Theorem isclm 22672

Description: A complex module is a left module over a subring of the complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)

Hypotheses

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

Assertion

Ref	Expression
isclm	⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))

Proof of Theorem isclm

Dummy variables 𝑓 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6113	. . . . 5 ⊢ (Scalar‘𝑤) ∈ V
2	1	a1i 11	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
3		fvex 6113	. . . . . 6 ⊢ (Base‘𝑓) ∈ V
4	3	a1i 11	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) ∈ V)
5		id 22	. . . . . . . . 9 ⊢ (𝑓 = (Scalar‘𝑤) → 𝑓 = (Scalar‘𝑤))
6		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
7		isclm.f	. . . . . . . . . 10 ⊢ 𝐹 = (Scalar‘𝑊)
8	6, 7	syl6eqr 2662	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
9	5, 8	sylan9eqr 2666	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → 𝑓 = 𝐹)
10	9	adantr 480	. . . . . . 7 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = 𝐹)
11		id 22	. . . . . . . . 9 ⊢ (𝑘 = (Base‘𝑓) → 𝑘 = (Base‘𝑓))
12	9	fveq2d 6107	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = (Base‘𝐹))
13		isclm.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝐹)
14	12, 13	syl6eqr 2662	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = 𝐾)
15	11, 14	sylan9eqr 2666	. . . . . . . 8 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = 𝐾)
16	15	oveq2d 6565	. . . . . . 7 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (ℂfld ↾s 𝑘) = (ℂfld ↾s 𝐾))
17	10, 16	eqeq12d 2625	. . . . . 6 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑓 = (ℂfld ↾s 𝑘) ↔ 𝐹 = (ℂfld ↾s 𝐾)))
18	15	eleq1d 2672	. . . . . 6 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑘 ∈ (SubRing‘ℂfld) ↔ 𝐾 ∈ (SubRing‘ℂfld)))
19	17, 18	anbi12d 743	. . . . 5 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
20	4, 19	sbcied 3439	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ([(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
21	2, 20	sbcied 3439	. . 3 ⊢ (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
22		df-clm 22671	. . 3 ⊢ ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
23	21, 22	elrab2 3333	. 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
24		3anass 1035	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ↔ (𝑊 ∈ LMod ∧ (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
25	23, 24	bitr4i 266	1 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))

Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾_s cress 15696  Scalarcsca 15771  SubRingcsubrg 18599  LModclmod 18686  ℂ_fldccnfld 19567  ℂModcclm 22670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-clm 22671

This theorem is referenced by:  clmsca  22673  clmsubrg  22674  clmlmod  22675  isclmi  22685  lmhmclm  22695  isclmp  22705  cphclm  22797  tchclm  22839