Theorem efgcpbllema 18167

Description: Lemma for efgrelex 18164. Define an auxiliary equivalence relation L such that A L B if there are sequences from A to B passing through the same reduced word. (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 ) ) )
efgcpbllem.1	|- L = { <. i , j >. | ( { i , j } C_ W /\ ( ( A ++ i ) ++ B ) .~ ( ( A ++ j ) ++ B ) ) }

Assertion

Ref	Expression
efgcpbllema	|- ( X L Y <-> ( X e. W /\ Y e. W /\ ( ( A ++ X ) ++ B ) .~ ( ( A ++ Y ) ++ B ) ) )

Distinct variable groups:   i, j, A   y, z   t, n, v, w, y, z   i, m, n, t, v, w, x, M, j   i, k, T, j, m, t, x   i, X, j   y, i, z, W, j   k, n, v, w, y, z, W, m, t, x   .~ , i, j, m, t, x, y, z   B, i, j   S, i, j   i, Y, j   i, I, j, m, n, t, v, w, x, y, z   D, i, j, 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)   L( x, y, z, w, v, t, i, j, k, m, n)   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 efgcpbllema

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . 5 |- ( i = X -> ( A ++ i ) = ( A ++ X ) )
2	1	oveq1d 6665	. . . 4 |- ( i = X -> ( ( A ++ i ) ++ B ) = ( ( A ++ X ) ++ B ) )
3		oveq2 6658	. . . . 5 |- ( j = Y -> ( A ++ j ) = ( A ++ Y ) )
4	3	oveq1d 6665	. . . 4 |- ( j = Y -> ( ( A ++ j ) ++ B ) = ( ( A ++ Y ) ++ B ) )
5	2, 4	breqan12d 4669	. . 3 |- ( ( i = X /\ j = Y ) -> ( ( ( A ++ i ) ++ B ) .~ ( ( A ++ j ) ++ B ) <-> ( ( A ++ X ) ++ B ) .~ ( ( A ++ Y ) ++ B ) ) )
6		efgcpbllem.1	. . . 4 |- L = { <. i , j >. | ( { i , j } C_ W /\ ( ( A ++ i ) ++ B ) .~ ( ( A ++ j ) ++ B ) ) }
7		vex 3203	. . . . . . 7 |- i e. _V
8		vex 3203	. . . . . . 7 |- j e. _V
9	7, 8	prss 4351	. . . . . 6 |- ( ( i e. W /\ j e. W ) <-> { i , j } C_ W )
10	9	anbi1i 731	. . . . 5 |- ( ( ( i e. W /\ j e. W ) /\ ( ( A ++ i ) ++ B ) .~ ( ( A ++ j ) ++ B ) ) <-> ( { i , j } C_ W /\ ( ( A ++ i ) ++ B ) .~ ( ( A ++ j ) ++ B ) ) )
11	10	opabbii 4717	. . . 4 |- { <. i , j >. | ( ( i e. W /\ j e. W ) /\ ( ( A ++ i ) ++ B ) .~ ( ( A ++ j ) ++ B ) ) } = { <. i , j >. | ( { i , j } C_ W /\ ( ( A ++ i ) ++ B ) .~ ( ( A ++ j ) ++ B ) ) }
12	6, 11	eqtr4i 2647	. . 3 |- L = { <. i , j >. | ( ( i e. W /\ j e. W ) /\ ( ( A ++ i ) ++ B ) .~ ( ( A ++ j ) ++ B ) ) }
13	5, 12	brab2a 5194	. 2 |- ( X L Y <-> ( ( X e. W /\ Y e. W ) /\ ( ( A ++ X ) ++ B ) .~ ( ( A ++ Y ) ++ B ) ) )
14		df-3an 1039	. 2 |- ( ( X e. W /\ Y e. W /\ ( ( A ++ X ) ++ B ) .~ ( ( A ++ Y ) ++ B ) ) <-> ( ( X e. W /\ Y e. W ) /\ ( ( A ++ X ) ++ B ) .~ ( ( A ++ Y ) ++ B ) ) )
15	13, 14	bitr4i 267	1 |- ( X L Y <-> ( X e. W /\ Y e. W /\ ( ( A ++ X ) ++ B ) .~ ( ( A ++ Y ) ++ B ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   <.cop 4183   <.cotp 4185   U_ciun 4520   class class class wbr 4653   {copab 4712   |-> cmpt 4729   _I cid 5023   X. cxp 5112   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  efgcpbllemb  18168  efgcpbl  18169