Theorem cphnmfval 25256

Description: The value of the norm in a subcomplex pre-Hilbert space is the square root of the inner product of a vector with itself. (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
cphnmfval	⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))

Distinct variable groups:   𝑥, ,   𝑥,𝑉   𝑥,𝑊

Allowed substitution hint:   𝑁(𝑥)

Proof of Theorem cphnmfval

Step	Hyp	Ref	Expression
1		nmsq.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
2		nmsq.h	. . 3 ⊢ , = (·𝑖‘𝑊)
3		nmsq.n	. . 3 ⊢ 𝑁 = (norm‘𝑊)
4		eqid 2764	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2764	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	1, 2, 3, 4, 5	iscph 25234	. 2 ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))) ∧ (√ “ ((Base‘(Scalar‘𝑊)) ∩ (0[,)+∞))) ⊆ (Base‘(Scalar‘𝑊)) ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))
7	6	simp3bi 1161	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))

Syntax hints:   → wi 4   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ∩ cin 3905   ⊆ wss 3906   ↦ cmpt 5183   “ cima 5652  ‘cfv 6523  (class class class)co 7398  0cc0 11075  +∞cpnf 11215  [,)cico 13353  √csqrt 15262  Basecbs 17247   ↾_s cress 17268  Scalarcsca 17291  ·_𝑖cip 17293  ℂ_fldccnfld 21426  PreHilcphl 21678  normcnm 24638  NrmModcnlm 24642  ℂPreHilccph 25230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-nul 5258

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-xp 5655  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fv 6531  df-ov 7401  df-cph 25232

This theorem is referenced by:  cphnm  25257  cphnmf  25259  cphtcphnm  25294  cphsscph  25315