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Theorem lmhmco 16111
 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 2435 . 2
2 eqid 2435 . 2
3 eqid 2435 . 2
4 eqid 2435 . 2 Scalar Scalar
5 eqid 2435 . 2 Scalar Scalar
6 eqid 2435 . 2 Scalar Scalar
7 lmhmlmod1 16101 . . 3 LMHom
87adantl 453 . 2 LMHom LMHom
9 lmhmlmod2 16100 . . 3 LMHom
109adantr 452 . 2 LMHom LMHom
11 eqid 2435 . . . 4 Scalar Scalar
1211, 5lmhmsca 16098 . . 3 LMHom Scalar Scalar
134, 11lmhmsca 16098 . . 3 LMHom Scalar Scalar
1412, 13sylan9eq 2487 . 2 LMHom LMHom Scalar Scalar
15 lmghm 16099 . . 3 LMHom
16 lmghm 16099 . . 3 LMHom
17 ghmco 15017 . . 3
1815, 16, 17syl2an 464 . 2 LMHom LMHom
19 simplr 732 . . . . . 6 LMHom LMHom Scalar LMHom
20 simprl 733 . . . . . 6 LMHom LMHom Scalar Scalar
21 simprr 734 . . . . . 6 LMHom LMHom Scalar
22 eqid 2435 . . . . . . 7
234, 6, 1, 2, 22lmhmlin 16103 . . . . . 6 LMHom Scalar
2419, 20, 21, 23syl3anc 1184 . . . . 5 LMHom LMHom Scalar
2524fveq2d 5724 . . . 4 LMHom LMHom Scalar
26 simpll 731 . . . . 5 LMHom LMHom Scalar LMHom
2713fveq2d 5724 . . . . . . 7 LMHom Scalar Scalar
2827ad2antlr 708 . . . . . 6 LMHom LMHom Scalar Scalar Scalar
2920, 28eleqtrrd 2512 . . . . 5 LMHom LMHom Scalar Scalar
30 eqid 2435 . . . . . . . . 9
311, 30lmhmf 16102 . . . . . . . 8 LMHom
3231adantl 453 . . . . . . 7 LMHom LMHom
3332ffvelrnda 5862 . . . . . 6 LMHom LMHom
3433adantrl 697 . . . . 5 LMHom LMHom Scalar
35 eqid 2435 . . . . . 6 Scalar Scalar
3611, 35, 30, 22, 3lmhmlin 16103 . . . . 5 LMHom Scalar
3726, 29, 34, 36syl3anc 1184 . . . 4 LMHom LMHom Scalar
3825, 37eqtrd 2467 . . 3 LMHom LMHom Scalar
39 ffn 5583 . . . . . 6
4032, 39syl 16 . . . . 5 LMHom LMHom
4140adantr 452 . . . 4 LMHom LMHom Scalar
427ad2antlr 708 . . . . 5 LMHom LMHom Scalar
431, 4, 2, 6lmodvscl 15959 . . . . 5 Scalar
4442, 20, 21, 43syl3anc 1184 . . . 4 LMHom LMHom Scalar
45 fvco2 5790 . . . 4
4641, 44, 45syl2anc 643 . . 3 LMHom LMHom Scalar
47 fvco2 5790 . . . . 5
4841, 21, 47syl2anc 643 . . . 4 LMHom LMHom Scalar
4948oveq2d 6089 . . 3 LMHom LMHom Scalar
5038, 46, 493eqtr4d 2477 . 2 LMHom LMHom Scalar
511, 2, 3, 4, 5, 6, 8, 10, 14, 18, 50islmhmd 16107 1 LMHom LMHom LMHom
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725   ccom 4874   wfn 5441  wf 5442  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:  nmhmco  18782  mendrng  27458 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-rmo 2705  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-riota 6541  df-map 7012  df-0g 13719  df-mnd 14682  df-mhm 14730  df-grp 14804  df-ghm 14996  df-lmod 15944  df-lmhm 16090
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