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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 6906 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
4 | tcphval.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
5 | 3, 4 | eqtr4di 2792 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉) |
6 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = (·𝑖‘𝑊)) | |
7 | tcphval.h | . . . . . . . . 9 ⊢ , = (·𝑖‘𝑊) | |
8 | 6, 7 | eqtr4di 2792 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = , ) |
9 | 8 | oveqd 7447 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥(·𝑖‘𝑤)𝑥) = (𝑥 , 𝑥)) |
10 | 9 | fveq2d 6910 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (√‘(𝑥(·𝑖‘𝑤)𝑥)) = (√‘(𝑥 , 𝑥))) |
11 | 5, 10 | mpteq12dv 5238 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
12 | 2, 11 | oveq12d 7448 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
13 | df-tcph 25216 | . . . 4 ⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))) | |
14 | ovex 7463 | . . . 4 ⊢ (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) ∈ V | |
15 | 12, 13, 14 | fvmpt 7015 | . . 3 ⊢ (𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
16 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (toℂPreHil‘𝑊) = ∅) | |
17 | reldmtng 24666 | . . . . 5 ⊢ Rel dom toNrmGrp | |
18 | 17 | ovprc1 7469 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) = ∅) |
19 | 16, 18 | eqtr4d 2777 | . . 3 ⊢ (¬ 𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
20 | 15, 19 | pm2.61i 182 | . 2 ⊢ (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
21 | 1, 20 | eqtri 2762 | 1 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∅c0 4338 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 √csqrt 15268 Basecbs 17244 ·𝑖cip 17302 toNrmGrp ctng 24606 toℂPreHilctcph 25214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-tng 24612 df-tcph 25216 |
This theorem is referenced by: tcphbas 25266 tchplusg 25267 tcphmulr 25269 tcphsca 25270 tcphvsca 25271 tcphip 25272 tcphtopn 25273 tchnmfval 25275 tcphds 25278 tcphcph 25284 rrxsca 25443 rrx0 25444 rrxdim 33641 |
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