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Mirrors > Home > HSE Home > Th. List > hhsst | Structured version Visualization version GIF version |
Description: A member of Sℋ is a subspace. (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)〉 |
Ref | Expression |
---|---|
hhsst | ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ (SubSp‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhsst.2 | . . . 4 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
2 | 1 | hhssnvt 31192 | . . 3 ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ NrmCVec) |
3 | resss 6001 | . . . 4 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ | |
4 | resss 6001 | . . . 4 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ | |
5 | resss 6001 | . . . 4 ⊢ (normℎ ↾ 𝐻) ⊆ normℎ | |
6 | 3, 4, 5 | 3pm3.2i 1336 | . . 3 ⊢ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ) |
7 | 2, 6 | jctir 519 | . 2 ⊢ (𝐻 ∈ Sℋ → (𝑊 ∈ NrmCVec ∧ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ))) |
8 | hhsst.1 | . . . 4 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
9 | 8 | hhnv 31092 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
10 | 8 | hhva 31093 | . . . 4 ⊢ +ℎ = ( +𝑣 ‘𝑈) |
11 | 1 | hhssva 31184 | . . . 4 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) = ( +𝑣 ‘𝑊) |
12 | 8 | hhsm 31096 | . . . 4 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
13 | 1 | hhsssm 31185 | . . . 4 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) = ( ·𝑠OLD ‘𝑊) |
14 | 8 | hhnm 31098 | . . . 4 ⊢ normℎ = (normCV‘𝑈) |
15 | 1 | hhssnm 31186 | . . . 4 ⊢ (normℎ ↾ 𝐻) = (normCV‘𝑊) |
16 | eqid 2726 | . . . 4 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
17 | 10, 11, 12, 13, 14, 15, 16 | isssp 30651 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ (SubSp‘𝑈) ↔ (𝑊 ∈ NrmCVec ∧ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ)))) |
18 | 9, 17 | ax-mp 5 | . 2 ⊢ (𝑊 ∈ (SubSp‘𝑈) ↔ (𝑊 ∈ NrmCVec ∧ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ))) |
19 | 7, 18 | sylibr 233 | 1 ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ (SubSp‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ⊆ wss 3946 〈cop 4629 × cxp 5670 ↾ cres 5674 ‘cfv 6543 ℂcc 11144 NrmCVeccnv 30511 SubSpcss 30648 +ℎ cva 30847 ·ℎ csm 30848 normℎcno 30850 Sℋ csh 30855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226 ax-hilex 30926 ax-hfvadd 30927 ax-hvcom 30928 ax-hvass 30929 ax-hv0cl 30930 ax-hvaddid 30931 ax-hfvmul 30932 ax-hvmulid 30933 ax-hvmulass 30934 ax-hvdistr1 30935 ax-hvdistr2 30936 ax-hvmul0 30937 ax-hfi 31006 ax-his1 31009 ax-his2 31010 ax-his3 31011 ax-his4 31012 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-map 8846 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9475 df-inf 9476 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12256 df-2 12318 df-3 12319 df-4 12320 df-n0 12516 df-z 12602 df-uz 12866 df-q 12976 df-rp 13020 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-icc 13376 df-seq 14013 df-exp 14073 df-cj 15096 df-re 15097 df-im 15098 df-sqrt 15232 df-abs 15233 df-topgen 17450 df-psmet 21328 df-xmet 21329 df-met 21330 df-bl 21331 df-mopn 21332 df-top 22881 df-topon 22898 df-bases 22934 df-lm 23218 df-haus 23304 df-grpo 30420 df-gid 30421 df-ginv 30422 df-gdiv 30423 df-ablo 30472 df-vc 30486 df-nv 30519 df-va 30522 df-ba 30523 df-sm 30524 df-0v 30525 df-vs 30526 df-nmcv 30527 df-ims 30528 df-ssp 30649 df-hnorm 30895 df-hba 30896 df-hvsub 30898 df-hlim 30899 df-sh 31134 df-ch 31148 df-ch0 31180 |
This theorem is referenced by: hhsssh 31196 hhssba 31198 hhssvs 31199 pjhthlem2 31319 |
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