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Theorem cphnm 25227
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.)
Hypotheses
Ref Expression
nmsq.v 𝑉 = (Base‘𝑊)
nmsq.h , = (·𝑖𝑊)
nmsq.n 𝑁 = (norm‘𝑊)
Assertion
Ref Expression
cphnm ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝑁𝐴) = (√‘(𝐴 , 𝐴)))

Proof of Theorem cphnm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nmsq.v . . . 4 𝑉 = (Base‘𝑊)
2 nmsq.h . . . 4 , = (·𝑖𝑊)
3 nmsq.n . . . 4 𝑁 = (norm‘𝑊)
41, 2, 3cphnmfval 25226 . . 3 (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
54fveq1d 6908 . 2 (𝑊 ∈ ℂPreHil → (𝑁𝐴) = ((𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴))
6 oveq12 7440 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥 , 𝑥) = (𝐴 , 𝐴))
76anidms 566 . . . 4 (𝑥 = 𝐴 → (𝑥 , 𝑥) = (𝐴 , 𝐴))
87fveq2d 6910 . . 3 (𝑥 = 𝐴 → (√‘(𝑥 , 𝑥)) = (√‘(𝐴 , 𝐴)))
9 eqid 2737 . . 3 (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))
10 fvex 6919 . . 3 (√‘(𝐴 , 𝐴)) ∈ V
118, 9, 10fvmpt 7016 . 2 (𝐴𝑉 → ((𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴) = (√‘(𝐴 , 𝐴)))
125, 11sylan9eq 2797 1 ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝑁𝐴) = (√‘(𝐴 , 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cmpt 5225  cfv 6561  (class class class)co 7431  csqrt 15272  Basecbs 17247  ·𝑖cip 17302  normcnm 24589  ℂPreHilccph 25200
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-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  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-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-cph 25202
This theorem is referenced by:  nmsq  25228
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