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Theorem isclm 22787
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 fvex 6163 . . . . 5 (Scalar‘𝑤) ∈ V
21a1i 11 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
3 fvex 6163 . . . . . 6 (Base‘𝑓) ∈ V
43a1i 11 . . . . 5 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) ∈ V)
5 id 22 . . . . . . . . 9 (𝑓 = (Scalar‘𝑤) → 𝑓 = (Scalar‘𝑤))
6 fveq2 6153 . . . . . . . . . 10 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
7 isclm.f . . . . . . . . . 10 𝐹 = (Scalar‘𝑊)
86, 7syl6eqr 2673 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
95, 8sylan9eqr 2677 . . . . . . . 8 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → 𝑓 = 𝐹)
109adantr 481 . . . . . . 7 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = 𝐹)
11 id 22 . . . . . . . . 9 (𝑘 = (Base‘𝑓) → 𝑘 = (Base‘𝑓))
129fveq2d 6157 . . . . . . . . . 10 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = (Base‘𝐹))
13 isclm.k . . . . . . . . . 10 𝐾 = (Base‘𝐹)
1412, 13syl6eqr 2673 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = 𝐾)
1511, 14sylan9eqr 2677 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = 𝐾)
1615oveq2d 6626 . . . . . . 7 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (ℂflds 𝑘) = (ℂflds 𝐾))
1710, 16eqeq12d 2636 . . . . . 6 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑓 = (ℂflds 𝑘) ↔ 𝐹 = (ℂflds 𝐾)))
1815eleq1d 2683 . . . . . 6 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑘 ∈ (SubRing‘ℂfld) ↔ 𝐾 ∈ (SubRing‘ℂfld)))
1917, 18anbi12d 746 . . . . 5 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
204, 19sbcied 3458 . . . 4 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ([(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
212, 20sbcied 3458 . . 3 (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld)) ↔ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
22 df-clm 22786 . . 3 ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
2321, 22elrab2 3352 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
24 3anass 1040 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ↔ (𝑊 ∈ LMod ∧ (𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
2523, 24bitr4i 267 1 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3189  [wsbc 3421  cfv 5852  (class class class)co 6610  Basecbs 15792  s cress 15793  Scalarcsca 15876  SubRingcsubrg 18708  LModclmod 18795  fldccnfld 19678  ℂModcclm 22785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-ov 6613  df-clm 22786
This theorem is referenced by:  clmsca  22788  clmsubrg  22789  clmlmod  22790  isclmi  22800  lmhmclm  22810  isclmp  22820  cphclm  22912  tchclm  22954
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