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Theorem lmhmf1o 16114
 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 2435 . . 3
3 eqid 2435 . . 3
4 eqid 2435 . . 3 Scalar Scalar
5 eqid 2435 . . 3 Scalar Scalar
6 eqid 2435 . . 3 Scalar Scalar
7 lmhmlmod2 16100 . . . 4 LMHom
87adantr 452 . . 3 LMHom
9 lmhmlmod1 16101 . . . 4 LMHom
109adantr 452 . . 3 LMHom
115, 4lmhmsca 16098 . . . . 5 LMHom Scalar Scalar
1211eqcomd 2440 . . . 4 LMHom Scalar Scalar
1312adantr 452 . . 3 LMHom Scalar Scalar
14 lmghm 16099 . . . . 5 LMHom
15 lmhmf1o.x . . . . . 6
1615, 1ghmf1o 15027 . . . . 5
1714, 16syl 16 . . . 4 LMHom
1817biimpa 471 . . 3 LMHom
19 simpll 731 . . . . . 6 LMHom Scalar LMHom
2013fveq2d 5724 . . . . . . . . 9 LMHom Scalar Scalar
2120eleq2d 2502 . . . . . . . 8 LMHom Scalar Scalar
2221biimpar 472 . . . . . . 7 LMHom Scalar Scalar
2322adantrr 698 . . . . . 6 LMHom Scalar Scalar
24 f1ocnv 5679 . . . . . . . . . 10
25 f1of 5666 . . . . . . . . . 10
2624, 25syl 16 . . . . . . . . 9
2726adantl 453 . . . . . . . 8 LMHom
2827ffvelrnda 5862 . . . . . . 7 LMHom
2928adantrl 697 . . . . . 6 LMHom Scalar
30 eqid 2435 . . . . . . 7 Scalar Scalar
315, 30, 15, 3, 2lmhmlin 16103 . . . . . 6 LMHom Scalar
3219, 23, 29, 31syl3anc 1184 . . . . 5 LMHom Scalar
33 f1ocnvfv2 6007 . . . . . . 7
3433ad2ant2l 727 . . . . . 6 LMHom Scalar
3534oveq2d 6089 . . . . 5 LMHom Scalar
3632, 35eqtrd 2467 . . . 4 LMHom Scalar
37 simplr 732 . . . . 5 LMHom Scalar
3810adantr 452 . . . . . 6 LMHom Scalar
3915, 5, 3, 30lmodvscl 15959 . . . . . 6 Scalar
4038, 23, 29, 39syl3anc 1184 . . . . 5 LMHom Scalar
41 f1ocnvfv 6008 . . . . 5
4237, 40, 41syl2anc 643 . . . 4 LMHom Scalar
4336, 42mpd 15 . . 3 LMHom Scalar
441, 2, 3, 4, 5, 6, 8, 10, 13, 18, 43islmhmd 16107 . 2 LMHom LMHom
4515, 1lmhmf 16102 . . . . 5 LMHom
46 ffn 5583 . . . . 5
4745, 46syl 16 . . . 4 LMHom
4847adantr 452 . . 3 LMHom LMHom
491, 15lmhmf 16102 . . . . 5 LMHom
5049adantl 453 . . . 4 LMHom LMHom
51 ffn 5583 . . . 4
5250, 51syl 16 . . 3 LMHom LMHom
53 dff1o4 5674 . . 3
5448, 52, 53sylanbrc 646 . 2 LMHom LMHom
5544, 54impbida 806 1 LMHom LMHom
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  ccnv 4869   wfn 5441  wf 5442  wf1o 5445  cfv 5446  (class class class)co 6073  cbs 13461  Scalarcsca 13524  cvsca 13525   cghm 14995  clmod 15942   LMHom clmhm 16087 This theorem is referenced by:  islmim2  16130 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-mnd 14682  df-grp 14804  df-ghm 14996  df-lmod 15944  df-lmhm 16090
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