Theorem efgred 18161

Description: The reduced word that forms the base of the sequence in efgsval 18144 is uniquely determined, given the terminal point. (Contributed by Mario Carneiro, 28-Sep-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
efgred	|- ( ( A e. dom S /\ B e. dom S /\ ( S ` A ) = ( S ` B ) ) -> ( A ` 0 ) = ( B ` 0 ) )

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)

Proof of Theorem efgred

Dummy variables a b c d i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . . . 8 |- W = ( _I ` Word ( I X. 2o ) )
2		fviss 6256	. . . . . . . 8 |- ( _I ` Word ( I X. 2o ) ) C_ Word ( I X. 2o )
3	1, 2	eqsstri 3635	. . . . . . 7 |- W C_ Word ( I X. 2o )
4		efgval.r	. . . . . . . . . . 11 |- .~ = ( ~FG ` I )
5		efgval2.m	. . . . . . . . . . 11 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
6		efgval2.t	. . . . . . . . . . 11 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
7		efgred.d	. . . . . . . . . . 11 |- D = ( W \ U_ x e. W ran ( T ` x ) )
8		efgred.s	. . . . . . . . . . 11 |- 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 ) ) )
9	1, 4, 5, 6, 7, 8	efgsf 18142	. . . . . . . . . 10 |- S : { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } --> W
10	9	fdmi 6052	. . . . . . . . . . 11 |- dom S = { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) }
11	10	feq2i 6037	. . . . . . . . . 10 |- ( S : dom S --> W <-> S : { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } --> W )
12	9, 11	mpbir 221	. . . . . . . . 9 |- S : dom S --> W
13	12	ffvelrni 6358	. . . . . . . 8 |- ( A e. dom S -> ( S ` A ) e. W )
14	13	adantr 481	. . . . . . 7 |- ( ( A e. dom S /\ B e. dom S ) -> ( S ` A ) e. W )
15	3, 14	sseldi 3601	. . . . . 6 |- ( ( A e. dom S /\ B e. dom S ) -> ( S ` A ) e. Word ( I X. 2o ) )
16		lencl 13324	. . . . . 6 |- ( ( S ` A ) e. Word ( I X. 2o ) -> ( # ` ( S ` A ) ) e. NN0 )
17	15, 16	syl 17	. . . . 5 |- ( ( A e. dom S /\ B e. dom S ) -> ( # ` ( S ` A ) ) e. NN0 )
18		peano2nn0 11333	. . . . 5 |- ( ( # ` ( S ` A ) ) e. NN0 -> ( ( # ` ( S ` A ) ) + 1 ) e. NN0 )
19	17, 18	syl 17	. . . 4 |- ( ( A e. dom S /\ B e. dom S ) -> ( ( # ` ( S ` A ) ) + 1 ) e. NN0 )
20		breq2 4657	. . . . . . 7 |- ( c = 0 -> ( ( # ` ( S ` a ) ) < c <-> ( # ` ( S ` a ) ) < 0 ) )
21	20	imbi1d 331	. . . . . 6 |- ( c = 0 -> ( ( ( # ` ( S ` a ) ) < c -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> ( ( # ` ( S ` a ) ) < 0 -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
22	21	2ralbidv 2989	. . . . 5 |- ( c = 0 -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < c -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < 0 -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
23		breq2 4657	. . . . . . 7 |- ( c = i -> ( ( # ` ( S ` a ) ) < c <-> ( # ` ( S ` a ) ) < i ) )
24	23	imbi1d 331	. . . . . 6 |- ( c = i -> ( ( ( # ` ( S ` a ) ) < c -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
25	24	2ralbidv 2989	. . . . 5 |- ( c = i -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < c -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
26		breq2 4657	. . . . . . 7 |- ( c = ( i + 1 ) -> ( ( # ` ( S ` a ) ) < c <-> ( # ` ( S ` a ) ) < ( i + 1 ) ) )
27	26	imbi1d 331	. . . . . 6 |- ( c = ( i + 1 ) -> ( ( ( # ` ( S ` a ) ) < c -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> ( ( # ` ( S ` a ) ) < ( i + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
28	27	2ralbidv 2989	. . . . 5 |- ( c = ( i + 1 ) -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < c -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( i + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
