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Theorem lmhmplusg 17124
Description: The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypothesis
Ref Expression
lmhmplusg.p  |-  .+  =  ( +g  `  N )
Assertion
Ref Expression
lmhmplusg  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  oF  .+  G )  e.  ( M LMHom  N
) )

Proof of Theorem lmhmplusg
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
`  N )  =  ( .s `  N
)
4 eqid 2442 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 eqid 2442 . 2  |-  (Scalar `  N )  =  (Scalar `  N )
6 eqid 2442 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 17113 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  M  e.  LMod )
87adantr 465 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  M  e.  LMod )
9 lmhmlmod2 17112 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  N  e.  LMod )
109adantr 465 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  N  e.  LMod )
114, 5lmhmsca 17110 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  (Scalar `  N
)  =  (Scalar `  M ) )
1211adantr 465 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  (Scalar `  N
)  =  (Scalar `  M ) )
13 lmodabl 16991 . . . 4  |-  ( N  e.  LMod  ->  N  e. 
Abel )
1410, 13syl 16 . . 3  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  N  e.  Abel )
15 lmghm 17111 . . . 4  |-  ( F  e.  ( M LMHom  N
)  ->  F  e.  ( M  GrpHom  N ) )
1615adantr 465 . . 3  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  F  e.  ( M  GrpHom  N ) )
17 lmghm 17111 . . . 4  |-  ( G  e.  ( M LMHom  N
)  ->  G  e.  ( M  GrpHom  N ) )
1817adantl 466 . . 3  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  G  e.  ( M  GrpHom  N ) )
19 lmhmplusg.p . . . 4  |-  .+  =  ( +g  `  N )
2019ghmplusg 16327 . . 3  |-  ( ( N  e.  Abel  /\  F  e.  ( M  GrpHom  N )  /\  G  e.  ( M  GrpHom  N ) )  ->  ( F  oF  .+  G )  e.  ( M  GrpHom  N ) )
2114, 16, 18, 20syl3anc 1218 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  oF  .+  G )  e.  ( M  GrpHom  N ) )
22 simpll 753 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  F  e.  ( M LMHom  N ) )
23 simprl 755 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  x  e.  ( Base `  (Scalar `  M ) ) )
24 simprr 756 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  y  e.  ( Base `  M
) )
254, 6, 1, 2, 3lmhmlin 17115 . . . . . 6  |-  ( ( F  e.  ( M LMHom 
N )  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( F `  (
x ( .s `  M ) y ) )  =  ( x ( .s `  N
) ( F `  y ) ) )
2622, 23, 24, 25syl3anc 1218 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( F `  ( x
( .s `  M
) y ) )  =  ( x ( .s `  N ) ( F `  y
) ) )
27 simplr 754 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  G  e.  ( M LMHom  N ) )
284, 6, 1, 2, 3lmhmlin 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 ) ) )
2927, 23, 24, 28syl3anc 1218 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( G `  ( x
( .s `  M
) y ) )  =  ( x ( .s `  N ) ( G `  y
) ) )
3026, 29oveq12d 6108 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F `  (
x ( .s `  M ) y ) )  .+  ( G `
 ( x ( .s `  M ) y ) ) )  =  ( ( x ( .s `  N
) ( F `  y ) )  .+  ( x ( .s
`  N ) ( G `  y ) ) ) )
319ad2antrr 725 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  N  e.  LMod )
3211fveq2d 5694 . . . . . . 7  |-  ( F  e.  ( M LMHom  N
)  ->  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  M )
) )
3332ad2antrr 725 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( Base `  (Scalar `  N
) )  =  (
Base `  (Scalar `  M
) ) )
3423, 33eleqtrrd 2519 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  x  e.  ( Base `  (Scalar `  N ) ) )
35 eqid 2442 . . . . . . . 8  |-  ( Base `  N )  =  (
Base `  N )
361, 35lmhmf 17114 . . . . . . 7  |-  ( F  e.  ( M LMHom  N
)  ->  F :
( Base `  M ) --> ( Base `  N )
)
3736ad2antrr 725 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  F : ( Base `  M
) --> ( Base `  N
) )
3837, 24ffvelrnd 5843 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( F `  y )  e.  ( Base `  N
) )
391, 35lmhmf 17114 . . . . . . 7  |-  ( G  e.  ( M LMHom  N
)  ->  G :
( Base `  M ) --> ( Base `  N )
)
