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Theorem lmhmplusg 17490
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 2467 . 2  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2467 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2467 . 2  |-  ( .s
`  N )  =  ( .s `  N
)
4 eqid 2467 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 eqid 2467 . 2  |-  (Scalar `  N )  =  (Scalar `  N )
6 eqid 2467 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 17479 . . 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 17478 . . 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 17476 . . 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 17357 . . . 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 17477 . . . 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 17477 . . . 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 16655 . . 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 1228 . 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 17481 . . . . . 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 1228 . . . . 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 17481 . . . . . 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 1228 . . . . 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 6302 . . . 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 5870 . . . . . . 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 2558 . . . . 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 2467 . . . . . . . 8  |-  ( Base `  N )  =  (
Base `  N )
361, 35lmhmf 17480 . . . . . . 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 6022 . . . . 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 17480 . . . . . . 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 6022 . . . . 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 2467 . . . . . 6  |-  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  N )
)
4335, 19, 5, 3, 42lmodvsdi 17335 . . . . 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 1230 . . . 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 2511 . . 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 5731 . . . . 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 5731 . . . . 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 5876 . . . . 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 17329 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( x ( .s
`  M ) y )  e.  ( Base `  M ) )
5452, 23, 24, 53syl3anc 1228 . . . 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 6537 . . . 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 1229 . . 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 6537 . . . . 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 1229 . . . 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 6300 . . 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 2518 . 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 17485 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 1379    e. wcel 1767   _Vcvv 3113    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284    oFcof 6522   Basecbs 14490   +g cplusg 14555  Scalarcsca 14558   .scvsca 14559    GrpHom cghm 16069   Abelcabl 16605   LModclmod 17312   LMHom clmhm 17465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-plusg 14568  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-ghm 16070  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-lmod 17314  df-lmhm 17468
This theorem is referenced by:  nmhmplusg  21027  mendrng  30774
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