29		breq2 4657	. . . . . . 7 |- ( c = ( ( # ` ( S ` A ) ) + 1 ) -> ( ( # ` ( S ` a ) ) < c <-> ( # ` ( S ` a ) ) < ( ( # ` ( S ` A ) ) + 1 ) ) )
30	29	imbi1d 331	. . . . . 6 |- ( c = ( ( # ` ( S ` A ) ) + 1 ) -> ( ( ( # ` ( S ` a ) ) < c -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> ( ( # ` ( S ` a ) ) < ( ( # ` ( S ` A ) ) + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
31	30	2ralbidv 2989	. . . . 5 |- ( c = ( ( # ` ( S ` A ) ) + 1 ) -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < c -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( ( # ` ( S ` A ) ) + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
32	12	ffvelrni 6358	. . . . . . . . . . 11 |- ( a e. dom S -> ( S ` a ) e. W )
33	3, 32	sseldi 3601	. . . . . . . . . 10 |- ( a e. dom S -> ( S ` a ) e. Word ( I X. 2o ) )
34		lencl 13324	. . . . . . . . . 10 |- ( ( S ` a ) e. Word ( I X. 2o ) -> ( # ` ( S ` a ) ) e. NN0 )
35	33, 34	syl 17	. . . . . . . . 9 |- ( a e. dom S -> ( # ` ( S ` a ) ) e. NN0 )
36		nn0nlt0 11319	. . . . . . . . 9 |- ( ( # ` ( S ` a ) ) e. NN0 -> -. ( # ` ( S ` a ) ) < 0 )
37	35, 36	syl 17	. . . . . . . 8 |- ( a e. dom S -> -. ( # ` ( S ` a ) ) < 0 )
38	37	pm2.21d 118	. . . . . . 7 |- ( a e. dom S -> ( ( # ` ( S ` a ) ) < 0 -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) )
39	38	adantr 481	. . . . . 6 |- ( ( a e. dom S /\ b e. dom S ) -> ( ( # ` ( S ` a ) ) < 0 -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) )
40	39	rgen2a 2977	. . . . 5 |- A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < 0 -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) )
41		simpl1 1064	. . . . . . . . . . . . . 14 |- ( ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) /\ ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) ) /\ -. ( c ` 0 ) = ( d ` 0 ) ) -> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) )
42		simpl3l 1116	. . . . . . . . . . . . . . 15 |- ( ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) /\ ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) ) /\ -. ( c ` 0 ) = ( d ` 0 ) ) -> ( # ` ( S ` c ) ) = i )
43		breq2 4657	. . . . . . . . . . . . . . . . 17 |- ( ( # ` ( S ` c ) ) = i -> ( ( # ` ( S ` a ) ) < ( # ` ( S ` c ) ) <-> ( # ` ( S ` a ) ) < i ) )
44	43	imbi1d 331	. . . . . . . . . . . . . . . 16 |- ( ( # ` ( S ` c ) ) = i -> ( ( ( # ` ( S ` a ) ) < ( # ` ( S ` c ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
45	44	2ralbidv 2989	. . . . . . . . . . . . . . 15 |- ( ( # ` ( S ` c ) ) = i -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( # ` ( S ` c ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
46	42, 45	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) /\ ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) ) /\ -. ( c ` 0 ) = ( d ` 0 ) ) -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( # ` ( S ` c ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
47	41, 46	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) /\ ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) ) /\ -. ( c ` 0 ) = ( d ` 0 ) ) -> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( # ` ( S ` c ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) )
48		simpl2l 1114	. . . . . . . . . . . . 13 |- ( ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) /\ ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) ) /\ -. ( c ` 0 ) = ( d ` 0 ) ) -> c e. dom S )
49		simpl2r 1115	. . . . . . . . . . . . 13 |- ( ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) /\ ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) ) /\ -. ( c ` 0 ) = ( d ` 0 ) ) -> d e. dom S )
