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Theorem isclm 24227
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 6789 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
2 fvexd 6789 . . . . 5 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) ∈ V)
3 id 22 . . . . . . . . 9 (𝑓 = (Scalar‘𝑤) → 𝑓 = (Scalar‘𝑤))
4 fveq2 6774 . . . . . . . . . 10 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
5 isclm.f . . . . . . . . . 10 𝐹 = (Scalar‘𝑊)
64, 5eqtr4di 2796 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
73, 6sylan9eqr 2800 . . . . . . . 8 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → 𝑓 = 𝐹)
87adantr 481 . . . . . . 7 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = 𝐹)
9 id 22 . . . . . . . . 9 (𝑘 = (Base‘𝑓) → 𝑘 = (Base‘𝑓))
107fveq2d 6778 . . . . . . . . . 10 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = (Base‘𝐹))
11 isclm.k . . . . . . . . . 10 𝐾 = (Base‘𝐹)
1210, 11eqtr4di 2796 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = 𝐾)
139, 12sylan9eqr 2800 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = 𝐾)
1413oveq2d 7291 . . . . . . 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 2823 . . . . . 6 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑘 ∈ (SubRing‘ℂfld) ↔ 𝐾 ∈ (SubRing‘ℂfld)))
1715, 16anbi12d 631 . . . . 5 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
182, 17sbcied 3761 . . . 4 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ([(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
191, 18sbcied 3761 . . 3 (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
20 df-clm 24226 . . 3 ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
2119, 20elrab2 3627 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
22 3anass 1094 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ↔ (𝑊 ∈ LMod ∧ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
2321, 22bitr4i 277 1 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  Vcvv 3432  [wsbc 3716  cfv 6433  (class class class)co 7275  Basecbs 16912  s cress 16941  Scalarcsca 16965  SubRingcsubrg 20020  LModclmod 20123  fldccnfld 20597  ℂModcclm 24225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-clm 24226
This theorem is referenced by:  clmsca  24228  clmsubrg  24229  clmlmod  24230  isclmi  24240  lmhmclm  24250  isclmp  24260  cphclm  24353  phclm  24396  bj-isclm  35462
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