Theorem cphsca 25500

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 2760	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2760	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		eqid 2760	. . . 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 25491	. . 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 1163	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)))
8	7	simp3d 1162	1 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∩ cin 4089   ↦ cmpt 5185   “ cima 5653  ‘cfv 6538  (class class class)co 7420  0cc0 11200  +∞cpnf 11340  [,)cico 13478  √csqrt 15400  Basecbs 17387   ↾_s cress 17408  Scalarcsca 17431  ·_𝑖cip 17433  ℂ_fldccnfld 21678  PreHilcphl 21930  normcnm 24895  NrmModcnlm 24899  ℂPreHilccph 25487

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-xp 5656  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fv 6546  df-ov 7423  df-cph 25489

This theorem is used by:  cphsubrg  25501  cphreccl  25502  cphcjcl  25504  cphqss  25509  cphclm  25510  ipcau  25559  cphsscph  25572  hlprlem  25688  ishl2  25691