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Mirrors > Home > MPE Home > Th. List > tchval | Structured version Visualization version GIF version |
Description: Define a function to augment a pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
tchval.n | ⊢ 𝐺 = (toℂHil‘𝑊) |
tchval.v | ⊢ 𝑉 = (Base‘𝑊) |
tchval.h | ⊢ , = (·𝑖‘𝑊) |
Ref | Expression |
---|---|
tchval | ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tchval.n | . 2 ⊢ 𝐺 = (toℂHil‘𝑊) | |
2 | id 22 | . . . . 5 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊) | |
3 | fveq2 6103 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
4 | tchval.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
5 | 3, 4 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉) |
6 | fveq2 6103 | . . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = (·𝑖‘𝑊)) | |
7 | tchval.h | . . . . . . . . 9 ⊢ , = (·𝑖‘𝑊) | |
8 | 6, 7 | syl6eqr 2662 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = , ) |
9 | 8 | oveqd 6566 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥(·𝑖‘𝑤)𝑥) = (𝑥 , 𝑥)) |
10 | 9 | fveq2d 6107 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (√‘(𝑥(·𝑖‘𝑤)𝑥)) = (√‘(𝑥 , 𝑥))) |
11 | 5, 10 | mpteq12dv 4663 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
12 | 2, 11 | oveq12d 6567 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
13 | df-tch 22777 | . . . 4 ⊢ toℂHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))) | |
14 | ovex 6577 | . . . 4 ⊢ (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) ∈ V | |
15 | 12, 13, 14 | fvmpt 6191 | . . 3 ⊢ (𝑊 ∈ V → (toℂHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
16 | fvprc 6097 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (toℂHil‘𝑊) = ∅) | |
17 | reldmtng 22252 | . . . . 5 ⊢ Rel dom toNrmGrp | |
18 | 17 | ovprc1 6582 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) = ∅) |
19 | 16, 18 | eqtr4d 2647 | . . 3 ⊢ (¬ 𝑊 ∈ V → (toℂHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
20 | 15, 19 | pm2.61i 175 | . 2 ⊢ (toℂHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
21 | 1, 20 | eqtri 2632 | 1 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 √csqrt 13821 Basecbs 15695 ·𝑖cip 15773 toNrmGrp ctng 22193 toℂHilctch 22775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-tng 22199 df-tch 22777 |
This theorem is referenced by: tchbas 22826 tchplusg 22827 tchmulr 22829 tchsca 22830 tchvsca 22831 tchip 22832 tchtopn 22833 tchnmfval 22835 tchds 22838 tchcph 22844 |
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