Theorem cphsca 25232

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 2740	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2740	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		eqid 2740	. . . 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 25223	. . 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 1145	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)))
8	7	simp3d 1144	1 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976   ↦ cmpt 5249   “ cima 5703  ‘cfv 6573  (class class class)co 7448  0cc0 11184  +∞cpnf 11321  [,)cico 13409  √csqrt 15282  Basecbs 17258   ↾_s cress 17287  Scalarcsca 17314  ·_𝑖cip 17316  ℂ_fldccnfld 21387  PreHilcphl 21665  normcnm 24610  NrmModcnlm 24614  ℂPreHilccph 25219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fv 6581  df-ov 7451  df-cph 25221

This theorem is referenced by:  cphsubrg  25233  cphreccl  25234  cphcjcl  25236  cphqss  25241  cphclm  25242  ipcau  25291  cphsscph  25304  hlprlem  25420  ishl2  25423