Theorem cphnm 25179

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 25178	. . 3 ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))
5	4	fveq1d 6830	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑁‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴))
6		oveq12 7366	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 , 𝑥) = (𝐴 , 𝐴))
7	6	anidms 571	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 , 𝑥) = (𝐴 , 𝐴))
8	7	fveq2d 6832	. . 3 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 , 𝑥)) = (√‘(𝐴 , 𝐴)))
9		eqid 2739	. . 3 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))
10		fvex 6841	. . 3 ⊢ (√‘(𝐴 , 𝐴)) ∈ V
11	8, 9, 10	fvmpt 6936	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴) = (√‘(𝐴 , 𝐴)))
12	5, 11	sylan9eq 2794	1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) = (√‘(𝐴 , 𝐴)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ↦ cmpt 5154  ‘cfv 6486  (class class class)co 7357  √csqrt 15187  Basecbs 17171  ·_𝑖cip 17217  normcnm 24560  ℂPreHilccph 25152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7360  df-cph 25154

This theorem is referenced by:  nmsq  25180