Theorem cphnm 24367

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 24366	. . 3 ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))
5	4	fveq1d 6768	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑁‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴))
6		oveq12 7276	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 , 𝑥) = (𝐴 , 𝐴))
7	6	anidms 567	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 , 𝑥) = (𝐴 , 𝐴))
8	7	fveq2d 6770	. . 3 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 , 𝑥)) = (√‘(𝐴 , 𝐴)))
9		eqid 2738	. . 3 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))
10		fvex 6779	. . 3 ⊢ (√‘(𝐴 , 𝐴)) ∈ V
11	8, 9, 10	fvmpt 6867	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴) = (√‘(𝐴 , 𝐴)))
12	5, 11	sylan9eq 2798	1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) = (√‘(𝐴 , 𝐴)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ↦ cmpt 5156  ‘cfv 6426  (class class class)co 7267  √csqrt 14954  Basecbs 16922  ·_𝑖cip 16977  normcnm 23742  ℂPreHilccph 24340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fv 6434  df-ov 7270  df-cph 24342

This theorem is referenced by:  nmsq  24368