Theorem cphnmfval 25247

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 2740	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2740	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	1, 2, 3, 4, 5	iscph 25225	. 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 1087   = wceq 1537   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976   ↦ cmpt 5249   “ cima 5703  ‘cfv 6575  (class class class)co 7450  0cc0 11186  +∞cpnf 11323  [,)cico 13411  √csqrt 15284  Basecbs 17260   ↾_s cress 17289  Scalarcsca 17316  ·_𝑖cip 17318  ℂ_fldccnfld 21389  PreHilcphl 21667  normcnm 24612  NrmModcnlm 24616  ℂPreHilccph 25221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fv 6583  df-ov 7453  df-cph 25223

This theorem is referenced by:  cphnm  25248  cphnmf  25250  cphtcphnm  25285  cphsscph  25306