| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 2795 | . . . . . 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 2795 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = , ) |
| 9 | 8 | oveqd 7448 | . . . . . . 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 5233 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
| 12 | 2, 11 | oveq12d 7449 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
| 13 | df-tcph 25203 | . . . 4 ⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))) | |
| 14 | ovex 7464 | . . . 4 ⊢ (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) ∈ V | |
| 15 | 12, 13, 14 | fvmpt 7016 | . . 3 ⊢ (𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
| 16 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (toℂPreHil‘𝑊) = ∅) | |
| 17 | reldmtng 24651 | . . . . 5 ⊢ Rel dom toNrmGrp | |
| 18 | 17 | ovprc1 7470 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) = ∅) |
| 19 | 16, 18 | eqtr4d 2780 | . . 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 2765 | 1 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 √csqrt 15272 Basecbs 17247 ·𝑖cip 17302 toNrmGrp ctng 24591 toℂPreHilctcph 25201 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-tng 24597 df-tcph 25203 |
| This theorem is referenced by: tcphbas 25253 tchplusg 25254 tcphmulr 25256 tcphsca 25257 tcphvsca 25258 tcphip 25259 tcphtopn 25260 tchnmfval 25262 tcphds 25265 tcphcph 25271 rrxsca 25430 rrx0 25431 rrxdim 33665 |
| Copyright terms: Public domain | W3C validator |