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Theorem cphnm 25241
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 25240 . . 3 (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
54fveq1d 6909 . 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 6911 . . 3 (𝑥 = 𝐴 → (√‘(𝑥 , 𝑥)) = (√‘(𝐴 , 𝐴)))
9 eqid 2735 . . 3 (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))
10 fvex 6920 . . 3 (√‘(𝐴 , 𝐴)) ∈ V
118, 9, 10fvmpt 7016 . 2 (𝐴𝑉 → ((𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴) = (√‘(𝐴 , 𝐴)))
125, 11sylan9eq 2795 1 ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝑁𝐴) = (√‘(𝐴 , 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cmpt 5231  cfv 6563  (class class class)co 7431  csqrt 15269  Basecbs 17245  ·𝑖cip 17303  normcnm 24605  ℂPreHilccph 25214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-cph 25216
This theorem is referenced by:  nmsq  25242
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