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Theorem lmhmco 17815
 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 2457 . 2
2 eqid 2457 . 2
3 eqid 2457 . 2
4 eqid 2457 . 2 Scalar Scalar
5 eqid 2457 . 2 Scalar Scalar
6 eqid 2457 . 2 Scalar Scalar
7 lmhmlmod1 17805 . . 3 LMHom
87adantl 466 . 2 LMHom LMHom
9 lmhmlmod2 17804 . . 3 LMHom
109adantr 465 . 2 LMHom LMHom
11 eqid 2457 . . . 4 Scalar Scalar
1211, 5lmhmsca 17802 . . 3 LMHom Scalar Scalar
134, 11lmhmsca 17802 . . 3 LMHom Scalar Scalar
1412, 13sylan9eq 2518 . 2 LMHom LMHom Scalar Scalar
15 lmghm 17803 . . 3 LMHom
16 lmghm 17803 . . 3 LMHom
17 ghmco 16412 . . 3
1815, 16, 17syl2an 477 . 2 LMHom LMHom
19 simplr 755 . . . . . 6 LMHom LMHom Scalar LMHom
20 simprl 756 . . . . . 6 LMHom LMHom Scalar Scalar
21 simprr 757 . . . . . 6 LMHom LMHom Scalar
22 eqid 2457 . . . . . . 7
234, 6, 1, 2, 22lmhmlin 17807 . . . . . 6 LMHom Scalar
2419, 20, 21, 23syl3anc 1228 . . . . 5 LMHom LMHom Scalar
2524fveq2d 5876 . . . 4 LMHom LMHom Scalar
26 simpll 753 . . . . 5 LMHom LMHom Scalar LMHom
2713fveq2d 5876 . . . . . . 7 LMHom Scalar Scalar
2827ad2antlr 726 . . . . . 6 LMHom LMHom Scalar Scalar Scalar
2920, 28eleqtrrd 2548 . . . . 5 LMHom LMHom Scalar Scalar
30 eqid 2457 . . . . . . . . 9
311, 30lmhmf 17806 . . . . . . . 8 LMHom
3231adantl 466 . . . . . . 7 LMHom LMHom
3332ffvelrnda 6032 . . . . . 6 LMHom LMHom
3433adantrl 715 . . . . 5 LMHom LMHom Scalar
35 eqid 2457 . . . . . 6 Scalar Scalar
3611, 35, 30, 22, 3lmhmlin 17807 . . . . 5 LMHom Scalar
3726, 29, 34, 36syl3anc 1228 . . . 4 LMHom LMHom Scalar
3825, 37eqtrd 2498 . . 3 LMHom LMHom Scalar
39 ffn 5737 . . . . . 6
4032, 39syl 16 . . . . 5 LMHom LMHom
4140adantr 465 . . . 4 LMHom LMHom Scalar
427ad2antlr 726 . . . . 5 LMHom LMHom Scalar
431, 4, 2, 6lmodvscl 17655 . . . . 5 Scalar
4442, 20, 21, 43syl3anc 1228 . . . 4 LMHom LMHom Scalar
45 fvco2 5948 . . . 4
4641, 44, 45syl2anc 661 . . 3 LMHom LMHom Scalar
47 fvco2 5948 . . . . 5
4841, 21, 47syl2anc 661 . . . 4 LMHom LMHom Scalar
4948oveq2d 6312 . . 3 LMHom LMHom Scalar
5038, 46, 493eqtr4d 2508 . 2 LMHom LMHom Scalar
511, 2, 3, 4, 5, 6, 8, 10, 14, 18, 50islmhmd 17811 1 LMHom LMHom LMHom
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395   wcel 1819   ccom 5012   wfn 5589  wf 5590  cfv 5594  (class class class)co 6296  cbs 14643  Scalarcsca 14714  cvsca 14715   cghm 16390  clmod 17638   LMHom clmhm 17791 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-grp 16183  df-ghm 16391  df-lmod 17640  df-lmhm 17794 This theorem is referenced by:  lmimco  19005  nmhmco  21388  mendring  31303
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