Theorem cphnmfval 25162

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 2734	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2734	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	1, 2, 3, 4, 5	iscph 25140	. 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 1147	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ∩ cin 3930   ⊆ wss 3931   ↦ cmpt 5205   “ cima 5668  ‘cfv 6541  (class class class)co 7413  0cc0 11137  +∞cpnf 11274  [,)cico 13371  √csqrt 15254  Basecbs 17229   ↾_s cress 17252  Scalarcsca 17276  ·_𝑖cip 17278  ℂ_fldccnfld 21326  PreHilcphl 21596  normcnm 24533  NrmModcnlm 24537  ℂPreHilccph 25136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-nul 5286

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-xp 5671  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fv 6549  df-ov 7416  df-cph 25138

This theorem is referenced by:  cphnm  25163  cphnmf  25165  cphtcphnm  25200  cphsscph  25221