4039ad2antlr 726 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  G : ( Base `  M
) --> ( Base `  N
) )
4140, 24ffvelrnd 5843 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( G `  y )  e.  ( Base `  N
) )
42 eqid 2442 . . . . . 6  |-  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  N )
)
4335, 19, 5, 3, 42lmodvsdi 16970 . . . . 5  |-  ( ( N  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  N )
)  /\  ( F `  y )  e.  (
Base `  N )  /\  ( G `  y
)  e.  ( Base `  N ) ) )  ->  ( x ( .s `  N ) ( ( F `  y )  .+  ( G `  y )
) )  =  ( ( x ( .s
`  N ) ( F `  y ) )  .+  ( x ( .s `  N
) ( G `  y ) ) ) )
4431, 34, 38, 41, 43syl13anc 1220 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
x ( .s `  N ) ( ( F `  y ) 
.+  ( G `  y ) ) )  =  ( ( x ( .s `  N
) ( F `  y ) )  .+  ( x ( .s
`  N ) ( G `  y ) ) ) )
4530, 44eqtr4d 2477 . . 3  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F `  (
x ( .s `  M ) y ) )  .+  ( G `
 ( x ( .s `  M ) y ) ) )  =  ( x ( .s `  N ) ( ( F `  y )  .+  ( G `  y )
) ) )
46 ffn 5558 . . . . 5  |-  ( F : ( Base `  M
) --> ( Base `  N
)  ->  F  Fn  ( Base `  M )
)
4737, 46syl 16 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  F  Fn  ( Base `  M
) )
48 ffn 5558 . . . . 5  |-  ( G : ( Base `  M
) --> ( Base `  N
)  ->  G  Fn  ( Base `  M )
)
4940, 48syl 16 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  G  Fn  ( Base `  M
) )
50 fvex 5700 . . . . 5  |-  ( Base `  M )  e.  _V
5150a1i 11 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( Base `  M )  e. 
_V )
527ad2antrr 725 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  M  e.  LMod )
531, 4, 2, 6lmodvscl 16964 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( x ( .s
`  M ) y )  e.  ( Base `  M ) )
5452, 23, 24, 53syl3anc 1218 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
x ( .s `  M ) y )  e.  ( Base `  M
) )
55 fnfvof 6332 . . . 4  |-  ( ( ( F  Fn  ( Base `  M )  /\  G  Fn  ( Base `  M ) )  /\  ( ( Base `  M
)  e.  _V  /\  ( x ( .s
`  M ) y )  e.  ( Base `  M ) ) )  ->  ( ( F  oF  .+  G
) `  ( x
( .s `  M
) y ) )  =  ( ( F `
 ( x ( .s `  M ) y ) )  .+  ( G `  ( x ( .s `  M
) y ) ) ) )
5647, 49, 51, 54, 55syl22anc 1219 . . 3  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F  oF  .+  G ) `  ( x ( .s
`  M ) y ) )  =  ( ( F `  (
x ( .s `  M ) y ) )  .+  ( G `
 ( x ( .s `  M ) y ) ) ) )
57 fnfvof 6332 . . . . 5  |-  ( ( ( F  Fn  ( Base `  M )  /\  G  Fn  ( Base `  M ) )  /\  ( ( Base `  M
)  e.  _V  /\  y  e.  ( Base `  M ) ) )  ->  ( ( F  oF  .+  G
) `  y )  =  ( ( F `
 y )  .+  ( G `  y ) ) )
5847, 49, 51, 24, 57syl22anc 1219 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F  oF  .+  G ) `  y )  =  ( ( F `  y
)  .+  ( G `  y ) ) )
5958oveq2d 6106 . . 3  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
x ( .s `  N ) ( ( F  oF  .+  G ) `  y
) )  =  ( x ( .s `  N ) ( ( F `  y ) 
.+  ( G `  y ) ) ) )
6045, 56, 593eqtr4d 2484 . 2  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F  oF  .+  G ) `  ( x ( .s
`  M ) y ) )  =  ( x ( .s `  N ) ( ( F  oF  .+  G ) `  y
) ) )
611, 2, 3, 4, 5, 6, 8, 10, 12, 21, 60islmhmd 17119 1  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  oF  .+  G )  e.  ( M LMHom  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2971    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6090    oFcof 6317   Basecbs 14173   +g cplusg 14237  Scalarcsca 14240   .scvsca 14241    GrpHom cghm 15743   Abelcabel 16277   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  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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-nel 2608  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-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  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-of 6319  df-om 6476  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-plusg 14250  df-0g 14379  df-mnd 15414  df-grp 15544  df-minusg 15545  df-ghm 15744  df-cmn 16278  df-abl 16279  df-mgp 16591  df-ur 16603  df-rng 16646  df-lmod 16949  df-lmhm 17102
This theorem is referenced by:  nmhmplusg  20335  mendrng  29547
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