Theorem cphsca 25349

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 2762	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2762	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		eqid 2762	. . . 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 25340	. . 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 1162	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)))
8	7	simp3d 1161	1 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ∩ cin 3903   ⊆ wss 3904   ↦ cmpt 5191   “ cima 5663  ‘cfv 6536  (class class class)co 7412  0cc0 11106  +∞cpnf 11246  [,)cico 13380  √csqrt 15291  Basecbs 17275   ↾_s cress 17296  Scalarcsca 17319  ·_𝑖cip 17321  ℂ_fldccnfld 21533  PreHilcphl 21785  normcnm 24744  NrmModcnlm 24748  ℂPreHilccph 25336

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5268

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-xp 5666  df-cnv 5668  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fv 6544  df-ov 7415  df-cph 25338

This theorem is used by:  cphsubrg  25350  cphreccl  25351  cphcjcl  25353  cphqss  25358  cphclm  25359  ipcau  25408  cphsscph  25421  hlprlem  25537  ishl2  25540