Theorem cphsca 25307

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 2769	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2769	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		eqid 2769	. . . 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 25298	. . 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 1161	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)))
8	7	simp3d 1160	1 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ∩ cin 3910   ⊆ wss 3911   ↦ cmpt 5194   “ cima 5665  ‘cfv 6537  (class class class)co 7411  0cc0 11100  +∞cpnf 11240  [,)cico 13374  √csqrt 15284  Basecbs 17269   ↾_s cress 17290  Scalarcsca 17313  ·_𝑖cip 17315  ℂ_fldccnfld 21491  PreHilcphl 21743  normcnm 24702  NrmModcnlm 24706  ℂPreHilccph 25294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3423  df-v 3463  df-sbc 3752  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fv 6545  df-ov 7414  df-cph 25296

This theorem is referenced by:  cphsubrg  25308  cphreccl  25309  cphcjcl  25311  cphqss  25316  cphclm  25317  ipcau  25366  cphsscph  25379  hlprlem  25495  ishl2  25498