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Mirrors > Home > HSE Home > Th. List > hhsssh2 | Structured version Visualization version GIF version |
Description: The predicate "𝐻 is a subspace of Hilbert space." (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhsssh2.1 | ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 |
Ref | Expression |
---|---|
hhsssh2 | ⊢ (𝐻 ∈ Sℋ ↔ (𝑊 ∈ NrmCVec ∧ 𝐻 ⊆ ℋ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . 3 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | hhsssh2.1 | . . 3 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
3 | 1, 2 | hhsssh 28698 | . 2 ⊢ (𝐻 ∈ Sℋ ↔ (𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∧ 𝐻 ⊆ ℋ)) |
4 | resss 5671 | . . . . 5 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ | |
5 | resss 5671 | . . . . 5 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ | |
6 | resss 5671 | . . . . 5 ⊢ (normℎ ↾ 𝐻) ⊆ normℎ | |
7 | 4, 5, 6 | 3pm3.2i 1395 | . . . 4 ⊢ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ) |
8 | 1 | hhnv 28594 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
9 | 1 | hhva 28595 | . . . . . 6 ⊢ +ℎ = ( +𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
10 | 2 | hhssva 28686 | . . . . . 6 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) = ( +𝑣 ‘𝑊) |
11 | 1 | hhsm 28598 | . . . . . 6 ⊢ ·ℎ = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
12 | 2 | hhsssm 28687 | . . . . . 6 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) = ( ·𝑠OLD ‘𝑊) |
13 | 1 | hhnm 28600 | . . . . . 6 ⊢ normℎ = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
14 | 2 | hhssnm 28688 | . . . . . 6 ⊢ (normℎ ↾ 𝐻) = (normCV‘𝑊) |
15 | eqid 2778 | . . . . . 6 ⊢ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
16 | 9, 10, 11, 12, 13, 14, 15 | isssp 28151 | . . . . 5 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec → (𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ↔ (𝑊 ∈ NrmCVec ∧ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ)))) |
17 | 8, 16 | ax-mp 5 | . . . 4 ⊢ (𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ↔ (𝑊 ∈ NrmCVec ∧ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ))) |
18 | 7, 17 | mpbiran2 700 | . . 3 ⊢ (𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ↔ 𝑊 ∈ NrmCVec) |
19 | 18 | anbi1i 617 | . 2 ⊢ ((𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∧ 𝐻 ⊆ ℋ) ↔ (𝑊 ∈ NrmCVec ∧ 𝐻 ⊆ ℋ)) |
20 | 3, 19 | bitri 267 | 1 ⊢ (𝐻 ∈ Sℋ ↔ (𝑊 ∈ NrmCVec ∧ 𝐻 ⊆ ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 〈cop 4404 × cxp 5353 ↾ cres 5357 ‘cfv 6135 ℂcc 10270 NrmCVeccnv 28011 SubSpcss 28148 ℋchba 28348 +ℎ cva 28349 ·ℎ csm 28350 normℎcno 28352 Sℋ csh 28357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 ax-hilex 28428 ax-hfvadd 28429 ax-hvcom 28430 ax-hvass 28431 ax-hv0cl 28432 ax-hvaddid 28433 ax-hfvmul 28434 ax-hvmulid 28435 ax-hvmulass 28436 ax-hvdistr1 28437 ax-hvdistr2 28438 ax-hvmul0 28439 ax-hfi 28508 ax-his1 28511 ax-his2 28512 ax-his3 28513 ax-his4 28514 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-n0 11643 df-z 11729 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-icc 12494 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-topgen 16490 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-top 21106 df-topon 21123 df-bases 21158 df-lm 21441 df-haus 21527 df-grpo 27920 df-gid 27921 df-ginv 27922 df-gdiv 27923 df-ablo 27972 df-vc 27986 df-nv 28019 df-va 28022 df-ba 28023 df-sm 28024 df-0v 28025 df-vs 28026 df-nmcv 28027 df-ims 28028 df-ssp 28149 df-hnorm 28397 df-hba 28398 df-hvsub 28400 df-hlim 28401 df-sh 28636 df-ch 28650 df-ch0 28682 |
This theorem is referenced by: (None) |
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