Theorem cphnmfval 25320

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 2769	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2769	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	1, 2, 3, 4, 5	iscph 25298	. 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 1163	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ∩ cin 3912   ⊆ wss 3913   ↦ cmpt 5196   “ cima 5665  ‘cfv 6537  (class class class)co 7411  0cc0 11100  +∞cpnf 11240  [,)cico 13374  √csqrt 15284  Basecbs 17269   ↾_s cress 17290  Scalarcsca 17313  ·_𝑖cip 17315  ℂ_fldccnfld 21491  PreHilcphl 21743  normcnm 24702  NrmModcnlm 24706  ℂPreHilccph 25294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fv 6545  df-ov 7414  df-cph 25296

This theorem is referenced by:  cphnm  25321  cphnmf  25323  cphtcphnm  25358  cphsscph  25379