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Theorem lmimcnv 20912
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 2726 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2726 . . . . . . 7 (Base‘𝑇) = (Base‘𝑇)
31, 2lmhmf 20879 . . . . . 6 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4 frel 6715 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
53, 4syl 17 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → Rel 𝐹)
6 dfrel2 6181 . . . . 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 2827 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇))
109anim1ci 615 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)) → (𝐹 ∈ (𝑇 LMHom 𝑆) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)))
11 islmim2 20911 . 2 (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑇 LMHom 𝑆)))
12 islmim2 20911 . 2 (𝐹 ∈ (𝑇 LMIso 𝑆) ↔ (𝐹 ∈ (𝑇 LMHom 𝑆) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)))
1310, 11, 123imtr4i 292 1 (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹 ∈ (𝑇 LMIso 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  ccnv 5668  Rel wrel 5674  wf 6532  cfv 6536  (class class class)co 7404  Basecbs 17150   LMHom clmhm 20864   LMIso clmim 20865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  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 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-grp 18863  df-ghm 19136  df-lmod 20705  df-lmhm 20867  df-lmim 20868
This theorem is referenced by:  lmicsym  20917  lbslcic  21731
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