Theorem efgcpbl2 18170

Description: Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)

Hypotheses

Ref	Expression
efgval.w	|- W = ( _I ` Word ( I X. 2o ) )
efgval.r	|- .~ = ( ~FG ` I )
efgval2.m	|- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
efgval2.t	|- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
efgred.d	|- D = ( W \ U_ x e. W ran ( T ` x ) )
efgred.s	|- S = ( m e. { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) )

Assertion

Ref	Expression
efgcpbl2	|- ( ( A .~ X /\ B .~ Y ) -> ( A ++ B ) .~ ( X ++ Y ) )

Distinct variable groups:   y, z   t, n, v, w, y, z, m, x   m, M   x, n, M, t, v, w   k, m, t, x, T   k, n, v, w, y, z, W, m, t, x   .~ , m, t, x, y, z   m, I, n, t, v, w, x, y, z   D, m, t

Allowed substitution hints:   A( x, y, z, w, v, t, k, m, n)   B( x, y, z, w, v, t, k, m, n)   D( x, y, z, w, v, k, n)   .~ ( w, v, k, n)   S( x, y, z, w, v, t, k, m, n)   T( y, z, w, v, n)   I( k)   M( y, z, k)   X( x, y, z, w, v, t, k, m, n)   Y( x, y, z, w, v, t, k, m, n)

