Theorem bj-isclm 35441

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 2739	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2		eqid 2739	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
3	1, 2	isclm 24208	. 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 2745	. . . 4 ⊢ (𝜑 → (Scalar‘𝑊) = 𝐹)
6		bj-isclm.base	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (Base‘𝐹))
7		fveq2 6768	. . . . . . 7 ⊢ (𝐹 = (Scalar‘𝑊) → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
8		eqtr 2762	. . . . . . . . 9 ⊢ ((𝐾 = (Base‘𝐹) ∧ (Base‘𝐹) = (Base‘(Scalar‘𝑊))) → 𝐾 = (Base‘(Scalar‘𝑊)))
9	8	eqcomd 2745	. . . . . . . 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 7284	. . . 4 ⊢ (𝜑 → (ℂfld ↾s (Base‘(Scalar‘𝑊))) = (ℂfld ↾s 𝐾))
14	5, 13	eqeq12d 2755	. . 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 1437	. 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	syl5bb 282	1 ⊢ (𝜑 → (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  ‘cfv 6430  (class class class)co 7268  Basecbs 16893   ↾_s cress 16922  Scalarcsca 16946  SubRingcsubrg 20001  LModclmod 20104  ℂ_fldccnfld 20578  ℂModcclm 24206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-nul 5233

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438  df-ov 7271  df-clm 24207

This theorem is referenced by:  bj-rveccmod  35452