Theorem frgpcpbl 18172

Description: Compatibility of the group operation with the free group equivalence relation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
frgpval.m	|- G = (freeGrp ` I )
frgpval.b	|- M = (freeMnd ` ( I X. 2o ) )
frgpval.r	|- .~ = ( ~FG ` I )
frgpcpbl.p	|- .+ = ( +g ` M )

Assertion

Ref	Expression
frgpcpbl	|- ( ( A .~ C /\ B .~ D ) -> ( A .+ B ) .~ ( C .+ D ) )

Proof of Theorem frgpcpbl

Dummy variables k m n t v w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( _I ` Word ( I X. 2o ) ) = ( _I ` Word ( I X. 2o ) )
2		frgpval.r	. . 3 |- .~ = ( ~FG ` I )
3		eqid 2622	. . 3 |- ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
4		eqid 2622	. . 3 |- ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) = ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) )
5		eqid 2622	. . 3 |- ( ( _I ` Word ( I X. 2o ) ) \ U_ x e. ( _I ` Word ( I X. 2o ) ) ran ( ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) ` x ) ) = ( ( _I ` Word ( I X. 2o ) ) \ U_ x e. ( _I ` Word ( I X. 2o ) ) ran ( ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) ` x ) )
6		eqid 2622	. . 3 |- ( m e. { t e. (Word ( _I ` Word ( I X. 2o ) ) \ { (/) } ) | ( ( t ` 0 ) e. ( ( _I ` Word ( I X. 2o ) ) \ U_ x e. ( _I ` Word ( I X. 2o ) ) ran ( ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) ` x ) ) /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) ) = ( m e. { t e. (Word ( _I ` Word ( I X. 2o ) ) \ { (/) } ) | ( ( t ` 0 ) e. ( ( _I ` Word ( I X. 2o ) ) \ U_ x e. ( _I ` Word ( I X. 2o ) ) ran ( ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) ` x ) ) /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) )
7	1, 2, 3, 4, 5, 6	efgcpbl2 18170	. 2 |- ( ( A .~ C /\ B .~ D ) -> ( A ++ B ) .~ ( C ++ D ) )
8	1, 2	efger 18131	. . . . . 6 |- .~ Er ( _I ` Word ( I X. 2o ) )
9	8	a1i 11	. . . . 5 |- ( ( A .~ C /\ B .~ D ) -> .~ Er ( _I ` Word ( I X. 2o ) ) )
10		simpl 473	. . . . 5 |- ( ( A .~ C /\ B .~ D ) -> A .~ C )
11	9, 10	ercl 7753	. . . 4 |- ( ( A .~ C /\ B .~ D ) -> A e. ( _I ` Word ( I X. 2o ) ) )
12	1	efgrcl 18128	. . . . . . 7 |- ( A e. ( _I ` Word ( I X. 2o ) ) -> ( I e. _V /\ ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) ) )
13	11, 12	syl 17	. . . . . 6 |- ( ( A .~ C /\ B .~ D ) -> ( I e. _V /\ ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) ) )
14	13	simprd 479	. . . . 5 |- ( ( A .~ C /\ B .~ D ) -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
15	13	simpld 475	. . . . . . 7 |- ( ( A .~ C /\ B .~ D ) -> I e. _V )
16		2on 7568	. . . . . . 7 |- 2o e. On
17		xpexg 6960	. . . . . . 7 |- ( ( I e. _V /\ 2o e. On ) -> ( I X. 2o ) e. _V )
18	15, 16, 17	sylancl 694	. . . . . 6 |- ( ( A .~ C /\ B .~ D ) -> ( I X. 2o ) e. _V )
19		frgpval.b	. . . . . . 7 |- M = (freeMnd ` ( I X. 2o ) )
20		eqid 2622	. . . . . . 7 |- ( Base ` M ) = ( Base ` M )
21	19, 20	frmdbas 17389	. . . . . 6 |- ( ( I X. 2o ) e. _V -> ( Base ` M ) = Word ( I X. 2o ) )
22	18, 21	syl 17	. . . . 5 |- ( ( A .~ C /\ B .~ D ) -> ( Base ` M ) = Word ( I X. 2o ) )
23	14, 22	eqtr4d 2659	. . . 4 |- ( ( A .~ C /\ B .~ D ) -> ( _I ` Word ( I X. 2o ) ) = ( Base ` M ) )
24	11, 23	eleqtrd 2703	. . 3 |- ( ( A .~ C /\ B .~ D ) -> A e. ( Base ` M ) )
25		simpr 477	. . . . 5 |- ( ( A .~ C /\ B .~ D ) -> B .~ D )
26	9, 25	ercl 7753	. . . 4 |- ( ( A .~ C /\ B .~ D ) -> B e. ( _I ` Word ( I X. 2o ) ) )
27	26, 23	eleqtrd 2703	. . 3 |- ( ( A .~ C /\ B .~ D ) -> B e. ( Base ` M ) )
28		frgpcpbl.p	. . . 4 |- .+ = ( +g ` M )
29	19, 20, 28	frmdadd 17392	. . 3 |- ( ( A e. ( Base ` M ) /\ B e. ( Base ` M ) ) -> ( A .+ B ) = ( A ++ B ) )
30	24, 27, 29	syl2anc 693	. 2 |- ( ( A .~ C /\ B .~ D ) -> ( A .+ B ) = ( A ++ B ) )
31	9, 10	ercl2 7755	. . . 4 |- ( ( A .~ C /\ B .~ D ) -> C e. ( _I ` Word ( I X. 2o ) ) )
32	31, 23	eleqtrd 2703	. . 3 |- ( ( A .~ C /\ B .~ D ) -> C e. ( Base ` M ) )
33	9, 25	ercl2 7755	. . . 4 |- ( ( A .~ C /\ B .~ D ) -> D e. ( _I ` Word ( I X. 2o ) ) )
34	33, 23	eleqtrd 2703	. . 3 |- ( ( A .~ C /\ B .~ D ) -> D e. ( Base ` M ) )
35	19, 20, 28	frmdadd 17392	. . 3 |- ( ( C e. ( Base ` M ) /\ D e. ( Base ` M ) ) -> ( C .+ D ) = ( C ++ D ) )
36	32, 34, 35	syl2anc 693	. 2 |- ( ( A .~ C /\ B .~ D ) -> ( C .+ D ) = ( C ++ D ) )
37	7, 30, 36	3brtr4d 4685	1 |- ( ( A .~ C /\ B .~ D ) -> ( A .+ B ) .~ ( C .+ D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   (/)c0 3915   {csn 4177   <.cop 4183   <.cotp 4185   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   ran crn 5115   Oncon0 5723   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   Er wer 7739   0cc0 9936   1c1 9937   - cmin 10266   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   splice csplice 13296   <"cs2 13586   Basecbs 15857   +g cplusg 15941  freeMndcfrmd 17384   ~_FG cefg 18119  freeGrpcfrgp 18120

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-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-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  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-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  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-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-frmd 17386  df-efg 18122

This theorem is referenced by:  frgp0  18173  frgpadd  18176