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Mirrors > Home > HSE Home > Th. List > hhshsslem2 | Structured version Visualization version GIF version |
Description: Lemma for hhsssh 29052. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.) |
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 |
Ref | Expression |
---|---|
hhshsslem2 | ⊢ 𝐻 ∈ Sℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhssp3.4 | . . 3 ⊢ 𝐻 ⊆ ℋ | |
2 | hhsst.1 | . . . . . 6 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
3 | 2 | hhnv 28948 | . . . . 5 ⊢ 𝑈 ∈ NrmCVec |
4 | hhssp3.3 | . . . . 5 ⊢ 𝑊 ∈ (SubSp‘𝑈) | |
5 | 2 | hh0v 28951 | . . . . . 6 ⊢ 0ℎ = (0vec‘𝑈) |
6 | eqid 2798 | . . . . . 6 ⊢ (0vec‘𝑊) = (0vec‘𝑊) | |
7 | eqid 2798 | . . . . . 6 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
8 | 5, 6, 7 | sspz 28518 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → (0vec‘𝑊) = 0ℎ) |
9 | 3, 4, 8 | mp2an 691 | . . . 4 ⊢ (0vec‘𝑊) = 0ℎ |
10 | 7 | sspnv 28509 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑊 ∈ NrmCVec) |
11 | 3, 4, 10 | mp2an 691 | . . . . . 6 ⊢ 𝑊 ∈ NrmCVec |
12 | eqid 2798 | . . . . . . 7 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
13 | 12, 6 | nvzcl 28417 | . . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (0vec‘𝑊) ∈ (BaseSet‘𝑊)) |
14 | 11, 13 | ax-mp 5 | . . . . 5 ⊢ (0vec‘𝑊) ∈ (BaseSet‘𝑊) |
15 | hhsst.2 | . . . . . 6 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
16 | 2, 15, 4, 1 | hhshsslem1 29050 | . . . . 5 ⊢ 𝐻 = (BaseSet‘𝑊) |
17 | 14, 16 | eleqtrri 2889 | . . . 4 ⊢ (0vec‘𝑊) ∈ 𝐻 |
18 | 9, 17 | eqeltrri 2887 | . . 3 ⊢ 0ℎ ∈ 𝐻 |
19 | 1, 18 | pm3.2i 474 | . 2 ⊢ (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) |
20 | 2 | hhva 28949 | . . . . . . 7 ⊢ +ℎ = ( +𝑣 ‘𝑈) |
21 | eqid 2798 | . . . . . . 7 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
22 | 16, 20, 21, 7 | sspgval 28512 | . . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥( +𝑣 ‘𝑊)𝑦) = (𝑥 +ℎ 𝑦)) |
23 | 3, 4, 22 | mpanl12 701 | . . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +𝑣 ‘𝑊)𝑦) = (𝑥 +ℎ 𝑦)) |
24 | 16, 21 | nvgcl 28403 | . . . . . 6 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +𝑣 ‘𝑊)𝑦) ∈ 𝐻) |
25 | 11, 24 | mp3an1 1445 | . . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +𝑣 ‘𝑊)𝑦) ∈ 𝐻) |
26 | 23, 25 | eqeltrrd 2891 | . . . 4 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +ℎ 𝑦) ∈ 𝐻) |
27 | 26 | rgen2 3168 | . . 3 ⊢ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 |
28 | 2 | hhsm 28952 | . . . . . . 7 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
29 | eqid 2798 | . . . . . . 7 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
30 | 16, 28, 29, 7 | sspsval 28514 | . . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻)) → (𝑥( ·𝑠OLD ‘𝑊)𝑦) = (𝑥 ·ℎ 𝑦)) |
31 | 3, 4, 30 | mpanl12 701 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·𝑠OLD ‘𝑊)𝑦) = (𝑥 ·ℎ 𝑦)) |
32 | 16, 29 | nvscl 28409 | . . . . . 6 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·𝑠OLD ‘𝑊)𝑦) ∈ 𝐻) |
33 | 11, 32 | mp3an1 1445 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·𝑠OLD ‘𝑊)𝑦) ∈ 𝐻) |
34 | 31, 33 | eqeltrrd 2891 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥 ·ℎ 𝑦) ∈ 𝐻) |
35 | 34 | rgen2 3168 | . . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻 |
36 | 27, 35 | pm3.2i 474 | . 2 ⊢ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻) |
37 | issh2 28992 | . 2 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))) | |
38 | 19, 36, 37 | mpbir2an 710 | 1 ⊢ 𝐻 ∈ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 〈cop 4531 × cxp 5517 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 NrmCVeccnv 28367 +𝑣 cpv 28368 BaseSetcba 28369 ·𝑠OLD cns 28370 0veccn0v 28371 SubSpcss 28504 ℋchba 28702 +ℎ cva 28703 ·ℎ csm 28704 normℎcno 28706 0ℎc0v 28707 Sℋ csh 28711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-hilex 28782 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvmulass 28790 ax-hvdistr1 28791 ax-hvdistr2 28792 ax-hvmul0 28793 ax-hfi 28862 ax-his1 28865 ax-his2 28866 ax-his3 28867 ax-his4 28868 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ssp 28505 df-hnorm 28751 df-hvsub 28754 df-sh 28990 |
This theorem is referenced by: hhsssh 29052 |
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