Theorem cphsca 25198

Description: A subcomplex pre-Hilbert space is a vector space over a subfield of ℂ_fld. (Contributed by Mario Carneiro, 8-Oct-2015.)

Hypotheses

Ref	Expression
cphsca.f	⊢ 𝐹 = (Scalar‘𝑊)
cphsca.k	⊢ 𝐾 = (Base‘𝐹)

Assertion

Ref	Expression
cphsca	⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾))

Proof of Theorem cphsca

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2726	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2726	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		eqid 2726	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
4		cphsca.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
5		cphsca.k	. . . 4 ⊢ 𝐾 = (Base‘𝐹)
6	1, 2, 3, 4, 5	iscph 25189	. . 3 ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖‘𝑊)𝑥)))))
7	6	simp1bi 1142	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)))
8	7	simp3d 1141	1 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ∩ cin 3946   ⊆ wss 3947   ↦ cmpt 5236   “ cima 5685  ‘cfv 6554  (class class class)co 7424  0cc0 11158  +∞cpnf 11295  [,)cico 13380  √csqrt 15238  Basecbs 17213   ↾_s cress 17242  Scalarcsca 17269  ·_𝑖cip 17271  ℂ_fldccnfld 21343  PreHilcphl 21620  normcnm 24576  NrmModcnlm 24580  ℂPreHilccph 25185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5311

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fv 6562  df-ov 7427  df-cph 25187

This theorem is referenced by:  cphsubrg  25199  cphreccl  25200  cphcjcl  25202  cphqss  25207  cphclm  25208  ipcau  25257  cphsscph  25270  hlprlem  25386  ishl2  25389