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Theorem isclm 23816
Description: A subcomplex module is a left module over a subring of the field of complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
isclm.f 𝐹 = (Scalar‘𝑊)
isclm.k 𝐾 = (Base‘𝐹)
Assertion
Ref Expression
isclm (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))

Proof of Theorem isclm
Dummy variables 𝑓 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6689 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
2 fvexd 6689 . . . . 5 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) ∈ V)
3 id 22 . . . . . . . . 9 (𝑓 = (Scalar‘𝑤) → 𝑓 = (Scalar‘𝑤))
4 fveq2 6674 . . . . . . . . . 10 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
5 isclm.f . . . . . . . . . 10 𝐹 = (Scalar‘𝑊)
64, 5eqtr4di 2791 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
73, 6sylan9eqr 2795 . . . . . . . 8 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → 𝑓 = 𝐹)
87adantr 484 . . . . . . 7 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = 𝐹)
9 id 22 . . . . . . . . 9 (𝑘 = (Base‘𝑓) → 𝑘 = (Base‘𝑓))
107fveq2d 6678 . . . . . . . . . 10 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = (Base‘𝐹))
11 isclm.k . . . . . . . . . 10 𝐾 = (Base‘𝐹)
1210, 11eqtr4di 2791 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = 𝐾)
139, 12sylan9eqr 2795 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = 𝐾)
1413oveq2d 7186 . . . . . . 7 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (ℂflds 𝑘) = (ℂflds 𝐾))
158, 14eqeq12d 2754 . . . . . 6 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑓 = (ℂflds 𝑘) ↔ 𝐹 = (ℂflds 𝐾)))
1613eleq1d 2817 . . . . . 6 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑘 ∈ (SubRing‘ℂfld) ↔ 𝐾 ∈ (SubRing‘ℂfld)))
1715, 16anbi12d 634 . . . . 5 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
182, 17sbcied 3724 . . . 4 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ([(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
191, 18sbcied 3724 . . 3 (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
20 df-clm 23815 . . 3 ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
2119, 20elrab2 3591 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
22 3anass 1096 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ↔ (𝑊 ∈ LMod ∧ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
2321, 22bitr4i 281 1 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  Vcvv 3398  [wsbc 3680  cfv 6339  (class class class)co 7170  Basecbs 16586  s cress 16587  Scalarcsca 16671  SubRingcsubrg 19650  LModclmod 19753  fldccnfld 20217  ℂModcclm 23814
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 2710  ax-nul 5174
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 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-iota 6297  df-fv 6347  df-ov 7173  df-clm 23815
This theorem is referenced by:  clmsca  23817  clmsubrg  23818  clmlmod  23819  isclmi  23829  lmhmclm  23839  isclmp  23849  cphclm  23941  phclm  23984  bj-isclm  35082
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