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Mirrors > Home > MPE Home > Th. List > tcphval | Structured version Visualization version GIF version |
Description: Define a function to augment a subcomplex pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
tcphval.n | ⊢ 𝐺 = (toℂPreHil‘𝑊) |
tcphval.v | ⊢ 𝑉 = (Base‘𝑊) |
tcphval.h | ⊢ , = (·𝑖‘𝑊) |
Ref | Expression |
---|---|
tcphval | ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tcphval.n | . 2 ⊢ 𝐺 = (toℂPreHil‘𝑊) | |
2 | id 22 | . . . . 5 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊) | |
3 | fveq2 6645 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
4 | tcphval.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
5 | 3, 4 | eqtr4di 2851 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉) |
6 | fveq2 6645 | . . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = (·𝑖‘𝑊)) | |
7 | tcphval.h | . . . . . . . . 9 ⊢ , = (·𝑖‘𝑊) | |
8 | 6, 7 | eqtr4di 2851 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = , ) |
9 | 8 | oveqd 7152 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥(·𝑖‘𝑤)𝑥) = (𝑥 , 𝑥)) |
10 | 9 | fveq2d 6649 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (√‘(𝑥(·𝑖‘𝑤)𝑥)) = (√‘(𝑥 , 𝑥))) |
11 | 5, 10 | mpteq12dv 5115 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
12 | 2, 11 | oveq12d 7153 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
13 | df-tcph 23774 | . . . 4 ⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))) | |
14 | ovex 7168 | . . . 4 ⊢ (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) ∈ V | |
15 | 12, 13, 14 | fvmpt 6745 | . . 3 ⊢ (𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
16 | fvprc 6638 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (toℂPreHil‘𝑊) = ∅) | |
17 | reldmtng 23244 | . . . . 5 ⊢ Rel dom toNrmGrp | |
18 | 17 | ovprc1 7174 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) = ∅) |
19 | 16, 18 | eqtr4d 2836 | . . 3 ⊢ (¬ 𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
20 | 15, 19 | pm2.61i 185 | . 2 ⊢ (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
21 | 1, 20 | eqtri 2821 | 1 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 √csqrt 14584 Basecbs 16475 ·𝑖cip 16562 toNrmGrp ctng 23185 toℂPreHilctcph 23772 |
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-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-tng 23191 df-tcph 23774 |
This theorem is referenced by: tcphbas 23823 tchplusg 23824 tcphmulr 23826 tcphsca 23827 tcphvsca 23828 tcphip 23829 tcphtopn 23830 tchnmfval 23832 tcphds 23835 tcphcph 23841 rrxsca 24000 rrx0 24001 rrxdim 31100 |
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