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Theorem lmhmco 15800
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 2283 . 2  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2283 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2283 . 2  |-  ( .s
`  O )  =  ( .s `  O
)
4 eqid 2283 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 eqid 2283 . 2  |-  (Scalar `  O )  =  (Scalar `  O )
6 eqid 2283 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 15790 . . 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 15789 . . 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 2283 . . . 4  |-  (Scalar `  N )  =  (Scalar `  N )
1211, 5lmhmsca 15787 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  (Scalar `  O
)  =  (Scalar `  N ) )
134, 11lmhmsca 15787 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  (Scalar `  N
)  =  (Scalar `  M ) )
1412, 13sylan9eq 2335 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  (Scalar `  O
)  =  (Scalar `  M ) )
15 lmghm 15788 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  F  e.  ( N  GrpHom  O ) )
16 lmghm 15788 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  G  e.  ( M  GrpHom  N ) )
17 ghmco 14702 . . 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 2283 . . . . . . 7  |-  ( .s
`  N )  =  ( .s `  N
)
234, 6, 1, 2, 22lmhmlin 15792 . . . . . 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 5529 . . . 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 5529 . . . . . . 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 2360 . . . . 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 2283 . . . . . . . . 9  |-  ( Base `  N )  =  (
Base `  N )
311, 30lmhmf 15791 . . . . . . . 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 5663 . . . . . . 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 2283 . . . . . 6  |-  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  N )
)
3711, 36, 30, 22, 3lmhmlin 15792 . . . . 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 2315 . . 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 5389 . . . . . 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 15644 . . . . 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 5594 . . . 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 5594 . . . . 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 5874 . . 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 2325 . 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 15796 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 1623    e. wcel 1684    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212    GrpHom cghm 14680   LModclmod 15627   LMHom clmhm 15776
This theorem is referenced by:  nmhmco  18265  mendrng  27500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-map 6774  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-ghm 14681  df-lmod 15629  df-lmhm 15779
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