Theorem cphnlm 23770

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 2821	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2821	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		eqid 2821	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
4		eqid 2821	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2821	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	1, 2, 3, 4, 5	iscph 23768	. . 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 1141	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))))
8	7	simp2d 1139	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ∩ cin 3934   ⊆ wss 3935   ↦ cmpt 5138   “ cima 5552  ‘cfv 6349  (class class class)co 7150  0cc0 10531  +∞cpnf 10666  [,)cico 12734  √csqrt 14586  Basecbs 16477   ↾_s cress 16478  Scalarcsca 16562  ·_𝑖cip 16564  ℂ_fldccnfld 20539  PreHilcphl 20762  normcnm 23180  NrmModcnlm 23184  ℂPreHilccph 23764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-xp 5555  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fv 6357  df-ov 7153  df-cph 23766

This theorem is referenced by:  cphngp  23771  cphlmod  23772  cphnvc  23774  cphnmvs  23788  ipcnlem2  23841  ipcnlem1  23842  csscld  23846  cphsscph  23848