50		simpl3r 1117	. . . . . . . . . . . . 13 |- ( ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) /\ ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) ) /\ -. ( c ` 0 ) = ( d ` 0 ) ) -> ( S ` c ) = ( S ` d ) )
51		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) /\ ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) ) /\ -. ( c ` 0 ) = ( d ` 0 ) ) -> -. ( c ` 0 ) = ( d ` 0 ) )
52	1, 4, 5, 6, 7, 8, 47, 48, 49, 50, 51	efgredlem 18160	. . . . . . . . . . . 12 |- -. ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) /\ ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) ) /\ -. ( c ` 0 ) = ( d ` 0 ) )
53		iman 440	. . . . . . . . . . . 12 |- ( ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) /\ ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) ) -> ( c ` 0 ) = ( d ` 0 ) ) <-> -. ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) /\ ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) ) /\ -. ( c ` 0 ) = ( d ` 0 ) ) )
54	52, 53	mpbir 221	. . . . . . . . . . 11 |- ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) /\ ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) ) -> ( c ` 0 ) = ( d ` 0 ) )
55	54	3expia 1267	. . . . . . . . . 10 |- ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) ) -> ( ( ( # ` ( S ` c ) ) = i /\ ( S ` c ) = ( S ` d ) ) -> ( c ` 0 ) = ( d ` 0 ) ) )
56	55	expd 452	. . . . . . . . 9 |- ( ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( c e. dom S /\ d e. dom S ) ) -> ( ( # ` ( S ` c ) ) = i -> ( ( S ` c ) = ( S ` d ) -> ( c ` 0 ) = ( d ` 0 ) ) ) )
57	56	ralrimivva 2971	. . . . . . . 8 |- ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) -> A. c e. dom S A. d e. dom S ( ( # ` ( S ` c ) ) = i -> ( ( S ` c ) = ( S ` d ) -> ( c ` 0 ) = ( d ` 0 ) ) ) )
58		fveq2 6191	. . . . . . . . . . . 12 |- ( c = a -> ( S ` c ) = ( S ` a ) )
59	58	fveq2d 6195	. . . . . . . . . . 11 |- ( c = a -> ( # ` ( S ` c ) ) = ( # ` ( S ` a ) ) )
60	59	eqeq1d 2624	. . . . . . . . . 10 |- ( c = a -> ( ( # ` ( S ` c ) ) = i <-> ( # ` ( S ` a ) ) = i ) )
61	58	eqeq1d 2624	. . . . . . . . . . 11 |- ( c = a -> ( ( S ` c ) = ( S ` d ) <-> ( S ` a ) = ( S ` d ) ) )
62		fveq1 6190	. . . . . . . . . . . 12 |- ( c = a -> ( c ` 0 ) = ( a ` 0 ) )
63	62	eqeq1d 2624	. . . . . . . . . . 11 |- ( c = a -> ( ( c ` 0 ) = ( d ` 0 ) <-> ( a ` 0 ) = ( d ` 0 ) ) )
64	61, 63	imbi12d 334	. . . . . . . . . 10 |- ( c = a -> ( ( ( S ` c ) = ( S ` d ) -> ( c ` 0 ) = ( d ` 0 ) ) <-> ( ( S ` a ) = ( S ` d ) -> ( a ` 0 ) = ( d ` 0 ) ) ) )
65	60, 64	imbi12d 334	. . . . . . . . 9 |- ( c = a -> ( ( ( # ` ( S ` c ) ) = i -> ( ( S ` c ) = ( S ` d ) -> ( c ` 0 ) = ( d ` 0 ) ) ) <-> ( ( # ` ( S ` a ) ) = i -> ( ( S ` a ) = ( S ` d ) -> ( a ` 0 ) = ( d ` 0 ) ) ) ) )
66		fveq2 6191	. . . . . . . . . . . 12 |- ( d = b -> ( S ` d ) = ( S ` b ) )
67	66	eqeq2d 2632	. . . . . . . . . . 11 |- ( d = b -> ( ( S ` a ) = ( S ` d ) <-> ( S ` a ) = ( S ` b ) ) )
68		fveq1 6190	. . . . . . . . . . . 12 |- ( d = b -> ( d ` 0 ) = ( b ` 0 ) )
69	68	eqeq2d 2632	. . . . . . . . . . 11 |- ( d = b -> ( ( a ` 0 ) = ( d ` 0 ) <-> ( a ` 0 ) = ( b ` 0 ) ) )
70	67, 69	imbi12d 334	. . . . . . . . . 10 |- ( d = b -> ( ( ( S ` a ) = ( S ` d ) -> ( a ` 0 ) = ( d ` 0 ) ) <-> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) )
71	70	imbi2d 330	. . . . . . . . 9 |- ( d = b -> ( ( ( # ` ( S ` a ) ) = i -> ( ( S ` a ) = ( S ` d ) -> ( a ` 0 ) = ( d ` 0 ) ) ) <-> ( ( # ` ( S ` a ) ) = i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
