447	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +_𝑣 ‘𝑊)𝑦) ∈ 𝐻)
26	23, 25	eqeltrrd 2833	. . . 4 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +_ℎ 𝑦) ∈ 𝐻)
27	26	rgen2 3196	. . 3 ⊢ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +_ℎ 𝑦) ∈ 𝐻
28	2	hhsm 30686	. . . . . . 7 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
29		eqid 2731	. . . . . . 7 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
30	16, 28, 29, 7	sspsval 30248	. . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻)) → (𝑥( ·_𝑠OLD ‘𝑊)𝑦) = (𝑥 ·_ℎ 𝑦))
31	3, 4, 30	mpanl12 699	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·_𝑠OLD ‘𝑊)𝑦) = (𝑥 ·_ℎ 𝑦))
32	16, 29	nvscl 30143	. . . . . 6 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·_𝑠OLD ‘𝑊)𝑦) ∈ 𝐻)
33	11, 32	mp3an1 1447	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·_𝑠OLD ‘𝑊)𝑦) ∈ 𝐻)
34	31, 33	eqeltrrd 2833	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥 ·_ℎ 𝑦) ∈ 𝐻)
35	34	rgen2 3196	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·_ℎ 𝑦) ∈ 𝐻
36	27, 35	pm3.2i 470	. 2 ⊢ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +_ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·_ℎ 𝑦) ∈ 𝐻)
37		issh2 30726	. 2 ⊢ (𝐻 ∈ S_ℋ ↔ ((𝐻 ⊆ ℋ ∧ 0_ℎ ∈ 𝐻) ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +_ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·_ℎ 𝑦) ∈ 𝐻)))
38	19, 36, 37	mpbir2an 708	1 ⊢ 𝐻 ∈ S_ℋ

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ⊆ wss 3949 ⟨cop 4635 × cxp 5675 ↾ cres 5679 ‘cfv 6544 (class class class)co 7412 ℂcc 11111 NrmCVeccnv 30101 +_𝑣 cpv 30102 BaseSetcba 30103 ·_𝑠OLD cns 30104 0_veccn0v 30105 SubSpcss 30238 ℋchba 30436 +_ℎ cva 30437 ·_ℎ csm 30438 norm_ℎcno 30440 0_ℎc0v 30441 S_ℋ csh 30445

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-hilex 30516 ax-hfvadd 30517 ax-hvcom 30518 ax-hvass 30519 ax-hv0cl 30520 ax-hvaddid 30521 ax-hfvmul 30522 ax-hvmulid 30523 ax-hvmulass 30524 ax-hvdistr1 30525 ax-hvdistr2 30526 ax-hvmul0 30527 ax-hfi 30596 ax-his1 30599 ax-his2 30600 ax-his3 30601 ax-his4 30602

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-sup 9440 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-seq 13972 df-exp 14033 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-grpo 30010 df-gid 30011 df-ginv 30012 df-gdiv 30013 df-ablo 30062 df-vc 30076 df-nv 30109 df-va 30112 df-ba 30113 df-sm 30114 df-0v 30115 df-vs 30116 df-nmcv 30117 df-ssp 30239 df-hnorm 30485 df-hvsub 30488 df-sh 30724

This theorem is referenced by: hhsssh 30786

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