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Theorem lmimcnv 19841
Description: The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
lmimcnv (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹 ∈ (𝑇 LMIso 𝑆))

Proof of Theorem lmimcnv
StepHypRef Expression
1 eqid 2824 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2824 . . . . . . 7 (Base‘𝑇) = (Base‘𝑇)
31, 2lmhmf 19808 . . . . . 6 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4 frel 6510 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
53, 4syl 17 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → Rel 𝐹)
6 dfrel2 6035 . . . . 5 (Rel 𝐹𝐹 = 𝐹)
75, 6sylib 221 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 = 𝐹)
8 id 22 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇))
97, 8eqeltrd 2916 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇))
109anim1ci 618 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → (𝐹 ∈ (𝑇 LMHom 𝑆) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)))
11 islmim2 19840 . 2 (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)))
12 islmim2 19840 . 2 (𝐹 ∈ (𝑇 LMIso 𝑆) ↔ (𝐹 ∈ (𝑇 LMHom 𝑆) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)))
1310, 11, 123imtr4i 295 1 (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹 ∈ (𝑇 LMIso 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  ccnv 5542  Rel wrel 5548  wf 6341  cfv 6345  (class class class)co 7151  Basecbs 16485   LMHom clmhm 19793   LMIso clmim 19794
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 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
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 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-ghm 18358  df-lmod 19638  df-lmhm 19796  df-lmim 19797
This theorem is referenced by:  lmicsym  19846  lbslcic  20539
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