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Theorem isclmi 25017
Description: Reverse direction of isclm 25004. (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 1134 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ LMod)
2 simp2 1135 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂflds 𝐾))
3 eqid 2728 . . . . . . 7 (ℂflds 𝐾) = (ℂflds 𝐾)
43subrgbas 20520 . . . . . 6 (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 = (Base‘(ℂflds 𝐾)))
543ad2ant3 1133 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘(ℂflds 𝐾)))
62fveq2d 6901 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) = (Base‘(ℂflds 𝐾)))
75, 6eqtr4d 2771 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘𝐹))
87oveq2d 7436 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (ℂflds 𝐾) = (ℂflds (Base‘𝐹)))
92, 8eqtrd 2768 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂflds (Base‘𝐹)))
10 simp3 1136 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 ∈ (SubRing‘ℂfld))
117, 10eqeltrrd 2830 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) ∈ (SubRing‘ℂfld))
12 clm0.f . . 3 𝐹 = (Scalar‘𝑊)
13 eqid 2728 . . 3 (Base‘𝐹) = (Base‘𝐹)
1412, 13isclm 25004 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘ℂfld)))
151, 9, 11, 14syl3anbrc 1341 1 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1534  wcel 2099  cfv 6548  (class class class)co 7420  Basecbs 17180  s cress 17209  Scalarcsca 17236  SubRingcsubrg 20506  LModclmod 20743  fldccnfld 21279  ℂModcclm 25002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-1cn 11197  ax-addcl 11199
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-nn 12244  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-ress 17210  df-subg 19078  df-ring 20175  df-subrg 20508  df-clm 25003
This theorem is referenced by:  zlmclm  25052  cnstrcvs  25081  cncvs  25085  recvs  25086  recvsOLD  25087  qcvs  25088  zclmncvs  25089
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