Theorem cphnm 25093

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 25092	. . 3 ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))
5	4	fveq1d 6860	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑁‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴))
6		oveq12 7396	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 , 𝑥) = (𝐴 , 𝐴))
7	6	anidms 566	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 , 𝑥) = (𝐴 , 𝐴))
8	7	fveq2d 6862	. . 3 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 , 𝑥)) = (√‘(𝐴 , 𝐴)))
9		eqid 2729	. . 3 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))
10		fvex 6871	. . 3 ⊢ (√‘(𝐴 , 𝐴)) ∈ V
11	8, 9, 10	fvmpt 6968	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝐴) = (√‘(𝐴 , 𝐴)))
12	5, 11	sylan9eq 2784	1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) = (√‘(𝐴 , 𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387  √csqrt 15199  Basecbs 17179  ·_𝑖cip 17225  normcnm 24464  ℂPreHilccph 25066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-cph 25068

This theorem is referenced by:  nmsq  25094