72	65, 71	cbvral2v 3179	. . . . . . . 8 |- ( A. c e. dom S A. d e. dom S ( ( # ` ( S ` c ) ) = i -> ( ( S ` c ) = ( S ` d ) -> ( c ` 0 ) = ( d ` 0 ) ) ) <-> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) = i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) )
73	57, 72	sylib 208	. . . . . . 7 |- ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) -> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) = i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) )
74	73	ancli 574	. . . . . 6 |- ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) = i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
75	35	adantr 481	. . . . . . . . . . 11 |- ( ( a e. dom S /\ b e. dom S ) -> ( # ` ( S ` a ) ) e. NN0 )
76		nn0leltp1 11436	. . . . . . . . . . . . 13 |- ( ( ( # ` ( S ` a ) ) e. NN0 /\ i e. NN0 ) -> ( ( # ` ( S ` a ) ) <_ i <-> ( # ` ( S ` a ) ) < ( i + 1 ) ) )
77		nn0re 11301	. . . . . . . . . . . . . 14 |- ( ( # ` ( S ` a ) ) e. NN0 -> ( # ` ( S ` a ) ) e. RR )
78		nn0re 11301	. . . . . . . . . . . . . 14 |- ( i e. NN0 -> i e. RR )
79		leloe 10124	. . . . . . . . . . . . . 14 |- ( ( ( # ` ( S ` a ) ) e. RR /\ i e. RR ) -> ( ( # ` ( S ` a ) ) <_ i <-> ( ( # ` ( S ` a ) ) < i \/ ( # ` ( S ` a ) ) = i ) ) )
80	77, 78, 79	syl2an 494	. . . . . . . . . . . . 13 |- ( ( ( # ` ( S ` a ) ) e. NN0 /\ i e. NN0 ) -> ( ( # ` ( S ` a ) ) <_ i <-> ( ( # ` ( S ` a ) ) < i \/ ( # ` ( S ` a ) ) = i ) ) )
81	76, 80	bitr3d 270	. . . . . . . . . . . 12 |- ( ( ( # ` ( S ` a ) ) e. NN0 /\ i e. NN0 ) -> ( ( # ` ( S ` a ) ) < ( i + 1 ) <-> ( ( # ` ( S ` a ) ) < i \/ ( # ` ( S ` a ) ) = i ) ) )
82	81	ancoms 469	. . . . . . . . . . 11 |- ( ( i e. NN0 /\ ( # ` ( S ` a ) ) e. NN0 ) -> ( ( # ` ( S ` a ) ) < ( i + 1 ) <-> ( ( # ` ( S ` a ) ) < i \/ ( # ` ( S ` a ) ) = i ) ) )
83	75, 82	sylan2 491	. . . . . . . . . 10 |- ( ( i e. NN0 /\ ( a e. dom S /\ b e. dom S ) ) -> ( ( # ` ( S ` a ) ) < ( i + 1 ) <-> ( ( # ` ( S ` a ) ) < i \/ ( # ` ( S ` a ) ) = i ) ) )
84	83	imbi1d 331	. . . . . . . . 9 |- ( ( i e. NN0 /\ ( a e. dom S /\ b e. dom S ) ) -> ( ( ( # ` ( S ` a ) ) < ( i + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> ( ( ( # ` ( S ` a ) ) < i \/ ( # ` ( S ` a ) ) = i ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
85		jaob 822	. . . . . . . . 9 |- ( ( ( ( # ` ( S ` a ) ) < i \/ ( # ` ( S ` a ) ) = i ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> ( ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( ( # ` ( S ` a ) ) = i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
86	84, 85	syl6bb 276	. . . . . . . 8 |- ( ( i e. NN0 /\ ( a e. dom S /\ b e. dom S ) ) -> ( ( ( # ` ( S ` a ) ) < ( i + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> ( ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( ( # ` ( S ` a ) ) = i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) ) )
87	86	2ralbidva 2988	. . . . . . 7 |- ( i e. NN0 -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( i + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> A. a e. dom S A. b e. dom S ( ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( ( # ` ( S ` a ) ) = i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) ) )
88		r19.26-2 3065	. . . . . . 7 |- ( A. a e. dom S A. b e. dom S ( ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ ( ( # ` ( S ` a ) ) = i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) <-> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) = i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
89	87, 88	syl6bb 276	. . . . . 6 |- ( i e. NN0 -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( i + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) /\ A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) = i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) ) )
