Theorem mrsubco 31418

Description: The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypothesis

Ref	Expression
mrsubco.s	|- S = (mRSubst ` T )

Assertion

Ref	Expression
mrsubco	|- ( ( F e. ran S /\ G e. ran S ) -> ( F o. G ) e. ran S )

Proof of Theorem mrsubco

Dummy variables c x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrsubco.s	. . . . 5 |- S = (mRSubst ` T )
2		eqid 2622	. . . . 5 |- (mREx ` T ) = (mREx ` T )
3	1, 2	mrsubf 31414	. . . 4 |- ( F e. ran S -> F : (mREx ` T ) --> (mREx ` T ) )
4	3	adantr 481	. . 3 |- ( ( F e. ran S /\ G e. ran S ) -> F : (mREx ` T ) --> (mREx ` T ) )
5	1, 2	mrsubf 31414	. . . 4 |- ( G e. ran S -> G : (mREx ` T ) --> (mREx ` T ) )
6	5	adantl 482	. . 3 |- ( ( F e. ran S /\ G e. ran S ) -> G : (mREx ` T ) --> (mREx ` T ) )
7		fco 6058	. . 3 |- ( ( F : (mREx ` T ) --> (mREx ` T ) /\ G : (mREx ` T ) --> (mREx ` T ) ) -> ( F o. G ) : (mREx ` T ) --> (mREx ` T ) )
8	4, 6, 7	syl2anc 693	. 2 |- ( ( F e. ran S /\ G e. ran S ) -> ( F o. G ) : (mREx ` T ) --> (mREx ` T ) )
9	6	adantr 481	. . . . 5 |- ( ( ( F e. ran S /\ G e. ran S ) /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> G : (mREx ` T ) --> (mREx ` T ) )
10		eldifi 3732	. . . . . . . . 9 |- ( c e. ( (mCN ` T ) \ (mVR ` T ) ) -> c e. (mCN ` T ) )
11		elun1 3780	. . . . . . . . 9 |- ( c e. (mCN ` T ) -> c e. ( (mCN ` T ) u. (mVR ` T ) ) )
12	10, 11	syl 17	. . . . . . . 8 |- ( c e. ( (mCN ` T ) \ (mVR ` T ) ) -> c e. ( (mCN ` T ) u. (mVR ` T ) ) )
13	12	adantl 482	. . . . . . 7 |- ( ( ( F e. ran S /\ G e. ran S ) /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> c e. ( (mCN ` T ) u. (mVR ` T ) ) )
14	13	s1cld 13383	. . . . . 6 |- ( ( ( F e. ran S /\ G e. ran S ) /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> <" c "> e. Word ( (mCN ` T ) u. (mVR ` T ) ) )
15		n0i 3920	. . . . . . . . . 10 |- ( F e. ran S -> -. ran S = (/) )
16		fvprc 6185	. . . . . . . . . . . . 13 |- ( -. T e. _V -> (mRSubst ` T ) = (/) )
17	1, 16	syl5eq 2668	. . . . . . . . . . . 12 |- ( -. T e. _V -> S = (/) )
18	17	rneqd 5353	. . . . . . . . . . 11 |- ( -. T e. _V -> ran S = ran (/) )
19		rn0 5377	. . . . . . . . . . 11 |- ran (/) = (/)
20	18, 19	syl6eq 2672	. . . . . . . . . 10 |- ( -. T e. _V -> ran S = (/) )
21	15, 20	nsyl2 142	. . . . . . . . 9 |- ( F e. ran S -> T e. _V )
22	21	adantr 481	. . . . . . . 8 |- ( ( F e. ran S /\ G e. ran S ) -> T e. _V )
23	22	adantr 481	. . . . . . 7 |- ( ( ( F e. ran S /\ G e. ran S ) /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> T e. _V )
24		eqid 2622	. . . . . . . 8 |- (mCN ` T ) = (mCN ` T )
25		eqid 2622	. . . . . . . 8 |- (mVR ` T ) = (mVR ` T )
26	24, 25, 2	mrexval 31398	. . . . . . 7 |- ( T e. _V -> (mREx ` T ) = Word ( (mCN ` T ) u. (mVR ` T ) ) )
27	23, 26	syl 17	. . . . . 6 |- ( ( ( F e. ran S /\ G e. ran S ) /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> (mREx ` T ) = Word ( (mCN ` T ) u. (mVR ` T ) ) )
28	14, 27	eleqtrrd 2704	. . . . 5 |- ( ( ( F e. ran S /\ G e. ran S ) /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> <" c "> e. (mREx ` T ) )
29		fvco3 6275	. . . . 5 |- ( ( G : (mREx ` T ) --> (mREx ` T ) /\ <" c "> e. (mREx ` T ) ) -> ( ( F o. G ) ` <" c "> ) = ( F ` ( G ` <" c "> ) ) )
