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Theorem isclmi 22858
Description: Reverse direction of isclm 22845. (Contributed by Mario Carneiro, 30-Oct-2015.)
Hypothesis
Ref Expression
clm0.f 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
isclmi ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)

Proof of Theorem isclmi
StepHypRef Expression
1 simp1 1059 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ LMod)
2 simp2 1060 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂflds 𝐾))
3 eqid 2620 . . . . . . 7 (ℂflds 𝐾) = (ℂflds 𝐾)
43subrgbas 18770 . . . . . 6 (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 = (Base‘(ℂflds 𝐾)))
543ad2ant3 1082 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘(ℂflds 𝐾)))
62fveq2d 6182 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) = (Base‘(ℂflds 𝐾)))
75, 6eqtr4d 2657 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘𝐹))
87oveq2d 6651 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (ℂflds 𝐾) = (ℂflds (Base‘𝐹)))
92, 8eqtrd 2654 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂflds (Base‘𝐹)))
10 simp3 1061 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 ∈ (SubRing‘ℂfld))
117, 10eqeltrrd 2700 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) ∈ (SubRing‘ℂfld))
12 clm0.f . . 3 𝐹 = (Scalar‘𝑊)
13 eqid 2620 . . 3 (Base‘𝐹) = (Base‘𝐹)
1412, 13isclm 22845 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘ℂfld)))
151, 9, 11, 14syl3anbrc 1244 1 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  wcel 1988  cfv 5876  (class class class)co 6635  Basecbs 15838  s cress 15839  Scalarcsca 15925  SubRingcsubrg 18757  LModclmod 18844  fldccnfld 19727  ℂModcclm 22843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-i2m1 9989  ax-1ne0 9990  ax-rrecex 9993  ax-cnre 9994
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-nn 11006  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-subg 17572  df-ring 18530  df-subrg 18759  df-clm 22844
This theorem is referenced by:  zlmclm  22893  cnstrcvs  22922  cncvs  22926  recvs  22927  qcvs  22928  zclmncvs  22929
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