Theorem bj-isclm 37784

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 25127	. 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 6868	. . . . . . 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 416	. . . . . . 7 ⊢ (𝐾 = (Base‘𝐹) → ((Base‘𝐹) = (Base‘(Scalar‘𝑊)) → (Base‘(Scalar‘𝑊)) = 𝐾))
11	6, 7, 10	syl2im 40	. . . . . 6 ⊢ (𝜑 → (𝐹 = (Scalar‘𝑊) → (Base‘(Scalar‘𝑊)) = 𝐾))
12	4, 11	mpd 15	. . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝑊)) = 𝐾)
13	12	oveq2d 7413	. . . 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 2848	. . 3 ⊢ (𝜑 → ((Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld) ↔ 𝐾 ∈ (SubRing‘ℂfld)))
16	14, 15	3anbi23d 1461	. 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 285	1 ⊢ (𝜑 → (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ‘cfv 6522  (class class class)co 7397  Basecbs 17246   ↾_s cress 17267  Scalarcsca 17290  SubRingcsubrg 20620  LModclmod 20928  ℂ_fldccnfld 21425  ℂModcclm 25125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5257

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3416  df-v 3457  df-sbc 3746  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-iota 6478  df-fv 6530  df-ov 7400  df-clm 25126

This theorem is referenced by:  bj-rveccmod  37795