30	9, 28, 29	syl2anc 693	. . . 4 |- ( ( ( F e. ran S /\ G e. ran S ) /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> ( ( F o. G ) ` <" c "> ) = ( F ` ( G ` <" c "> ) ) )
31	1, 2, 25, 24	mrsubcn 31416	. . . . . 6 |- ( ( G e. ran S /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> ( G ` <" c "> ) = <" c "> )
32	31	adantll 750	. . . . 5 |- ( ( ( F e. ran S /\ G e. ran S ) /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> ( G ` <" c "> ) = <" c "> )
33	32	fveq2d 6195	. . . 4 |- ( ( ( F e. ran S /\ G e. ran S ) /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> ( F ` ( G ` <" c "> ) ) = ( F ` <" c "> ) )
34	1, 2, 25, 24	mrsubcn 31416	. . . . 5 |- ( ( F e. ran S /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> ( F ` <" c "> ) = <" c "> )
35	34	adantlr 751	. . . 4 |- ( ( ( F e. ran S /\ G e. ran S ) /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> ( F ` <" c "> ) = <" c "> )
36	30, 33, 35	3eqtrd 2660	. . 3 |- ( ( ( F e. ran S /\ G e. ran S ) /\ c e. ( (mCN ` T ) \ (mVR ` T ) ) ) -> ( ( F o. G ) ` <" c "> ) = <" c "> )
37	36	ralrimiva 2966	. 2 |- ( ( F e. ran S /\ G e. ran S ) -> A. c e. ( (mCN ` T ) \ (mVR ` T ) ) ( ( F o. G ) ` <" c "> ) = <" c "> )
38	1, 2	mrsubccat 31415	. . . . . . . 8 |- ( ( G e. ran S /\ x e. (mREx ` T ) /\ y e. (mREx ` T ) ) -> ( G ` ( x ++ y ) ) = ( ( G ` x ) ++ ( G ` y ) ) )
39	38	3expb 1266	. . . . . . 7 |- ( ( G e. ran S /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( G ` ( x ++ y ) ) = ( ( G ` x ) ++ ( G ` y ) ) )
40	39	adantll 750	. . . . . 6 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( G ` ( x ++ y ) ) = ( ( G ` x ) ++ ( G ` y ) ) )
41	40	fveq2d 6195	. . . . 5 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( F ` ( G ` ( x ++ y ) ) ) = ( F ` ( ( G ` x ) ++ ( G ` y ) ) ) )
42		simpll 790	. . . . . 6 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> F e. ran S )
43	6	adantr 481	. . . . . . 7 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> G : (mREx ` T ) --> (mREx ` T ) )
44		simprl 794	. . . . . . 7 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> x e. (mREx ` T ) )
45	43, 44	ffvelrnd 6360	. . . . . 6 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( G ` x ) e. (mREx ` T ) )
46		simprr 796	. . . . . . 7 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> y e. (mREx ` T ) )
47	43, 46	ffvelrnd 6360	. . . . . 6 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( G ` y ) e. (mREx ` T ) )
48	1, 2	mrsubccat 31415	. . . . . 6 |- ( ( F e. ran S /\ ( G ` x ) e. (mREx ` T ) /\ ( G ` y ) e. (mREx ` T ) ) -> ( F ` ( ( G ` x ) ++ ( G ` y ) ) ) = ( ( F ` ( G ` x ) ) ++ ( F ` ( G ` y ) ) ) )
49	42, 45, 47, 48	syl3anc 1326	. . . . 5 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( F ` ( ( G ` x ) ++ ( G ` y ) ) ) = ( ( F ` ( G ` x ) ) ++ ( F ` ( G ` y ) ) ) )
50	41, 49	eqtrd 2656	. . . 4 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( F ` ( G ` ( x ++ y ) ) ) = ( ( F ` ( G ` x ) ) ++ ( F ` ( G ` y ) ) ) )
51	22, 26	syl 17	. . . . . . . . 9 |- ( ( F e. ran S /\ G e. ran S ) -> (mREx ` T ) = Word ( (mCN ` T ) u. (mVR ` T ) ) )
52	51	adantr 481	. . . . . . . 8 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> (mREx ` T ) = Word ( (mCN ` T ) u. (mVR ` T ) ) )
