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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  |-  ( ( 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 2435 . 2  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2435 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2435 . 2  |-  ( .s
`  O )  =  ( .s `  O
)
4 eqid 2435 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 eqid 2435 . 2  |-  (Scalar `  O )  =  (Scalar `  O )
6 eqid 2435 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 16101 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  M  e.  LMod )
87adantl 453 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  M  e.  LMod )
9 lmhmlmod2 16100 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  O  e.  LMod )
109adantr 452 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  O  e.  LMod )
11 eqid 2435 . . . 4  |-  (Scalar `  N )  =  (Scalar `  N )
1211, 5lmhmsca 16098 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  (Scalar `  O
)  =  (Scalar `  N ) )
134, 11lmhmsca 16098 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  (Scalar `  N
)  =  (Scalar `  M ) )
1412, 13sylan9eq 2487 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  (Scalar `  O
)  =  (Scalar `  M ) )
15 lmghm 16099 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  F  e.  ( N  GrpHom  O ) )
16 lmghm 16099 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  G  e.  ( M  GrpHom  N ) )
17 ghmco 15017 . . 3  |-  ( ( F  e.  ( N 
GrpHom  O )  /\  G  e.  ( M  GrpHom  N ) )  ->  ( F  o.  G )  e.  ( M  GrpHom  O ) )
1815, 16, 17syl2an 464 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o.  G )  e.  ( M  GrpHom  O ) )
19 simplr 732 . . . . . 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 733 . . . . . 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 734 . . . . . 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 2435 . . . . . . 7  |-  ( .s
`  N )  =  ( .s `  N
)
234, 6, 1, 2, 22lmhmlin 16103 . . . . . 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 1184 . . . . 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 5724 . . . 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 731 . . . . 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 5724 . . . . . . 7  |-  ( G  e.  ( M LMHom  N
)  ->  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  M )
) )
2827ad2antlr 708 . . . . . 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 2512 . . . . 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 2435 . . . . . . . . 9  |-  ( Base `  N )  =  (
Base `  N )
311, 30lmhmf 16102 . . . . . . . 8  |-  ( G  e.  ( M LMHom  N
)  ->  G :
( Base `  M ) --> ( Base `  N )
)
3231adantl 453 . . . . . . 7  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  G :
( Base `  M ) --> ( Base `  N )
)
3332ffvelrnda 5862 . . . . . 6  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  y  e.  ( Base `  M
) )  ->  ( G `  y )  e.  ( Base `  N
) )
3433adantrl 697 . . . . 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 2435 . . . . . 6  |-  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  N )
)
3611, 35, 30, 22, 3lmhmlin 16103 . . . . 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 1184 . . . 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 2467 . . 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 5583 . . . . . 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 452 . . . 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 708 . . . . 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 15959 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( x ( .s
`  M ) y )  e.  ( Base `  M ) )
4442, 20, 21, 43syl3anc 1184 . . . 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 5790 . . . 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 643 . . 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 5790 . . . . 5  |-  ( ( G  Fn  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
( F  o.  G
) `  y )  =  ( F `  ( G `  y ) ) )
4841, 21, 47syl2anc 643 . . . 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 6089 . . 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 2477 . 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 16107 1  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o.  G )  e.  ( M LMHom  O ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    o. ccom 4874    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .scvsca 13525    GrpHom cghm 14995   LModclmod 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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