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Theorem lmhmco 15816
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 2296 . 2  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2296 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2296 . 2  |-  ( .s
`  O )  =  ( .s `  O
)
4 eqid 2296 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 eqid 2296 . 2  |-  (Scalar `  O )  =  (Scalar `  O )
6 eqid 2296 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 15806 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  M  e.  LMod )
87adantl 452 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  M  e.  LMod )
9 lmhmlmod2 15805 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  O  e.  LMod )
109adantr 451 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  O  e.  LMod )
11 eqid 2296 . . . 4  |-  (Scalar `  N )  =  (Scalar `  N )
1211, 5lmhmsca 15803 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  (Scalar `  O
)  =  (Scalar `  N ) )
134, 11lmhmsca 15803 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  (Scalar `  N
)  =  (Scalar `  M ) )
1412, 13sylan9eq 2348 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  (Scalar `  O
)  =  (Scalar `  M ) )
15 lmghm 15804 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  F  e.  ( N  GrpHom  O ) )
16 lmghm 15804 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  G  e.  ( M  GrpHom  N ) )
17 ghmco 14718 . . 3  |-  ( ( F  e.  ( N 
GrpHom  O )  /\  G  e.  ( M  GrpHom  N ) )  ->  ( F  o.  G )  e.  ( M  GrpHom  O ) )
1815, 16, 17syl2an 463 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o.  G )  e.  ( M  GrpHom  O ) )
19 simplr 731 . . . . . 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 732 . . . . . 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 733 . . . . . 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 2296 . . . . . . 7  |-  ( .s
`  N )  =  ( .s `  N
)
234, 6, 1, 2, 22lmhmlin 15808 . . . . . 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 1182 . . . . 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 5545 . . . 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 730 . . . . 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 5545 . . . . . . 7  |-  ( G  e.  ( M LMHom  N
)  ->  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  M )
) )
2827ad2antlr 707 . . . . . 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 2373 . . . . 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 2296 . . . . . . . . 9  |-  ( Base `  N )  =  (
Base `  N )
311, 30lmhmf 15807 . . . . . . . 8  |-  ( G  e.  ( M LMHom  N
)  ->  G :
( Base `  M ) --> ( Base `  N )
)
3231adantl 452 . . . . . . 7  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  G :
( Base `  M ) --> ( Base `  N )
)
33 ffvelrn 5679 . . . . . . 7  |-  ( ( G : ( Base `  M ) --> ( Base `  N )  /\  y  e.  ( Base `  M
) )  ->  ( G `  y )  e.  ( Base `  N
) )
3432, 33sylan 457 . . . . . 6  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  y  e.  ( Base `  M
) )  ->  ( G `  y )  e.  ( Base `  N
) )
3534adantrl 696 . . . . 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
) )
36 eqid 2296 . . . . . 6  |-  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  N )
)
3711, 36, 30, 22, 3lmhmlin 15808 . . . . 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 )
) ) )
3826, 29, 35, 37syl3anc 1182 . . . 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 )
) ) )
3925, 38eqtrd 2328 . . 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 ) ) ) )
40 ffn 5405 . . . . . 6  |-  ( G : ( Base `  M
) --> ( Base `  N
)  ->  G  Fn  ( Base `  M )
)
4132, 40syl 15 . . . . 5  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  G  Fn  ( Base `  M )
)
4241adantr 451 . . . 4  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  G  Fn  ( Base `  M
) )
437ad2antlr 707 . . . . 5  |-  ( ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  M  e.  LMod )
441, 4, 2, 6lmodvscl 15660 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( x ( .s
`  M ) y )  e.  ( Base `  M ) )
4543, 20, 21, 44syl3anc 1182 . . . 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
) )
46 fvco2 5610 . . . 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 ) ) ) )
4742, 45, 46syl2anc 642 . . 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 ) ) ) )
48 fvco2 5610 . . . . 5  |-  ( ( G  Fn  ( Base `  M )  /\  y  e.  ( Base `  M
) )  ->  (
( F  o.  G
) `  y )  =  ( F `  ( G `  y ) ) )
4942, 21, 48syl2anc 642 . . . 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 ) ) )
5049oveq2d 5890 . . 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 )
) ) )
5139, 47, 503eqtr4d 2338 . 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
) ) )
521, 2, 3, 4, 5, 6, 8, 10, 14, 18, 51islmhmd 15812 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 358    = wceq 1632    e. wcel 1696    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228    GrpHom cghm 14696   LModclmod 15643   LMHom clmhm 15792
This theorem is referenced by:  nmhmco  18281  mendrng  27603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-map 6790  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-ghm 14697  df-lmod 15645  df-lmhm 15795
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