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Theorem lmimcnv 20627
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 2731 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2731 . . . . . . 7 (Base‘𝑇) = (Base‘𝑇)
31, 2lmhmf 20594 . . . . . 6 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4 frel 6709 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
53, 4syl 17 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → Rel 𝐹)
6 dfrel2 6177 . . . . 5 (Rel 𝐹𝐹 = 𝐹)
75, 6sylib 217 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 = 𝐹)
8 id 22 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇))
97, 8eqeltrd 2832 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇))
109anim1ci 616 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → (𝐹 ∈ (𝑇 LMHom 𝑆) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)))
11 islmim2 20626 . 2 (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)))
12 islmim2 20626 . 2 (𝐹 ∈ (𝑇 LMIso 𝑆) ↔ (𝐹 ∈ (𝑇 LMHom 𝑆) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)))
1310, 11, 123imtr4i 291 1 (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹 ∈ (𝑇 LMIso 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ccnv 5668  Rel wrel 5674  wf 6528  cfv 6532  (class class class)co 7393  Basecbs 17126   LMHom clmhm 20579   LMIso clmim 20580
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-grp 18797  df-ghm 19056  df-lmod 20422  df-lmhm 20582  df-lmim 20583
This theorem is referenced by:  lmicsym  20632  lbslcic  21329
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