Theorem cphnlm 25143

Description: A subcomplex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)

Assertion

Ref	Expression
cphnlm	⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)

Proof of Theorem cphnlm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2737	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		eqid 2737	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
4		eqid 2737	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2737	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	1, 2, 3, 4, 5	iscph 25141	. . 3 ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))) ∧ (√ “ ((Base‘(Scalar‘𝑊)) ∩ (0[,)+∞))) ⊆ (Base‘(Scalar‘𝑊)) ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖‘𝑊)𝑥)))))
7	6	simp1bi 1146	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))))
8	7	simp2d 1144	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∩ cin 3902   ⊆ wss 3903   ↦ cmpt 5181   “ cima 5635  ‘cfv 6500  (class class class)co 7368  0cc0 11038  +∞cpnf 11175  [,)cico 13275  √csqrt 15168  Basecbs 17148   ↾_s cress 17169  Scalarcsca 17192  ·_𝑖cip 17194  ℂ_fldccnfld 21324  PreHilcphl 21594  normcnm 24535  NrmModcnlm 24539  ℂPreHilccph 25137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fv 6508  df-ov 7371  df-cph 25139

This theorem is referenced by:  cphngp  25144  cphlmod  25145  cphnvc  25147  cphnmvs  25161  ipcnlem2  25215  ipcnlem1  25216  csscld  25220  cphsscph  25222