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Theorem isclm 23083
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 6346 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
2 fvexd 6346 . . . . 5 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) ∈ V)
3 id 22 . . . . . . . . 9 (𝑓 = (Scalar‘𝑤) → 𝑓 = (Scalar‘𝑤))
4 fveq2 6333 . . . . . . . . . 10 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
5 isclm.f . . . . . . . . . 10 𝐹 = (Scalar‘𝑊)
64, 5syl6eqr 2823 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
73, 6sylan9eqr 2827 . . . . . . . 8 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → 𝑓 = 𝐹)
87adantr 466 . . . . . . 7 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = 𝐹)
9 id 22 . . . . . . . . 9 (𝑘 = (Base‘𝑓) → 𝑘 = (Base‘𝑓))
107fveq2d 6337 . . . . . . . . . 10 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = (Base‘𝐹))
11 isclm.k . . . . . . . . . 10 𝐾 = (Base‘𝐹)
1210, 11syl6eqr 2823 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = 𝐾)
139, 12sylan9eqr 2827 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = 𝐾)
1413oveq2d 6812 . . . . . . 7 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (ℂflds 𝑘) = (ℂflds 𝐾))
158, 14eqeq12d 2786 . . . . . 6 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑓 = (ℂflds 𝑘) ↔ 𝐹 = (ℂflds 𝐾)))
1613eleq1d 2835 . . . . . 6 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑘 ∈ (SubRing‘ℂfld) ↔ 𝐾 ∈ (SubRing‘ℂfld)))
1715, 16anbi12d 616 . . . . 5 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
182, 17sbcied 3624 . . . 4 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ([(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
191, 18sbcied 3624 . . 3 (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
20 df-clm 23082 . . 3 ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
2119, 20elrab2 3518 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
22 3anass 1080 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ↔ (𝑊 ∈ LMod ∧ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
2321, 22bitr4i 267 1 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  [wsbc 3587  cfv 6030  (class class class)co 6796  Basecbs 16064  s cress 16065  Scalarcsca 16152  SubRingcsubrg 18986  LModclmod 19073  fldccnfld 19961  ℂModcclm 23081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-ov 6799  df-clm 23082
This theorem is referenced by:  clmsca  23084  clmsubrg  23085  clmlmod  23086  isclmi  23096  lmhmclm  23106  isclmp  23116  cphclm  23208  tchclm  23250
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