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Theorem lmhmco 15800
 Description: The composition of two module-linear functions is module-linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Assertion
Ref Expression
lmhmco LMHom LMHom LMHom

Proof of Theorem lmhmco
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . 2
2 eqid 2283 . 2
3 eqid 2283 . 2
4 eqid 2283 . 2 Scalar Scalar
5 eqid 2283 . 2 Scalar Scalar
6 eqid 2283 . 2 Scalar Scalar
7 lmhmlmod1 15790 . . 3 LMHom
87adantl 452 . 2 LMHom LMHom
9 lmhmlmod2 15789 . . 3 LMHom
109adantr 451 . 2 LMHom LMHom
11 eqid 2283 . . . 4 Scalar Scalar
1211, 5lmhmsca 15787 . . 3 LMHom Scalar Scalar
134, 11lmhmsca 15787 . . 3 LMHom Scalar Scalar
1412, 13sylan9eq 2335 . 2 LMHom LMHom Scalar Scalar
15 lmghm 15788 . . 3 LMHom
16 lmghm 15788 . . 3 LMHom
17 ghmco 14702 . . 3
1815, 16, 17syl2an 463 . 2 LMHom LMHom
19 simplr 731 . . . . . 6 LMHom LMHom Scalar LMHom
20 simprl 732 . . . . . 6 LMHom LMHom Scalar Scalar
21 simprr 733 . . . . . 6 LMHom LMHom Scalar
22 eqid 2283 . . . . . . 7
234, 6, 1, 2, 22lmhmlin 15792 . . . . . 6 LMHom Scalar
2419, 20, 21, 23syl3anc 1182 . . . . 5 LMHom LMHom Scalar
2524fveq2d 5529 . . . 4 LMHom LMHom Scalar
26 simpll 730 . . . . 5 LMHom LMHom Scalar LMHom
2713fveq2d 5529 . . . . . . 7 LMHom Scalar Scalar
2827ad2antlr 707 . . . . . 6 LMHom LMHom Scalar Scalar Scalar
2920, 28eleqtrrd 2360 . . . . 5 LMHom LMHom Scalar Scalar
30 eqid 2283 . . . . . . . . 9
311, 30lmhmf 15791 . . . . . . . 8 LMHom
3231adantl 452 . . . . . . 7 LMHom LMHom
33 ffvelrn 5663 . . . . . . 7
3432, 33sylan 457 . . . . . 6 LMHom LMHom
3534adantrl 696 . . . . 5 LMHom LMHom Scalar
36 eqid 2283 . . . . . 6 Scalar Scalar
3711, 36, 30, 22, 3lmhmlin 15792 . . . . 5 LMHom Scalar
3826, 29, 35, 37syl3anc 1182 . . . 4 LMHom LMHom Scalar
3925, 38eqtrd 2315 . . 3 LMHom LMHom Scalar
40 ffn 5389 . . . . . 6
4132, 40syl 15 . . . . 5 LMHom LMHom
4241adantr 451 . . . 4 LMHom LMHom Scalar
437ad2antlr 707 . . . . 5 LMHom LMHom Scalar
441, 4, 2, 6lmodvscl 15644 . . . . 5 Scalar
4543, 20, 21, 44syl3anc 1182 . . . 4 LMHom LMHom Scalar
46 fvco2 5594 . . . 4
4742, 45, 46syl2anc 642 . . 3 LMHom LMHom Scalar
48 fvco2 5594 . . . . 5
4942, 21, 48syl2anc 642 . . . 4 LMHom LMHom Scalar
5049oveq2d 5874 . . 3 LMHom LMHom Scalar
5139, 47, 503eqtr4d 2325 . 2 LMHom LMHom Scalar
521, 2, 3, 4, 5, 6, 8, 10, 14, 18, 51islmhmd 15796 1 LMHom LMHom LMHom
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684   ccom 4693   wfn 5250  wf 5251  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:  nmhmco  18265  mendrng  27500 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-rmo 2551  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-riota 6304  df-map 6774  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-ghm 14681  df-lmod 15629  df-lmhm 15779
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