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Theorem lmhmplusg 18202
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 2429 . 2  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2429 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2429 . 2  |-  ( .s
`  N )  =  ( .s `  N
)
4 eqid 2429 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 eqid 2429 . 2  |-  (Scalar `  N )  =  (Scalar `  N )
6 eqid 2429 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 18191 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  M  e.  LMod )
87adantr 466 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  M  e.  LMod )
9 lmhmlmod2 18190 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  N  e.  LMod )
109adantr 466 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  N  e.  LMod )
114, 5lmhmsca 18188 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  (Scalar `  N
)  =  (Scalar `  M ) )
1211adantr 466 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  (Scalar `  N
)  =  (Scalar `  M ) )
13 lmodabl 18070 . . . 4  |-  ( N  e.  LMod  ->  N  e. 
Abel )
1410, 13syl 17 . . 3  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  N  e.  Abel )
15 lmghm 18189 . . . 4  |-  ( F  e.  ( M LMHom  N
)  ->  F  e.  ( M  GrpHom  N ) )
1615adantr 466 . . 3  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  F  e.  ( M  GrpHom  N ) )
17 lmghm 18189 . . . 4  |-  ( G  e.  ( M LMHom  N
)  ->  G  e.  ( M  GrpHom  N ) )
1817adantl 467 . . 3  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  G  e.  ( M  GrpHom  N ) )
19 lmhmplusg.p . . . 4  |-  .+  =  ( +g  `  N )
2019ghmplusg 17419 . . 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 1264 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  oF  .+  G )  e.  ( M  GrpHom  N ) )
22 simpll 758 . . . . . 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 762 . . . . . 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 764 . . . . . 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 18193 . . . . . 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 1264 . . . . 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 760 . . . . . 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 18193 . . . . . 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 1264 . . . . 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 6323 . . . 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 730 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  N  e.  LMod )
3211fveq2d 5885 . . . . . . 7  |-  ( F  e.  ( M LMHom  N
)  ->  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  M )
) )
3332ad2antrr 730 . . . . . 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 2520 . . . . 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 2429 . . . . . . . 8  |-  ( Base `  N )  =  (
Base `  N )
361, 35lmhmf 18192 . . . . . . 7  |-  ( F  e.  ( M LMHom  N
)  ->  F :
( Base `  M ) --> ( Base `  N )
)
3736ad2antrr 730 . . . . . 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 6038 . . . . 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 18192 . . . . . . 7  |-  ( G  e.  ( M LMHom  N
)  ->  G :
( Base `  M ) --> ( Base `  N )
)
4039ad2antlr 731 . . . . . 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 6038 . . . . 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 2429 . . . . . 6  |-  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  N )
)
4335, 19, 5, 3, 42lmodvsdi 18049 . . . . 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 1266 . . . 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 2473 . . 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 5746 . . . . 5  |-  ( F : ( Base `  M
) --> ( Base `  N
)  ->  F  Fn  ( Base `  M )
)
4737, 46syl 17 . . . 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 5746 . . . . 5  |-  ( G : ( Base `  M
) --> ( Base `  N
)  ->  G  Fn  ( Base `  M )
)
4940, 48syl 17 . . . 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 5891 . . . . 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 730 . . . . 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 18043 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( x ( .s
`  M ) y )  e.  ( Base `  M ) )
5452, 23, 24, 53syl3anc 1264 . . . 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 6559 . . . 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 1265 . . 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 6559 . . . . 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 1265 . . . 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 6321 . . 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 2480 . 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 18197 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 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    oFcof 6543   Basecbs 15084   +g cplusg 15152  Scalarcsca 15155   .scvsca 15156    GrpHom cghm 16831   Abelcabl 17366   LModclmod 18026   LMHom clmhm 18177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-plusg 15165  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-ghm 16832  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-lmod 18028  df-lmhm 18180
This theorem is referenced by:  nmhmplusg  21689  mendring  35757
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