Theorem cphnm 25166

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	⊢ , = (·𝑖‘𝑊)
nmsq.n	⊢ 𝑁 = (norm‘𝑊)

Assertion

Ref	Expression
cphnm	⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) = (√‘(𝐴 , 𝐴)))

Proof of Theorem cphnm

Dummy variable 𝑥 is distinct from all other variables.

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 25165	. . 3 ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))
5	4	fveq1d 6846	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑁‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴))
6		oveq12 7379	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 , 𝑥) = (𝐴 , 𝐴))
7	6	anidms 566	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 , 𝑥) = (𝐴 , 𝐴))
8	7	fveq2d 6848	. . 3 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 , 𝑥)) = (√‘(𝐴 , 𝐴)))
9		eqid 2737	. . 3 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))
10		fvex 6857	. . 3 ⊢ (√‘(𝐴 , 𝐴)) ∈ V
11	8, 9, 10	fvmpt 6951	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴) = (√‘(𝐴 , 𝐴)))
12	5, 11	sylan9eq 2792	1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) = (√‘(𝐴 , 𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ↦ cmpt 5181  ‘cfv 6502  (class class class)co 7370  √csqrt 15170  Basecbs 17150  ·_𝑖cip 17196  normcnm 24537  ℂPreHilccph 25139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fv 6510  df-ov 7373  df-cph 25141

This theorem is referenced by:  nmsq  25167