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Theorem lmhmco 17123
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 2442 . 2  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2442 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2442 . 2  |-  ( .s
`  O )  =  ( .s `  O
)
4 eqid 2442 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 eqid 2442 . 2  |-  (Scalar `  O )  =  (Scalar `  O )
6 eqid 2442 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 17113 . . 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 17112 . . 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 2442 . . . 4  |-  (Scalar `  N )  =  (Scalar `  N )
1211, 5lmhmsca 17110 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  (Scalar `  O
)  =  (Scalar `  N ) )
134, 11lmhmsca 17110 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  (Scalar `  N
)  =  (Scalar `  M ) )
1412, 13sylan9eq 2494 . 2  |-  ( ( F  e.  ( N LMHom 
O )  /\  G  e.  ( M LMHom  N ) )  ->  (Scalar `  O
)  =  (Scalar `  M ) )
15 lmghm 17111 . . 3  |-  ( F  e.  ( N LMHom  O
)  ->  F  e.  ( N  GrpHom  O ) )
16 lmghm 17111 . . 3  |-  ( G  e.  ( M LMHom  N
)  ->  G  e.  ( M  GrpHom  N ) )
17 ghmco 15765 . . 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 754 . . . . . 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 755 . . . . . 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 756 . . . . . 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 2442 . . . . . . 7  |-  ( .s
`  N )  =  ( .s `  N
)
234, 6, 1, 2, 22lmhmlin 17115 . . . . . 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 1218 . . . . 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 5694 . . . 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 5694 . . . . . . 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 2519 . . . . 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 2442 . . . . . . . . 9  |-  ( Base `  N )  =  (
Base `  N )
311, 30lmhmf 17114 . . . . . . . 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 5842 . . . . . 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 2442 . . . . . 6  |-  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  N )
)
3611, 35, 30, 22, 3lmhmlin 17115 . . . . 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 1218 . . . 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 2474 . . 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 5558 . . . . . 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 16964 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( x ( .s
`  M ) y )  e.  ( Base `  M ) )
4442, 20, 21, 43syl3anc 1218 . . . 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 5765 . . . 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 5765 . . . . 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 6106 . . 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 2484 . 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 17119 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 1369    e. wcel 1756    o. ccom 4843    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6090   Basecbs 14173  Scalarcsca 14240   .scvsca 14241    GrpHom cghm 15743   LModclmod 16947   LMHom clmhm 17099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7215  df-0g 14379  df-mnd 15414  df-mhm 15463  df-grp 15544  df-ghm 15744  df-lmod 16949  df-lmhm 17102
This theorem is referenced by:  lmimco  18272  nmhmco  20334  mendrng  29547
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