Theorem cphnlm 25331

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 2763	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2763	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		eqid 2763	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
4		eqid 2763	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2763	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	1, 2, 3, 4, 5	iscph 25329	. . 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 1163	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))))
8	7	simp2d 1161	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ∩ cin 3904   ⊆ wss 3905   ↦ cmpt 5192   “ cima 5664  ‘cfv 6536  (class class class)co 7410  0cc0 11095  +∞cpnf 11235  [,)cico 13369  √csqrt 15280  Basecbs 17264   ↾_s cress 17285  Scalarcsca 17308  ·_𝑖cip 17310  ℂ_fldccnfld 21522  PreHilcphl 21774  normcnm 24733  NrmModcnlm 24737  ℂPreHilccph 25325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fv 6544  df-ov 7413  df-cph 25327

This theorem is referenced by:  cphngp  25332  cphlmod  25333  cphnvc  25335  cphnmvs  25349  ipcnlem2  25403  ipcnlem1  25404  csscld  25408  cphsscph  25410