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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 6433 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
4 | tcphval.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
5 | 3, 4 | syl6eqr 2879 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉) |
6 | fveq2 6433 | . . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = (·𝑖‘𝑊)) | |
7 | tcphval.h | . . . . . . . . 9 ⊢ , = (·𝑖‘𝑊) | |
8 | 6, 7 | syl6eqr 2879 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = , ) |
9 | 8 | oveqd 6922 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥(·𝑖‘𝑤)𝑥) = (𝑥 , 𝑥)) |
10 | 9 | fveq2d 6437 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (√‘(𝑥(·𝑖‘𝑤)𝑥)) = (√‘(𝑥 , 𝑥))) |
11 | 5, 10 | mpteq12dv 4956 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
12 | 2, 11 | oveq12d 6923 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
13 | df-tcph 23338 | . . . 4 ⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))) | |
14 | ovex 6937 | . . . 4 ⊢ (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) ∈ V | |
15 | 12, 13, 14 | fvmpt 6529 | . . 3 ⊢ (𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
16 | fvprc 6426 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (toℂPreHil‘𝑊) = ∅) | |
17 | reldmtng 22812 | . . . . 5 ⊢ Rel dom toNrmGrp | |
18 | 17 | ovprc1 6943 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) = ∅) |
19 | 16, 18 | eqtr4d 2864 | . . 3 ⊢ (¬ 𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
20 | 15, 19 | pm2.61i 177 | . 2 ⊢ (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
21 | 1, 20 | eqtri 2849 | 1 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1658 ∈ wcel 2166 Vcvv 3414 ∅c0 4144 ↦ cmpt 4952 ‘cfv 6123 (class class class)co 6905 √csqrt 14350 Basecbs 16222 ·𝑖cip 16310 toNrmGrp ctng 22753 toℂPreHilctcph 23336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-tng 22759 df-tcph 23338 |
This theorem is referenced by: tcphbas 23387 tchplusg 23388 tcphmulr 23390 tcphsca 23391 tcphvsca 23392 tcphip 23393 tcphtopn 23394 tchnmfval 23396 tcphds 23399 tcphcph 23405 rrxsca 23564 rrx0 23565 |
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