Theorem bj-isclm 38355

Description: The predicate "is a subcomplex module". (Contributed by BJ, 6-Jan-2024.)

Hypotheses

Ref	Expression
bj-isclm.scal	⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))
bj-isclm.base	⊢ (𝜑 → 𝐾 = (Base‘𝐹))

Assertion

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

Proof of Theorem bj-isclm

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2		eqid 2763	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
3	1, 2	isclm 25400	. 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld)))
4		bj-isclm.scal	. . . . 5 ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))
5	4	eqcomd 2769	. . . 4 ⊢ (𝜑 → (Scalar‘𝑊) = 𝐹)
6		bj-isclm.base	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (Base‘𝐹))
7		fveq2 6896	. . . . . . 7 ⊢ (𝐹 = (Scalar‘𝑊) → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
8		eqtr 2783	. . . . . . . . 9 ⊢ ((𝐾 = (Base‘𝐹) ∧ (Base‘𝐹) = (Base‘(Scalar‘𝑊))) → 𝐾 = (Base‘(Scalar‘𝑊)))
9	8	eqcomd 2769	. . . . . . . 8 ⊢ ((𝐾 = (Base‘𝐹) ∧ (Base‘𝐹) = (Base‘(Scalar‘𝑊))) → (Base‘(Scalar‘𝑊)) = 𝐾)
10	9	ex 418	. . . . . . 7 ⊢ (𝐾 = (Base‘𝐹) → ((Base‘𝐹) = (Base‘(Scalar‘𝑊)) → (Base‘(Scalar‘𝑊)) = 𝐾))
11	6, 7, 10	syl2im 41	. . . . . 6 ⊢ (𝜑 → (𝐹 = (Scalar‘𝑊) → (Base‘(Scalar‘𝑊)) = 𝐾))
12	4, 11	mpd 16	. . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝑊)) = 𝐾)
13	12	oveq2d 7446	. . . 4 ⊢ (𝜑 → (ℂfld ↾s (Base‘(Scalar‘𝑊))) = (ℂfld ↾s 𝐾))
14	5, 13	eqeq12d 2779	. . 3 ⊢ (𝜑 → ((Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))) ↔ 𝐹 = (ℂfld ↾s 𝐾)))
15	12	eleq1d 2849	. . 3 ⊢ (𝜑 → ((Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld) ↔ 𝐾 ∈ (SubRing‘ℂfld)))
16	14, 15	3anbi23d 1467	. 2 ⊢ (𝜑 → ((𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld)) ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
17	3, 16	bitrid 286	1 ⊢ (𝜑 → (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ‘cfv 6547  (class class class)co 7430  Basecbs 17402   ↾_s cress 17423  Scalarcsca 17446  SubRingcsubrg 20836  LModclmod 21150  ℂ_fldccnfld 21693  ℂModcclm 25398

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-nul 5263

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 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6501  df-fv 6555  df-ov 7433  df-clm 25399

This theorem is used by:  bj-rveccmod  38366