53	44, 52	eleqtrd 2703	. . . . . . 7 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> x e. Word ( (mCN ` T ) u. (mVR ` T ) ) )
54	46, 52	eleqtrd 2703	. . . . . . 7 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> y e. Word ( (mCN ` T ) u. (mVR ` T ) ) )
55		ccatcl 13359	. . . . . . 7 |- ( ( x e. Word ( (mCN ` T ) u. (mVR ` T ) ) /\ y e. Word ( (mCN ` T ) u. (mVR ` T ) ) ) -> ( x ++ y ) e. Word ( (mCN ` T ) u. (mVR ` T ) ) )
56	53, 54, 55	syl2anc 693	. . . . . 6 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( x ++ y ) e. Word ( (mCN ` T ) u. (mVR ` T ) ) )
57	56, 52	eleqtrrd 2704	. . . . 5 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( x ++ y ) e. (mREx ` T ) )
58		fvco3 6275	. . . . 5 |- ( ( G : (mREx ` T ) --> (mREx ` T ) /\ ( x ++ y ) e. (mREx ` T ) ) -> ( ( F o. G ) ` ( x ++ y ) ) = ( F ` ( G ` ( x ++ y ) ) ) )
59	43, 57, 58	syl2anc 693	. . . 4 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( ( F o. G ) ` ( x ++ y ) ) = ( F ` ( G ` ( x ++ y ) ) ) )
60		fvco3 6275	. . . . . 6 |- ( ( G : (mREx ` T ) --> (mREx ` T ) /\ x e. (mREx ` T ) ) -> ( ( F o. G ) ` x ) = ( F ` ( G ` x ) ) )
61	43, 44, 60	syl2anc 693	. . . . 5 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( ( F o. G ) ` x ) = ( F ` ( G ` x ) ) )
62		fvco3 6275	. . . . . 6 |- ( ( G : (mREx ` T ) --> (mREx ` T ) /\ y e. (mREx ` T ) ) -> ( ( F o. G ) ` y ) = ( F ` ( G ` y ) ) )
63	43, 46, 62	syl2anc 693	. . . . 5 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( ( F o. G ) ` y ) = ( F ` ( G ` y ) ) )
64	61, 63	oveq12d 6668	. . . 4 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( ( ( F o. G ) ` x ) ++ ( ( F o. G ) ` y ) ) = ( ( F ` ( G ` x ) ) ++ ( F ` ( G ` y ) ) ) )
65	50, 59, 64	3eqtr4d 2666	. . 3 |- ( ( ( F e. ran S /\ G e. ran S ) /\ ( x e. (mREx ` T ) /\ y e. (mREx ` T ) ) ) -> ( ( F o. G ) ` ( x ++ y ) ) = ( ( ( F o. G ) ` x ) ++ ( ( F o. G ) ` y ) ) )
66	65	ralrimivva 2971	. 2 |- ( ( F e. ran S /\ G e. ran S ) -> A. x e. (mREx ` T ) A. y e. (mREx ` T ) ( ( F o. G ) ` ( x ++ y ) ) = ( ( ( F o. G ) ` x ) ++ ( ( F o. G ) ` y ) ) )
67	1, 2, 25, 24	elmrsubrn 31417	. . 3 |- ( T e. _V -> ( ( F o. G ) e. ran S <-> ( ( F o. G ) : (mREx ` T ) --> (mREx ` T ) /\ A. c e. ( (mCN ` T ) \ (mVR ` T ) ) ( ( F o. G ) ` <" c "> ) = <" c "> /\ A. x e. (mREx ` T ) A. y e. (mREx ` T ) ( ( F o. G ) ` ( x ++ y ) ) = ( ( ( F o. G ) ` x ) ++ ( ( F o. G ) ` y ) ) ) ) )
68	22, 67	syl 17	. 2 |- ( ( F e. ran S /\ G e. ran S ) -> ( ( F o. G ) e. ran S <-> ( ( F o. G ) : (mREx ` T ) --> (mREx ` T ) /\ A. c e. ( (mCN ` T ) \ (mVR ` T ) ) ( ( F o. G ) ` <" c "> ) = <" c "> /\ A. x e. (mREx ` T ) A. y e. (mREx ` T ) ( ( F o. G ) ` ( x ++ y ) ) = ( ( ( F o. G ) ` x ) ++ ( ( F o. G ) ` y ) ) ) ) )
69	8, 37, 66, 68	mpbir3and 1245	1 |- ( ( F e. ran S /\ G e. ran S ) -> ( F o. G ) e. ran S )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   u. cun 3572   (/)c0 3915   ran crn 5115   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650  Word cword 13291   ++ cconcat 13293   <"cs1 13294  mCNcmcn 31357  mVRcmvar 31358  mRExcmrex 31363  mRSubstcmrsub 31367

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-int 4476  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-frmd 17386  df-vrmd 17387  df-mrex 31383  df-mrsub 31387

This theorem is referenced by:  msubco  31428