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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  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o.  G )  e.  ( M LMHom  O ) )

Proof of Theorem lmhmco
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2457 . 2  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2457 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2457 . 2  |-  ( .s
`  O )  =  ( .s `  O
)
4 eqid 2457 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 eqid 2457 . 2  |-  (Scalar `  O )  =  (Scalar `  O )
6 eqid 2457 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 17805 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  M  e.  LMod )
87adantl 466 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  M  e.  LMod )
9 lmhmlmod2 17804 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  O  e.  LMod )
109adantr 465 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  O  e.  LMod )
11 eqid 2457 . . . 4  |-  (Scalar `  N )  =  (Scalar `  N )
1211, 5lmhmsca 17802 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  (Scalar `  O
)  =  (Scalar `  N ) )
134, 11lmhmsca 17802 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  (Scalar `  N
)  =  (Scalar `  M ) )
1412, 13sylan9eq 2518 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  (Scalar `  O
)  =  (Scalar `  M ) )
15 lmghm 17803 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  F  e.  ( N  GrpHom  O ) )
16 lmghm 17803 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  G  e.  ( M  GrpHom  N ) )
17 ghmco 16412 . . 3  |-  ( ( F  e.  ( N 
GrpHom  O )  /\  G  e.  ( M  GrpHom  N ) )  ->  ( F  o.  G )  e.  ( M  GrpHom  O ) )
1815, 16, 17syl2an 477 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o.  G )  e.  ( M  GrpHom  O ) )
19 simplr 755 . . . . . 6  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  G  e.  ( M LMHom  N ) )
20 simprl 756 . . . . . 6  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  x  e.  ( Base `  (Scalar `  M ) ) )
21 simprr 757 . . . . . 6  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  y  e.  ( Base `  M
) )
22 eqid 2457 . . . . . . 7  |-  ( .s
`  N )  =  ( .s `  N
)
234, 6, 1, 2, 22lmhmlin 17807 . . . . . 6  |-  ( ( G  e.  ( M LMHom 
N )  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( G `  (
x ( .s `  M ) y ) )  =  ( x ( .s `  N
) ( G `  y ) ) )
2419, 20, 21, 23syl3anc 1228 . . . . 5  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( G `  ( x
( .s `  M
) y ) )  =  ( x ( .s `  N ) ( G `  y
) ) )
2524fveq2d 5876 . . . 4  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( F `  ( G `  ( x ( .s
`  M ) y ) ) )  =  ( F `  (
x ( .s `  N ) ( G `
 y ) ) ) )
26 simpll 753 . . . . 5  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  F  e.  ( N LMHom  O ) )
2713fveq2d 5876 . . . . . . 7  |-  ( G  e.  ( M LMHom  N
)  ->  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  M )
) )
2827ad2antlr 726 . . . . . 6  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( Base `  (Scalar `  N
) )  =  (
Base `  (Scalar `  M
) ) )
2920, 28eleqtrrd 2548 . . . . 5  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  x  e.  ( Base `  (Scalar `  N ) ) )
30 eqid 2457 . . . . . . . . 9  |-  ( Base `  N )  =  (
Base `  N )
311, 30lmhmf 17806 . . . . . . . 8  |-  ( G  e.  ( M LMHom  N
)  ->  G :
( Base `  M ) --> ( Base `  N )
)
3231adantl 466 . . . . . . 7  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  G :
( Base `  M ) --> ( Base `  N )
)
3332ffvelrnda 6032 . . . . . 6  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  y  e.  ( Base `  M
) )  ->  ( G `  y )  e.  ( Base `  N
) )
3433adantrl 715 . . . . 5  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( G `  y )  e.  ( Base `  N
) )
35 eqid 2457 . . . . . 6  |-  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  N )
)
3611, 35, 30, 22, 3lmhmlin 17807 . . . . 5  |-  ( ( F  e.  ( N LMHom 
O )  /\  x  e.  ( Base `  (Scalar `  N ) )  /\  ( G `  y )  e.  ( Base `  N
) )  ->  ( F `  ( x
( .s `  N
) ( G `  y ) ) )  =  ( x ( .s `  O ) ( F `  ( G `  y )
) ) )
3726, 29, 34, 36syl3anc 1228 . . . 4  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( F `  ( x
( .s `  N
) ( G `  y ) ) )  =  ( x ( .s `  O ) ( F `  ( G `  y )
) ) )
3825, 37eqtrd 2498 . . 3  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( F `  ( G `  ( x ( .s
`  M ) y ) ) )  =  ( x ( .s
`  O ) ( F `  ( G `
 y ) ) ) )
39 ffn 5737 . . . . . 6  |-  ( G : ( Base `  M
) --> ( Base `  N
)  ->  G  Fn  ( Base `  M )
)
4032, 39syl 16 . . . . 5  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  G  Fn  ( Base `  M )
)
4140adantr 465 . . . 4  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  G  Fn  ( Base `  M
) )
427ad2antlr 726 . . . . 5  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  M  e.  LMod )
431, 4, 2, 6lmodvscl 17655 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( x ( .s
`  M ) y )  e.  ( Base `  M ) )
4442, 20, 21, 43syl3anc 1228 . . . 4  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
x ( .s `  M ) y )  e.  ( Base `  M
) )
45 fvco2 5948 . . . 4  |-  ( ( G  Fn  ( Base `  M )  /\  (
x ( .s `  M ) y )  e.  ( Base `  M
) )  ->  (
( F  o.  G
) `  ( x
( .s `  M
) y ) )  =  ( F `  ( G `  ( x ( .s `  M
) y ) ) ) )
4641, 44, 45syl2anc 661 . . 3  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F  o.  G
) `  ( x
( .s `  M
) y ) )  =  ( F `  ( G `  ( x ( .s `  M
) y ) ) ) )
47 fvco2 5948 . . . . 5  |-  ( ( G  Fn  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
( F  o.  G
) `  y )  =  ( F `  ( G `  y ) ) )
4841, 21, 47syl2anc 661 . . . 4  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F  o.  G
) `  y )  =  ( F `  ( G `  y ) ) )
4948oveq2d 6312 . . 3  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
x ( .s `  O ) ( ( F  o.  G ) `
 y ) )  =  ( x ( .s `  O ) ( F `  ( G `  y )
) ) )
5038, 46, 493eqtr4d 2508 . 2  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F  o.  G
) `  ( x
( .s `  M
) y ) )  =  ( x ( .s `  O ) ( ( F  o.  G ) `  y
) ) )
511, 2, 3, 4, 5, 6, 8, 10, 14, 18, 50islmhmd 17811 1  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o.  G )  e.  ( M LMHom  O ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    o. ccom 5012    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   Basecbs 14643  Scalarcsca 14714   .scvsca 14715    GrpHom cghm 16390   LModclmod 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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