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Theorem islmim 16139
 Description: An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Hypotheses
Ref Expression
islmim.b
islmim.c
Assertion
Ref Expression
islmim LMIso LMHom

Proof of Theorem islmim
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lmim 16104 . . 3 LMIso LMHom
2 ovex 6109 . . . 4 LMHom
32rabex 4357 . . 3 LMHom
4 oveq12 6093 . . . 4 LMHom LMHom
5 fveq2 5731 . . . . . 6
6 islmim.b . . . . . 6
75, 6syl6eqr 2488 . . . . 5
8 fveq2 5731 . . . . . 6
9 islmim.c . . . . . 6
108, 9syl6eqr 2488 . . . . 5
11 f1oeq23 5671 . . . . 5
127, 10, 11syl2an 465 . . . 4
134, 12rabeqbidv 2953 . . 3 LMHom LMHom
141, 3, 13elovmpt2 6294 . 2 LMIso LMHom
15 df-3an 939 . 2 LMHom LMHom
16 f1oeq1 5668 . . . . 5
1716elrab 3094 . . . 4 LMHom LMHom
1817anbi2i 677 . . 3 LMHom LMHom
19 lmhmlmod1 16114 . . . . . 6 LMHom
20 lmhmlmod2 16113 . . . . . 6 LMHom
2119, 20jca 520 . . . . 5 LMHom
2221adantr 453 . . . 4 LMHom
2322pm4.71ri 616 . . 3 LMHom LMHom
2418, 23bitr4i 245 . 2 LMHom LMHom
2514, 15, 243bitri 264 1 LMIso LMHom
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726  crab 2711  wf1o 5456  cfv 5457  (class class class)co 6084  cbs 13474  clmod 15955   LMHom clmhm 16100   LMIso clmim 16101 This theorem is referenced by:  lmimf1o  16140  lmimlmhm  16141  islmim2  16143  pwssplit4  27181  indlcim  27300 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-lmhm 16103  df-lmim 16104
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