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Mirrors > Home > HSE Home > Th. List > hhshsslem2 | Structured version Visualization version GIF version |
Description: Lemma for hhsssh 29049. (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 28945 | . . . . 5 ⊢ 𝑈 ∈ NrmCVec |
4 | hhssp3.3 | . . . . 5 ⊢ 𝑊 ∈ (SubSp‘𝑈) | |
5 | 2 | hh0v 28948 | . . . . . 6 ⊢ 0ℎ = (0vec‘𝑈) |
6 | eqid 2824 | . . . . . 6 ⊢ (0vec‘𝑊) = (0vec‘𝑊) | |
7 | eqid 2824 | . . . . . 6 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
8 | 5, 6, 7 | sspz 28515 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → (0vec‘𝑊) = 0ℎ) |
9 | 3, 4, 8 | mp2an 690 | . . . 4 ⊢ (0vec‘𝑊) = 0ℎ |
10 | 7 | sspnv 28506 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑊 ∈ NrmCVec) |
11 | 3, 4, 10 | mp2an 690 | . . . . . 6 ⊢ 𝑊 ∈ NrmCVec |
12 | eqid 2824 | . . . . . . 7 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
13 | 12, 6 | nvzcl 28414 | . . . . . 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 29047 | . . . . 5 ⊢ 𝐻 = (BaseSet‘𝑊) |
17 | 14, 16 | eleqtrri 2915 | . . . 4 ⊢ (0vec‘𝑊) ∈ 𝐻 |
18 | 9, 17 | eqeltrri 2913 | . . 3 ⊢ 0ℎ ∈ 𝐻 |
19 | 1, 18 | pm3.2i 473 | . 2 ⊢ (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) |
20 | 2 | hhva 28946 | . . . . . . 7 ⊢ +ℎ = ( +𝑣 ‘𝑈) |
21 | eqid 2824 | . . . . . . 7 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
22 | 16, 20, 21, 7 | sspgval 28509 | . . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥( +𝑣 ‘𝑊)𝑦) = (𝑥 +ℎ 𝑦)) |
23 | 3, 4, 22 | mpanl12 700 | . . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +𝑣 ‘𝑊)𝑦) = (𝑥 +ℎ 𝑦)) |
24 | 16, 21 | nvgcl 28400 | . . . . . 6 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +𝑣 ‘𝑊)𝑦) ∈ 𝐻) |
25 | 11, 24 | mp3an1 1444 | . . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +𝑣 ‘𝑊)𝑦) ∈ 𝐻) |
26 | 23, 25 | eqeltrrd 2917 | . . . 4 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +ℎ 𝑦) ∈ 𝐻) |
27 | 26 | rgen2 3206 | . . 3 ⊢ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 |
28 | 2 | hhsm 28949 | . . . . . . 7 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
29 | eqid 2824 | . . . . . . 7 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
30 | 16, 28, 29, 7 | sspsval 28511 | . . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻)) → (𝑥( ·𝑠OLD ‘𝑊)𝑦) = (𝑥 ·ℎ 𝑦)) |
31 | 3, 4, 30 | mpanl12 700 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·𝑠OLD ‘𝑊)𝑦) = (𝑥 ·ℎ 𝑦)) |
32 | 16, 29 | nvscl 28406 | . . . . . 6 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·𝑠OLD ‘𝑊)𝑦) ∈ 𝐻) |
33 | 11, 32 | mp3an1 1444 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·𝑠OLD ‘𝑊)𝑦) ∈ 𝐻) |
34 | 31, 33 | eqeltrrd 2917 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥 ·ℎ 𝑦) ∈ 𝐻) |
35 | 34 | rgen2 3206 | . . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻 |
36 | 27, 35 | pm3.2i 473 | . 2 ⊢ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻) |
37 | issh2 28989 | . 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 709 | 1 ⊢ 𝐻 ∈ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ⊆ wss 3939 〈cop 4576 × cxp 5556 ↾ cres 5560 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 NrmCVeccnv 28364 +𝑣 cpv 28365 BaseSetcba 28366 ·𝑠OLD cns 28367 0veccn0v 28368 SubSpcss 28501 ℋchba 28699 +ℎ cva 28700 ·ℎ csm 28701 normℎcno 28703 0ℎc0v 28704 Sℋ csh 28708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-hilex 28779 ax-hfvadd 28780 ax-hvcom 28781 ax-hvass 28782 ax-hv0cl 28783 ax-hvaddid 28784 ax-hfvmul 28785 ax-hvmulid 28786 ax-hvmulass 28787 ax-hvdistr1 28788 ax-hvdistr2 28789 ax-hvmul0 28790 ax-hfi 28859 ax-his1 28862 ax-his2 28863 ax-his3 28864 ax-his4 28865 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-grpo 28273 df-gid 28274 df-ginv 28275 df-gdiv 28276 df-ablo 28325 df-vc 28339 df-nv 28372 df-va 28375 df-ba 28376 df-sm 28377 df-0v 28378 df-vs 28379 df-nmcv 28380 df-ssp 28502 df-hnorm 28748 df-hvsub 28751 df-sh 28987 |
This theorem is referenced by: hhsssh 29049 |
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