Proof of Theorem efgcpbl2

Step	Hyp	Ref	Expression
1		efgval.w	. . . 4 |- W = ( _I ` Word ( I X. 2o ) )
2		efgval.r	. . . 4 |- .~ = ( ~FG ` I )
3	1, 2	efger 18131	. . 3 |- .~ Er W
4	3	a1i 11	. 2 |- ( ( A .~ X /\ B .~ Y ) -> .~ Er W )
5		simpl 473	. . . . 5 |- ( ( A .~ X /\ B .~ Y ) -> A .~ X )
6	4, 5	ercl 7753	. . . 4 |- ( ( A .~ X /\ B .~ Y ) -> A e. W )
7		wrd0 13330	. . . . 5 |- (/) e. Word ( I X. 2o )
8	1	efgrcl 18128	. . . . . . 7 |- ( A e. W -> ( I e. _V /\ W = Word ( I X. 2o ) ) )
9	6, 8	syl 17	. . . . . 6 |- ( ( A .~ X /\ B .~ Y ) -> ( I e. _V /\ W = Word ( I X. 2o ) ) )
10	9	simprd 479	. . . . 5 |- ( ( A .~ X /\ B .~ Y ) -> W = Word ( I X. 2o ) )
11	7, 10	syl5eleqr 2708	. . . 4 |- ( ( A .~ X /\ B .~ Y ) -> (/) e. W )
12		simpr 477	. . . 4 |- ( ( A .~ X /\ B .~ Y ) -> B .~ Y )
13		efgval2.m	. . . . 5 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
14		efgval2.t	. . . . 5 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
15		efgred.d	. . . . 5 |- D = ( W \ U_ x e. W ran ( T ` x ) )
16		efgred.s	. . . . 5 |- S = ( m e. { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) )
17	1, 2, 13, 14, 15, 16	efgcpbl 18169	. . . 4 |- ( ( A e. W /\ (/) e. W /\ B .~ Y ) -> ( ( A ++ B ) ++ (/) ) .~ ( ( A ++ Y ) ++ (/) ) )
18	6, 11, 12, 17	syl3anc 1326	. . 3 |- ( ( A .~ X /\ B .~ Y ) -> ( ( A ++ B ) ++ (/) ) .~ ( ( A ++ Y ) ++ (/) ) )
19	6, 10	eleqtrd 2703	. . . . 5 |- ( ( A .~ X /\ B .~ Y ) -> A e. Word ( I X. 2o ) )
20	4, 12	ercl 7753	. . . . . 6 |- ( ( A .~ X /\ B .~ Y ) -> B e. W )
21	20, 10	eleqtrd 2703	. . . . 5 |- ( ( A .~ X /\ B .~ Y ) -> B e. Word ( I X. 2o ) )
22		ccatcl 13359	. . . . 5 |- ( ( A e. Word ( I X. 2o ) /\ B e. Word ( I X. 2o ) ) -> ( A ++ B ) e. Word ( I X. 2o ) )
23	19, 21, 22	syl2anc 693	. . . 4 |- ( ( A .~ X /\ B .~ Y ) -> ( A ++ B ) e. Word ( I X. 2o ) )
24		ccatrid 13370	. . . 4 |- ( ( A ++ B ) e. Word ( I X. 2o ) -> ( ( A ++ B ) ++ (/) ) = ( A ++ B ) )
25	23, 24	syl 17	. . 3 |- ( ( A .~ X /\ B .~ Y ) -> ( ( A ++ B ) ++ (/) ) = ( A ++ B ) )
26	4, 12	ercl2 7755	. . . . . 6 |- ( ( A .~ X /\ B .~ Y ) -> Y e. W )
27	26, 10	eleqtrd 2703	. . . . 5 |- ( ( A .~ X /\ B .~ Y ) -> Y e. Word ( I X. 2o ) )
28		ccatcl 13359	. . . . 5 |- ( ( A e. Word ( I X. 2o ) /\ Y e. Word ( I X. 2o ) ) -> ( A ++ Y ) e. Word ( I X. 2o ) )
29	19, 27, 28	syl2anc 693	. . . 4 |- ( ( A .~ X /\ B .~ Y ) -> ( A ++ Y ) e. Word ( I X. 2o ) )
30		ccatrid 13370	. . . 4 |- ( ( A ++ Y ) e. Word ( I X. 2o ) -> ( ( A ++ Y ) ++ (/) ) = ( A ++ Y ) )
31	29, 30	syl 17	. . 3 |- ( ( A .~ X /\ B .~ Y ) -> ( ( A ++ Y ) ++ (/) ) = ( A ++ Y ) )
32	18, 25, 31	3brtr3d 4684	. 2 |- ( ( A .~ X /\ B .~ Y ) -> ( A ++ B ) .~ ( A ++ Y ) )
33	1, 2, 13, 14, 15, 16	efgcpbl 18169	. . . 4 |- ( ( (/) e. W /\ Y e. W /\ A .~ X ) -> ( ( (/) ++ A ) ++ Y ) .~ ( ( (/) ++ X ) ++ Y ) )
34	11, 26, 5, 33	syl3anc 1326	. . 3 |- ( ( A .~ X /\ B .~ Y ) -> ( ( (/) ++ A ) ++ Y ) .~ ( ( (/) ++ X ) ++ Y ) )
35		ccatlid 13369	. . . . 5 |- ( A e. Word ( I X. 2o ) -> ( (/) ++ A ) = A )
36	19, 35	syl 17	. . . 4 |- ( ( A .~ X /\ B .~ Y ) -> ( (/) ++ A ) = A )
37	36	oveq1d 6665	. . 3 |- ( ( A .~ X /\ B .~ Y ) -> ( ( (/) ++ A ) ++ Y ) = ( A ++ Y ) )
38	4, 5	ercl2 7755	. . . . . 6 |- ( ( A .~ X /\ B .~ Y ) -> X e. W )
39	38, 10	eleqtrd 2703	. . . . 5 |- ( ( A .~ X /\ B .~ Y ) -> X e. Word ( I X. 2o ) )
40		ccatlid 13369	. . . . 5 |- ( X e. Word ( I X. 2o ) -> ( (/) ++ X ) = X )
41	39, 40	syl 17	. . . 4 |- ( ( A .~ X /\ B .~ Y ) -> ( (/) ++ X ) = X )
42	41	oveq1d 6665	. . 3 |- ( ( A .~ X /\ B .~ Y ) -> ( ( (/) ++ X ) ++ Y ) = ( X ++ Y ) )
43	34, 37, 42	3brtr3d 4684	. 2 |- ( ( A .~ X /\ B .~ Y ) -> ( A ++ Y ) .~ ( X ++ Y ) )
44	4, 32, 43	ertrd 7758	1 |- ( ( A .~ X /\ B .~ Y ) -> ( A ++ B ) .~ ( X ++ Y ) )

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   ` 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   ~_FG cefg 18119

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-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-efg 18122

This theorem is referenced by:  frgpcpbl  18172