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Theorem cphnm 23724
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 23723 . . 3 (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
54fveq1d 6665 . 2 (𝑊 ∈ ℂPreHil → (𝑁𝐴) = ((𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴))
6 oveq12 7154 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥 , 𝑥) = (𝐴 , 𝐴))
76anidms 567 . . . 4 (𝑥 = 𝐴 → (𝑥 , 𝑥) = (𝐴 , 𝐴))
87fveq2d 6667 . . 3 (𝑥 = 𝐴 → (√‘(𝑥 , 𝑥)) = (√‘(𝐴 , 𝐴)))
9 eqid 2818 . . 3 (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))
10 fvex 6676 . . 3 (√‘(𝐴 , 𝐴)) ∈ V
118, 9, 10fvmpt 6761 . 2 (𝐴𝑉 → ((𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴) = (√‘(𝐴 , 𝐴)))
125, 11sylan9eq 2873 1 ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝑁𝐴) = (√‘(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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