Theorem lmhmco 19043

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.

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 ` O ) = ( .s ` O )
4		eqid 2622	. 2 |- (Scalar ` M ) = (Scalar ` M )
5		eqid 2622	. 2 |- (Scalar ` O ) = (Scalar ` O )
6		eqid 2622	. 2 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
7		lmhmlmod1 19033	. . 3 |- ( G e. ( M LMHom N ) -> M e. LMod )
8	7	adantl 482	. 2 |- ( ( F e. ( N LMHom O ) /\ G e. ( M LMHom N ) ) -> M e. LMod )
9		lmhmlmod2 19032	. . 3 |- ( F e. ( N LMHom O ) -> O e. LMod )
10	9	adantr 481	. 2 |- ( ( F e. ( N LMHom O ) /\ G e. ( M LMHom N ) ) -> O e. LMod )
11		eqid 2622	. . . 4 |- (Scalar ` N ) = (Scalar ` N )
12	11, 5	lmhmsca 19030	. . 3 |- ( F e. ( N LMHom O ) -> (Scalar ` O ) = (Scalar ` N ) )
13	4, 11	lmhmsca 19030	. . 3 |- ( G e. ( M LMHom N ) -> (Scalar ` N ) = (Scalar ` M ) )
14	12, 13	sylan9eq 2676	. 2 |- ( ( F e. ( N LMHom O ) /\ G e. ( M LMHom N ) ) -> (Scalar ` O ) = (Scalar ` M ) )
15		lmghm 19031	. . 3 |- ( F e. ( N LMHom O ) -> F e. ( N GrpHom O ) )
16		lmghm 19031	. . 3 |- ( G e. ( M LMHom N ) -> G e. ( M GrpHom N ) )
17		ghmco 17680	. . 3 |- ( ( F e. ( N GrpHom O ) /\ G e. ( M GrpHom N ) ) -> ( F o. G ) e. ( M GrpHom O ) )
18	15, 16, 17	syl2an 494	. 2 |- ( ( F e. ( N LMHom O ) /\ G e. ( M LMHom N ) ) -> ( F o. G ) e. ( M GrpHom O ) )
19		simplr 792	. . . . . 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 794	. . . . . 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 796	. . . . . 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 2622	. . . . . . 7 |- ( .s ` N ) = ( .s ` N )
23	4, 6, 1, 2, 22	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 ) ) )
24	19, 20, 21, 23	syl3anc 1326	. . . . 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 ) ) )
25	24	fveq2d 6195	. . . 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 790	. . . . 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 ) )
27	13	fveq2d 6195	. . . . . . 7 |- ( G e. ( M LMHom N ) -> ( Base ` (Scalar ` N ) ) = ( Base ` (Scalar ` M ) ) )
28	27	ad2antlr 763	. . . . . 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 ) ) )
29	20, 28	eleqtrrd 2704	. . . . 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 2622	. . . . . . . . 9 |- ( Base ` N ) = ( Base ` N )
31	1, 30	lmhmf 19034	. . . . . . . 8 |- ( G e. ( M LMHom N ) -> G : ( Base ` M ) --> ( Base ` N ) )
32	31	adantl 482	. . . . . . 7 |- ( ( F e. ( N LMHom O ) /\ G e. ( M LMHom N ) ) -> G : ( Base ` M ) --> ( Base ` N ) )
33	32	ffvelrnda 6359	. . . . . 6 |- ( ( ( F e. ( N LMHom O ) /\ G e. ( M LMHom N ) ) /\ y e. ( Base ` M ) ) -> ( G ` y ) e. ( Base ` N ) )
34	33	adantrl 752	. . . . 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 2622	. . . . . 6 |- ( Base ` (Scalar ` N ) ) = ( Base ` (Scalar ` N ) )
36	11, 35, 30, 22, 3	lmhmlin 19035	. . . . 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 ) ) ) )
37	26, 29, 34, 36	syl3anc 1326	. . . 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 ) ) ) )
38	25, 37	eqtrd 2656	. . 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 6045	. . . . . 6 |- ( G : ( Base ` M ) --> ( Base ` N ) -> G Fn ( Base ` M ) )
40	32, 39	syl 17	. . . . 5 |- ( ( F e. ( N LMHom O ) /\ G e. ( M LMHom N ) ) -> G Fn ( Base ` M ) )
41	40	adantr 481	. . . 4 |- ( ( ( F e. ( N LMHom O ) /\ G e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. ( Base ` M ) ) ) -> G Fn ( Base ` M ) )
42	7	ad2antlr 763	. . . . 5 |- ( ( ( F e. ( N LMHom O ) /\ G e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. ( Base ` M ) ) ) -> M e. LMod )
43	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 ) )
44	42, 20, 21, 43	syl3anc 1326	. . . 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 6273	. . . 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 ) ) ) )
46	41, 44, 45	syl2anc 693	. . 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 6273	. . . . 5 |- ( ( G Fn ( Base ` M ) /\ y e. ( Base ` M ) ) -> ( ( F o. G ) ` y ) = ( F ` ( G ` y ) ) )
48	41, 21, 47	syl2anc 693	. . . 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 ) ) )
49	48	oveq2d 6666	. . 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 ) ) ) )
50	38, 46, 49	3eqtr4d 2666	. 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 ) ) )
51	1, 2, 3, 4, 5, 6, 8, 10, 14, 18, 50	islmhmd 19039	1 |- ( ( F e. ( N LMHom O ) /\ G e. ( M LMHom N ) ) -> ( F o. G ) e. ( M LMHom O ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   GrpHom cghm 17657   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-lmod 18865  df-lmhm 19022

This theorem is referenced by:  lmimco  20183  nmhmco  22560  mendring  37762