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Theorem lmhmf1o 15803
 Description: A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypotheses
Ref Expression
lmhmf1o.x
lmhmf1o.y
Assertion
Ref Expression
lmhmf1o LMHom LMHom

Proof of Theorem lmhmf1o
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmf1o.y . . 3
2 eqid 2283 . . 3
3 eqid 2283 . . 3
4 eqid 2283 . . 3 Scalar Scalar
5 eqid 2283 . . 3 Scalar Scalar
6 eqid 2283 . . 3 Scalar Scalar
7 lmhmlmod2 15789 . . . 4 LMHom
87adantr 451 . . 3 LMHom
9 lmhmlmod1 15790 . . . 4 LMHom
109adantr 451 . . 3 LMHom
115, 4lmhmsca 15787 . . . . 5 LMHom Scalar Scalar
1211eqcomd 2288 . . . 4 LMHom Scalar Scalar
1312adantr 451 . . 3 LMHom Scalar Scalar
14 lmghm 15788 . . . . 5 LMHom
15 lmhmf1o.x . . . . . 6
1615, 1ghmf1o 14712 . . . . 5
1714, 16syl 15 . . . 4 LMHom
1817biimpa 470 . . 3 LMHom
19 simpll 730 . . . . . 6 LMHom Scalar LMHom
2013fveq2d 5529 . . . . . . . . 9 LMHom Scalar Scalar
2120eleq2d 2350 . . . . . . . 8 LMHom Scalar Scalar
2221biimpar 471 . . . . . . 7 LMHom Scalar Scalar
2322adantrr 697 . . . . . 6 LMHom Scalar Scalar
24 f1ocnv 5485 . . . . . . . . . 10
25 f1of 5472 . . . . . . . . . 10
2624, 25syl 15 . . . . . . . . 9
2726adantl 452 . . . . . . . 8 LMHom
28 ffvelrn 5663 . . . . . . . 8
2927, 28sylan 457 . . . . . . 7 LMHom
3029adantrl 696 . . . . . 6 LMHom Scalar
31 eqid 2283 . . . . . . 7 Scalar Scalar
325, 31, 15, 3, 2lmhmlin 15792 . . . . . 6 LMHom Scalar
3319, 23, 30, 32syl3anc 1182 . . . . 5 LMHom Scalar
34 f1ocnvfv2 5793 . . . . . . 7
3534ad2ant2l 726 . . . . . 6 LMHom Scalar
3635oveq2d 5874 . . . . 5 LMHom Scalar
3733, 36eqtrd 2315 . . . 4 LMHom Scalar
38 simplr 731 . . . . 5 LMHom Scalar
3910adantr 451 . . . . . 6 LMHom Scalar
4015, 5, 3, 31lmodvscl 15644 . . . . . 6 Scalar
4139, 23, 30, 40syl3anc 1182 . . . . 5 LMHom Scalar
42 f1ocnvfv 5794 . . . . 5
4338, 41, 42syl2anc 642 . . . 4 LMHom Scalar
4437, 43mpd 14 . . 3 LMHom Scalar
451, 2, 3, 4, 5, 6, 8, 10, 13, 18, 44islmhmd 15796 . 2 LMHom LMHom
4615, 1lmhmf 15791 . . . . 5 LMHom
47 ffn 5389 . . . . 5
4846, 47syl 15 . . . 4 LMHom
4948adantr 451 . . 3 LMHom LMHom
501, 15lmhmf 15791 . . . . 5 LMHom
5150adantl 452 . . . 4 LMHom LMHom
52 ffn 5389 . . . 4
5351, 52syl 15 . . 3 LMHom LMHom
54 dff1o4 5480 . . 3
5549, 53, 54sylanbrc 645 . 2 LMHom LMHom
5645, 55impbida 805 1 LMHom LMHom
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1623   wcel 1684  ccnv 4688   wfn 5250  wf 5251  wf1o 5254  cfv 5255  (class class class)co 5858  cbs 13148  Scalarcsca 13211  cvsca 13212   cghm 14680  clmod 15627   LMHom clmhm 15776 This theorem is referenced by:  islmim2  15819 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-mnd 14367  df-grp 14489  df-ghm 14681  df-lmod 15629  df-lmhm 15779
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