Theorem isclm 23816

Description: A subcomplex module is a left module over a subring of the field of 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		fvexd 6689	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
2		fvexd 6689	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) ∈ V)
3		id 22	. . . . . . . . 9 ⊢ (𝑓 = (Scalar‘𝑤) → 𝑓 = (Scalar‘𝑤))
4		fveq2 6674	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
5		isclm.f	. . . . . . . . . 10 ⊢ 𝐹 = (Scalar‘𝑊)
6	4, 5	eqtr4di 2791	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
7	3, 6	sylan9eqr 2795	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → 𝑓 = 𝐹)
8	7	adantr 484	. . . . . . 7 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = 𝐹)
9		id 22	. . . . . . . . 9 ⊢ (𝑘 = (Base‘𝑓) → 𝑘 = (Base‘𝑓))
10	7	fveq2d 6678	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = (Base‘𝐹))
11		isclm.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝐹)
12	10, 11	eqtr4di 2791	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = 𝐾)
13	9, 12	sylan9eqr 2795	. . . . . . . 8 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = 𝐾)
14	13	oveq2d 7186	. . . . . . 7 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (ℂfld ↾s 𝑘) = (ℂfld ↾s 𝐾))
15	8, 14	eqeq12d 2754	. . . . . 6 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑓 = (ℂfld ↾s 𝑘) ↔ 𝐹 = (ℂfld ↾s 𝐾)))
16	13	eleq1d 2817	. . . . . 6 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑘 ∈ (SubRing‘ℂfld) ↔ 𝐾 ∈ (SubRing‘ℂfld)))
17	15, 16	anbi12d 634	. . . . 5 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
18	2, 17	sbcied 3724	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ([(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
19	1, 18	sbcied 3724	. . 3 ⊢ (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
20		df-clm 23815	. . 3 ⊢ ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
21	19, 20	elrab2 3591	. 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
22		3anass 1096	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ↔ (𝑊 ∈ LMod ∧ (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
23	21, 22	bitr4i 281	1 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  Vcvv 3398  [wsbc 3680  ‘cfv 6339  (class class class)co 7170  Basecbs 16586   ↾_s cress 16587  Scalarcsca 16671  SubRingcsubrg 19650  LModclmod 19753  ℂ_fldccnfld 20217  ℂModcclm 23814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-iota 6297  df-fv 6347  df-ov 7173  df-clm 23815

This theorem is referenced by:  clmsca  23817  clmsubrg  23818  clmlmod  23819  isclmi  23829  lmhmclm  23839  isclmp  23849  cphclm  23941  phclm  23984  bj-isclm  35082