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Theorem bj-isclm 35105
Description: The predicate "is a subcomplex module". (Contributed by BJ, 6-Jan-2024.)
Hypotheses
Ref Expression
bj-isclm.scal (𝜑𝐹 = (Scalar‘𝑊))
bj-isclm.base (𝜑𝐾 = (Base‘𝐹))
Assertion
Ref Expression
bj-isclm (𝜑 → (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))

Proof of Theorem bj-isclm
StepHypRef Expression
1 eqid 2739 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2739 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
31, 2isclm 23819 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld)))
4 bj-isclm.scal . . . . 5 (𝜑𝐹 = (Scalar‘𝑊))
54eqcomd 2745 . . . 4 (𝜑 → (Scalar‘𝑊) = 𝐹)
6 bj-isclm.base . . . . . . 7 (𝜑𝐾 = (Base‘𝐹))
7 fveq2 6677 . . . . . . 7 (𝐹 = (Scalar‘𝑊) → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
8 eqtr 2759 . . . . . . . . 9 ((𝐾 = (Base‘𝐹) ∧ (Base‘𝐹) = (Base‘(Scalar‘𝑊))) → 𝐾 = (Base‘(Scalar‘𝑊)))
98eqcomd 2745 . . . . . . . 8 ((𝐾 = (Base‘𝐹) ∧ (Base‘𝐹) = (Base‘(Scalar‘𝑊))) → (Base‘(Scalar‘𝑊)) = 𝐾)
109ex 416 . . . . . . 7 (𝐾 = (Base‘𝐹) → ((Base‘𝐹) = (Base‘(Scalar‘𝑊)) → (Base‘(Scalar‘𝑊)) = 𝐾))
116, 7, 10syl2im 40 . . . . . 6 (𝜑 → (𝐹 = (Scalar‘𝑊) → (Base‘(Scalar‘𝑊)) = 𝐾))
124, 11mpd 15 . . . . 5 (𝜑 → (Base‘(Scalar‘𝑊)) = 𝐾)
1312oveq2d 7189 . . . 4 (𝜑 → (ℂflds (Base‘(Scalar‘𝑊))) = (ℂflds 𝐾))
145, 13eqeq12d 2755 . . 3 (𝜑 → ((Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊))) ↔ 𝐹 = (ℂflds 𝐾)))
1512eleq1d 2818 . . 3 (𝜑 → ((Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld) ↔ 𝐾 ∈ (SubRing‘ℂfld)))
1614, 153anbi23d 1440 . 2 (𝜑 → ((𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld)) ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
173, 16syl5bb 286 1 (𝜑 → (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  cfv 6340  (class class class)co 7173  Basecbs 16589  s cress 16590  Scalarcsca 16674  SubRingcsubrg 19653  LModclmod 19756  fldccnfld 20220  ℂModcclm 23817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-nul 5175
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-iota 6298  df-fv 6348  df-ov 7176  df-clm 23818
This theorem is referenced by:  bj-rveccmod  35116
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