Theorem bj-isclm 37274

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 2735	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2		eqid 2735	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
3	1, 2	isclm 25111	. 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 2741	. . . 4 ⊢ (𝜑 → (Scalar‘𝑊) = 𝐹)
6		bj-isclm.base	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (Base‘𝐹))
7		fveq2 6907	. . . . . . 7 ⊢ (𝐹 = (Scalar‘𝑊) → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
8		eqtr 2758	. . . . . . . . 9 ⊢ ((𝐾 = (Base‘𝐹) ∧ (Base‘𝐹) = (Base‘(Scalar‘𝑊))) → 𝐾 = (Base‘(Scalar‘𝑊)))
9	8	eqcomd 2741	. . . . . . . 8 ⊢ ((𝐾 = (Base‘𝐹) ∧ (Base‘𝐹) = (Base‘(Scalar‘𝑊))) → (Base‘(Scalar‘𝑊)) = 𝐾)
10	9	ex 412	. . . . . . 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 7447	. . . 4 ⊢ (𝜑 → (ℂfld ↾s (Base‘(Scalar‘𝑊))) = (ℂfld ↾s 𝐾))
14	5, 13	eqeq12d 2751	. . 3 ⊢ (𝜑 → ((Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))) ↔ 𝐹 = (ℂfld ↾s 𝐾)))
15	12	eleq1d 2824	. . 3 ⊢ (𝜑 → ((Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld) ↔ 𝐾 ∈ (SubRing‘ℂfld)))
16	14, 15	3anbi23d 1438	. 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 283	1 ⊢ (𝜑 → (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431  Basecbs 17245   ↾_s cress 17274  Scalarcsca 17301  SubRingcsubrg 20586  LModclmod 20875  ℂ_fldccnfld 21382  ℂModcclm 25109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-clm 25110

This theorem is referenced by:  bj-rveccmod  37285