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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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