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Theorem lmimgim 19837
Description: An isomorphism of modules is an isomorphism of groups. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
lmimgim (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆))

Proof of Theorem lmimgim
StepHypRef Expression
1 lmimlmhm 19836 . . 3 (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 LMHom 𝑆))
2 lmghm 19803 . . 3 (𝐹 ∈ (𝑅 LMHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
31, 2syl 17 . 2 (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
4 eqid 2824 . . 3 (Base‘𝑅) = (Base‘𝑅)
5 eqid 2824 . . 3 (Base‘𝑆) = (Base‘𝑆)
64, 5lmimf1o 19835 . 2 (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
74, 5isgim 18402 . 2 (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
83, 6, 7sylanbrc 586 1 (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  1-1-ontowf1o 6342  cfv 6343  (class class class)co 7149  Basecbs 16483   GrpHom cghm 18355   GrpIso cgim 18397   LMHom clmhm 19791   LMIso clmim 19792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-ghm 18356  df-gim 18399  df-lmhm 19794  df-lmim 19795
This theorem is referenced by: (None)
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