Theorem bj-isclm 34575

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 2821	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2		eqid 2821	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
3	1, 2	isclm 23668	. 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 2827	. . . 4 ⊢ (𝜑 → (Scalar‘𝑊) = 𝐹)
6		bj-isclm.base	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (Base‘𝐹))
7		fveq2 6670	. . . . . . 7 ⊢ (𝐹 = (Scalar‘𝑊) → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
8		eqtr 2841	. . . . . . . . 9 ⊢ ((𝐾 = (Base‘𝐹) ∧ (Base‘𝐹) = (Base‘(Scalar‘𝑊))) → 𝐾 = (Base‘(Scalar‘𝑊)))
9	8	eqcomd 2827	. . . . . . . 8 ⊢ ((𝐾 = (Base‘𝐹) ∧ (Base‘𝐹) = (Base‘(Scalar‘𝑊))) → (Base‘(Scalar‘𝑊)) = 𝐾)
10	9	ex 415	. . . . . . 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 7172	. . . 4 ⊢ (𝜑 → (ℂfld ↾s (Base‘(Scalar‘𝑊))) = (ℂfld ↾s 𝐾))
14	5, 13	eqeq12d 2837	. . 3 ⊢ (𝜑 → ((Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))) ↔ 𝐹 = (ℂfld ↾s 𝐾)))
15	12	eleq1d 2897	. . 3 ⊢ (𝜑 → ((Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld) ↔ 𝐾 ∈ (SubRing‘ℂfld)))
16	14, 15	3anbi23d 1435	. 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 285	1 ⊢ (𝜑 → (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ‘cfv 6355  (class class class)co 7156  Basecbs 16483   ↾_s cress 16484  Scalarcsca 16568  SubRingcsubrg 19531  LModclmod 19634  ℂ_fldccnfld 20545  ℂModcclm 23666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-nul 5210

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-ov 7159  df-clm 23667

This theorem is referenced by:  bj-rveccmod  34586