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Theorem cphnm 22801
 Description: The square of the norm is the norm of an inner product in a normed 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 22800 . . 3 (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
54fveq1d 6105 . 2 (𝑊 ∈ ℂPreHil → (𝑁𝐴) = ((𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴))
6 oveq12 6558 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥 , 𝑥) = (𝐴 , 𝐴))
76anidms 675 . . . 4 (𝑥 = 𝐴 → (𝑥 , 𝑥) = (𝐴 , 𝐴))
87fveq2d 6107 . . 3 (𝑥 = 𝐴 → (√‘(𝑥 , 𝑥)) = (√‘(𝐴 , 𝐴)))
9 eqid 2610 . . 3 (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))
10 fvex 6113 . . 3 (√‘(𝐴 , 𝐴)) ∈ V
118, 9, 10fvmpt 6191 . 2 (𝐴𝑉 → ((𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴) = (√‘(𝐴 , 𝐴)))
125, 11sylan9eq 2664 1 ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝑁𝐴) = (√‘(𝐴 , 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  √csqrt 13821  Basecbs 15695  ·𝑖cip 15773  normcnm 22191  ℂPreHilccph 22774 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  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-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-cph 22776 This theorem is referenced by:  nmsq  22802
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