Theorem lmhmplusg 19044

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.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( Base ` M ) = ( Base ` M )
2		eqid 2622	. 2 |- ( .s ` M ) = ( .s ` M )
3		eqid 2622	. 2 |- ( .s ` N ) = ( .s ` N )
4		eqid 2622	. 2 |- (Scalar ` M ) = (Scalar ` M )
5		eqid 2622	. 2 |- (Scalar ` N ) = (Scalar ` N )
6		eqid 2622	. 2 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
7		lmhmlmod1 19033	. . 3 |- ( F e. ( M LMHom N ) -> M e. LMod )
8	7	adantr 481	. 2 |- ( ( F e. ( M LMHom N ) /\ G e. ( M LMHom N ) ) -> M e. LMod )
9		lmhmlmod2 19032	. . 3 |- ( F e. ( M LMHom N ) -> N e. LMod )
10	9	adantr 481	. 2 |- ( ( F e. ( M LMHom N ) /\ G e. ( M LMHom N ) ) -> N e. LMod )
11	4, 5	lmhmsca 19030	. . 3 |- ( F e. ( M LMHom N ) -> (Scalar ` N ) = (Scalar ` M ) )
12	11	adantr 481	. 2 |- ( ( F e. ( M LMHom N ) /\ G e. ( M LMHom N ) ) -> (Scalar ` N ) = (Scalar ` M ) )
13		lmodabl 18910	. . . 4 |- ( N e. LMod -> N e. Abel )
14	10, 13	syl 17	. . 3 |- ( ( F e. ( M LMHom N ) /\ G e. ( M LMHom N ) ) -> N e. Abel )
15		lmghm 19031	. . . 4 |- ( F e. ( M LMHom N ) -> F e. ( M GrpHom N ) )
16	15	adantr 481	. . 3 |- ( ( F e. ( M LMHom N ) /\ G e. ( M LMHom N ) ) -> F e. ( M GrpHom N ) )
17		lmghm 19031	. . . 4 |- ( G e. ( M LMHom N ) -> G e. ( M GrpHom N ) )
18	17	adantl 482	. . 3 |- ( ( F e. ( M LMHom N ) /\ G e. ( M LMHom N ) ) -> G e. ( M GrpHom N ) )
19		lmhmplusg.p	. . . 4 |- .+ = ( +g ` N )
20	19	ghmplusg 18249	. . 3 |- ( ( N e. Abel /\ F e. ( M GrpHom N ) /\ G e. ( M GrpHom N ) ) -> ( F oF .+ G ) e. ( M GrpHom N ) )
21	14, 16, 18, 20	syl3anc 1326	. 2 |- ( ( F e. ( M LMHom N ) /\ G e. ( M LMHom N ) ) -> ( F oF .+ G ) e. ( M GrpHom N ) )
22		simpll 790	. . . . . 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 794	. . . . . 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 796	. . . . . 6 |- ( ( ( F e. ( M LMHom N ) /\ G e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. ( Base ` M ) ) ) -> y e. ( Base ` M ) )
25	4, 6, 1, 2, 3	lmhmlin 19035	. . . . . 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 ) ) )
26	22, 23, 24, 25	syl3anc 1326	. . . . 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 792	. . . . . 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 ) )
28	4, 6, 1, 2, 3	lmhmlin 19035	. . . . . 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 ) ) )
29	27, 23, 24, 28	syl3anc 1326	. . . . 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 ) ) )
30	26, 29	oveq12d 6668	. . . 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 ) ) ) )
31	9	ad2antrr 762	. . . . 5 |- ( ( ( F e. ( M LMHom N ) /\ G e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. ( Base ` M ) ) ) -> N e. LMod )
32	11	fveq2d 6195	. . . . . . 7 |- ( F e. ( M LMHom N ) -> ( Base ` (Scalar ` N ) ) = ( Base ` (Scalar ` M ) ) )
33	32	ad2antrr 762	. . . . . 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 ) ) )
34	23, 33	eleqtrrd 2704	. . . . 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 2622	. . . . . . . 8 |- ( Base ` N ) = ( Base ` N )
36	1, 35	lmhmf 19034	. . . . . . 7 |- ( F e. ( M LMHom N ) -> F : ( Base ` M ) --> ( Base ` N ) )
37	36	ad2antrr 762	. . . . . 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 ) )
38	37, 24	ffvelrnd 6360	. . . . 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 ) )
39	1, 35	lmhmf 19034	. . . . . . 7 |- ( G e. ( M LMHom N ) -> G : ( Base ` M ) --> ( Base ` N ) )
40	39	ad2antlr 763	. . . . . 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 ) )
41	40, 24	ffvelrnd 6360	. . . . 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 2622	. . . . . 6 |- ( Base ` (Scalar ` N ) ) = ( Base ` (Scalar ` N ) )
43	35, 19, 5, 3, 42	lmodvsdi 18886	. . . . 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 ) ) ) )
44	31, 34, 38, 41, 43	syl13anc 1328	. . . 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 ) ) ) )
45	30, 44	eqtr4d 2659	. . 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 6045	. . . . 5 |- ( F : ( Base ` M ) --> ( Base ` N ) -> F Fn ( Base ` M ) )
47	37, 46	syl 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 6045	. . . . 5 |- ( G : ( Base ` M ) --> ( Base ` N ) -> G Fn ( Base ` M ) )
49	40, 48	syl 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		fvexd 6203	. . . 4 |- ( ( ( F e. ( M LMHom N ) /\ G e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. ( Base ` M ) ) ) -> ( Base ` M ) e. _V )
51	7	ad2antrr 762	. . . . 5 |- ( ( ( F e. ( M LMHom N ) /\ G e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. ( Base ` M ) ) ) -> M e. LMod )
52	1, 4, 2, 6	lmodvscl 18880	. . . . 5 |- ( ( M e. LMod /\ x e. ( Base ` (Scalar ` M ) ) /\ y e. ( Base ` M ) ) -> ( x ( .s ` M ) y ) e. ( Base ` M ) )
53	51, 23, 24, 52	syl3anc 1326	. . . 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 ) )
54		fnfvof 6911	. . . 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 ) ) ) )
55	47, 49, 50, 53, 54	syl22anc 1327	. . 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 ) ) ) )
56		fnfvof 6911	. . . . 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 ) ) )
57	47, 49, 50, 24, 56	syl22anc 1327	. . . 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 ) ) )
58	57	oveq2d 6666	. . 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 ) ) ) )
59	45, 55, 58	3eqtr4d 2666	. 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 ) ) )
60	1, 2, 3, 4, 5, 6, 8, 10, 12, 21, 59	islmhmd 19039	1 |- ( ( F e. ( M LMHom N ) /\ G e. ( M LMHom N ) ) -> ( F oF .+ G ) e. ( M LMHom N ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   GrpHom cghm 17657   Abelcabl 18194   LModclmod 18863   LMHom clmhm 19019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-ghm 17658  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lmhm 19022

This theorem is referenced by:  nmhmplusg  22561  mendring  37762