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Mirrors > Home > MPE Home > Th. List > cphnm | Structured version Visualization version GIF version |
Description: The square of the norm is the norm of an inner product in a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
nmsq.v | ⊢ 𝑉 = (Base‘𝑊) |
nmsq.h | ⊢ , = (·𝑖‘𝑊) |
nmsq.n | ⊢ 𝑁 = (norm‘𝑊) |
Ref | Expression |
---|---|
cphnm | ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) = (√‘(𝐴 , 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmsq.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | nmsq.h | . . . 4 ⊢ , = (·𝑖‘𝑊) | |
3 | nmsq.n | . . . 4 ⊢ 𝑁 = (norm‘𝑊) | |
4 | 1, 2, 3 | cphnmfval 23723 | . . 3 ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
5 | 4 | fveq1d 6665 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑁‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴)) |
6 | oveq12 7154 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 , 𝑥) = (𝐴 , 𝐴)) | |
7 | 6 | anidms 567 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 , 𝑥) = (𝐴 , 𝐴)) |
8 | 7 | fveq2d 6667 | . . 3 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 , 𝑥)) = (√‘(𝐴 , 𝐴))) |
9 | eqid 2818 | . . 3 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) | |
10 | fvex 6676 | . . 3 ⊢ (√‘(𝐴 , 𝐴)) ∈ V | |
11 | 8, 9, 10 | fvmpt 6761 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴) = (√‘(𝐴 , 𝐴))) |
12 | 5, 11 | sylan9eq 2873 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) = (√‘(𝐴 , 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 √csqrt 14580 Basecbs 16471 ·𝑖cip 16558 normcnm 23113 ℂPreHilccph 23697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-cph 23699 |
This theorem is referenced by: nmsq 23725 |
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