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Mirrors > Home > HSE Home > Th. List > hhssnvt | Structured version Visualization version GIF version |
Description: Normed complex vector space property of a subspace. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhssnvt.1 | ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 |
Ref | Expression |
---|---|
hhssnvt | ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ NrmCVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhssnvt.1 | . . . 4 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
2 | xpeq1 5550 | . . . . . . . 8 ⊢ (𝐻 = if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) → (𝐻 × 𝐻) = (if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) × 𝐻)) | |
3 | xpeq2 5557 | . . . . . . . 8 ⊢ (𝐻 = if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) → (if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) × 𝐻) = (if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))) | |
4 | 2, 3 | eqtrd 2771 | . . . . . . 7 ⊢ (𝐻 = if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) → (𝐻 × 𝐻) = (if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))) |
5 | 4 | reseq2d 5836 | . . . . . 6 ⊢ (𝐻 = if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) → ( +ℎ ↾ (𝐻 × 𝐻)) = ( +ℎ ↾ (if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ)))) |
6 | xpeq2 5557 | . . . . . . 7 ⊢ (𝐻 = if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) → (ℂ × 𝐻) = (ℂ × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))) | |
7 | 6 | reseq2d 5836 | . . . . . 6 ⊢ (𝐻 = if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) → ( ·ℎ ↾ (ℂ × 𝐻)) = ( ·ℎ ↾ (ℂ × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ)))) |
8 | 5, 7 | opeq12d 4778 | . . . . 5 ⊢ (𝐻 = if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) → 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 = 〈( +ℎ ↾ (if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))), ( ·ℎ ↾ (ℂ × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ)))〉) |
9 | reseq2 5831 | . . . . 5 ⊢ (𝐻 = if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) → (normℎ ↾ 𝐻) = (normℎ ↾ if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))) | |
10 | 8, 9 | opeq12d 4778 | . . . 4 ⊢ (𝐻 = if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) → 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 = 〈〈( +ℎ ↾ (if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))), ( ·ℎ ↾ (ℂ × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ)))〉, (normℎ ↾ if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))〉) |
11 | 1, 10 | syl5eq 2783 | . . 3 ⊢ (𝐻 = if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) → 𝑊 = 〈〈( +ℎ ↾ (if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))), ( ·ℎ ↾ (ℂ × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ)))〉, (normℎ ↾ if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))〉) |
12 | 11 | eleq1d 2815 | . 2 ⊢ (𝐻 = if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) → (𝑊 ∈ NrmCVec ↔ 〈〈( +ℎ ↾ (if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))), ( ·ℎ ↾ (ℂ × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ)))〉, (normℎ ↾ if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))〉 ∈ NrmCVec)) |
13 | eqid 2736 | . . 3 ⊢ 〈〈( +ℎ ↾ (if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))), ( ·ℎ ↾ (ℂ × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ)))〉, (normℎ ↾ if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))〉 = 〈〈( +ℎ ↾ (if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))), ( ·ℎ ↾ (ℂ × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ)))〉, (normℎ ↾ if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))〉 | |
14 | h0elsh 29291 | . . . 4 ⊢ 0ℋ ∈ Sℋ | |
15 | 14 | elimel 4494 | . . 3 ⊢ if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) ∈ Sℋ |
16 | 13, 15 | hhssnv 29299 | . 2 ⊢ 〈〈( +ℎ ↾ (if(𝐻 ∈ Sℋ , 𝐻, 0ℋ) × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))), ( ·ℎ ↾ (ℂ × if(𝐻 ∈ Sℋ , 𝐻, 0ℋ)))〉, (normℎ ↾ if(𝐻 ∈ Sℋ , 𝐻, 0ℋ))〉 ∈ NrmCVec |
17 | 12, 16 | dedth 4483 | 1 ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ifcif 4425 〈cop 4533 × cxp 5534 ↾ cres 5538 ℂcc 10692 NrmCVeccnv 28619 +ℎ cva 28955 ·ℎ csm 28956 normℎcno 28958 Sℋ csh 28963 0ℋc0h 28970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774 ax-hilex 29034 ax-hfvadd 29035 ax-hvcom 29036 ax-hvass 29037 ax-hv0cl 29038 ax-hvaddid 29039 ax-hfvmul 29040 ax-hvmulid 29041 ax-hvmulass 29042 ax-hvdistr1 29043 ax-hvdistr2 29044 ax-hvmul0 29045 ax-hfi 29114 ax-his1 29117 ax-his2 29118 ax-his3 29119 ax-his4 29120 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-n0 12056 df-z 12142 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-icc 12907 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-topgen 16902 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-top 21745 df-topon 21762 df-bases 21797 df-lm 22080 df-haus 22166 df-grpo 28528 df-gid 28529 df-ginv 28530 df-gdiv 28531 df-ablo 28580 df-vc 28594 df-nv 28627 df-va 28630 df-ba 28631 df-sm 28632 df-0v 28633 df-vs 28634 df-nmcv 28635 df-ims 28636 df-hnorm 29003 df-hba 29004 df-hvsub 29006 df-hlim 29007 df-sh 29242 df-ch 29256 df-ch0 29288 |
This theorem is referenced by: hhsst 29301 |
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