Theorem cphnlm 25493

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 2761	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2761	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		eqid 2761	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
4		eqid 2761	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2761	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	1, 2, 3, 4, 5	iscph 25491	. . 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)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∩ cin 3898   ⊆ wss 3899   ↦ cmpt 5186   “ cima 5654  ‘cfv 6538  (class class class)co 7420  0cc0 11200  +∞cpnf 11340  [,)cico 13478  √csqrt 15400  Basecbs 17387   ↾_s cress 17408  Scalarcsca 17431  ·_𝑖cip 17433  ℂ_fldccnfld 21678  PreHilcphl 21930  normcnm 24895  NrmModcnlm 24899  ℂPreHilccph 25487

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fv 6546  df-ov 7423  df-cph 25489

This theorem is used by:  cphngp  25494  cphlmod  25495  cphnvc  25497  cphnmvs  25511  ipcnlem2  25565  ipcnlem1  25566  csscld  25570  cphsscph  25572