90	74, 89	syl5ibr 236	. . . . 5 |- ( i e. NN0 -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < i -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) -> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( i + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
91	22, 25, 28, 31, 40, 90	nn0ind 11472	. . . 4 |- ( ( ( # ` ( S ` A ) ) + 1 ) e. NN0 -> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( ( # ` ( S ` A ) ) + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) )
92	19, 91	syl 17	. . 3 |- ( ( A e. dom S /\ B e. dom S ) -> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( ( # ` ( S ` A ) ) + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) )
93	17	nn0red 11352	. . . 4 |- ( ( A e. dom S /\ B e. dom S ) -> ( # ` ( S ` A ) ) e. RR )
94	93	ltp1d 10954	. . 3 |- ( ( A e. dom S /\ B e. dom S ) -> ( # ` ( S ` A ) ) < ( ( # ` ( S ` A ) ) + 1 ) )
95		fveq2 6191	. . . . . . 7 |- ( a = A -> ( S ` a ) = ( S ` A ) )
96	95	fveq2d 6195	. . . . . 6 |- ( a = A -> ( # ` ( S ` a ) ) = ( # ` ( S ` A ) ) )
97	96	breq1d 4663	. . . . 5 |- ( a = A -> ( ( # ` ( S ` a ) ) < ( ( # ` ( S ` A ) ) + 1 ) <-> ( # ` ( S ` A ) ) < ( ( # ` ( S ` A ) ) + 1 ) ) )
98	95	eqeq1d 2624	. . . . . 6 |- ( a = A -> ( ( S ` a ) = ( S ` b ) <-> ( S ` A ) = ( S ` b ) ) )
99		fveq1 6190	. . . . . . 7 |- ( a = A -> ( a ` 0 ) = ( A ` 0 ) )
100	99	eqeq1d 2624	. . . . . 6 |- ( a = A -> ( ( a ` 0 ) = ( b ` 0 ) <-> ( A ` 0 ) = ( b ` 0 ) ) )
101	98, 100	imbi12d 334	. . . . 5 |- ( a = A -> ( ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) <-> ( ( S ` A ) = ( S ` b ) -> ( A ` 0 ) = ( b ` 0 ) ) ) )
102	97, 101	imbi12d 334	. . . 4 |- ( a = A -> ( ( ( # ` ( S ` a ) ) < ( ( # ` ( S ` A ) ) + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> ( ( # ` ( S ` A ) ) < ( ( # ` ( S ` A ) ) + 1 ) -> ( ( S ` A ) = ( S ` b ) -> ( A ` 0 ) = ( b ` 0 ) ) ) ) )
103		fveq2 6191	. . . . . . 7 |- ( b = B -> ( S ` b ) = ( S ` B ) )
104	103	eqeq2d 2632	. . . . . 6 |- ( b = B -> ( ( S ` A ) = ( S ` b ) <-> ( S ` A ) = ( S ` B ) ) )
105		fveq1 6190	. . . . . . 7 |- ( b = B -> ( b ` 0 ) = ( B ` 0 ) )
106	105	eqeq2d 2632	. . . . . 6 |- ( b = B -> ( ( A ` 0 ) = ( b ` 0 ) <-> ( A ` 0 ) = ( B ` 0 ) ) )
107	104, 106	imbi12d 334	. . . . 5 |- ( b = B -> ( ( ( S ` A ) = ( S ` b ) -> ( A ` 0 ) = ( b ` 0 ) ) <-> ( ( S ` A ) = ( S ` B ) -> ( A ` 0 ) = ( B ` 0 ) ) ) )
108	107	imbi2d 330	. . . 4 |- ( b = B -> ( ( ( # ` ( S ` A ) ) < ( ( # ` ( S ` A ) ) + 1 ) -> ( ( S ` A ) = ( S ` b ) -> ( A ` 0 ) = ( b ` 0 ) ) ) <-> ( ( # ` ( S ` A ) ) < ( ( # ` ( S ` A ) ) + 1 ) -> ( ( S ` A ) = ( S ` B ) -> ( A ` 0 ) = ( B ` 0 ) ) ) ) )
109	102, 108	rspc2v 3322	. . 3 |- ( ( A e. dom S /\ B e. dom S ) -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( ( # ` ( S ` A ) ) + 1 ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) -> ( ( # ` ( S ` A ) ) < ( ( # ` ( S ` A ) ) + 1 ) -> ( ( S ` A ) = ( S ` B ) -> ( A ` 0 ) = ( B ` 0 ) ) ) ) )
110	92, 94, 109	mp2d 49	. 2 |- ( ( A e. dom S /\ B e. dom S ) -> ( ( S ` A ) = ( S ` B ) -> ( A ` 0 ) = ( B ` 0 ) ) )
111	110	3impia 1261	1 |- ( ( A e. dom S /\ B e. dom S /\ ( S ` A ) = ( S ` B ) ) -> ( A ` 0 ) = ( B ` 0 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ 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   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   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-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-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-rp 11833  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

This theorem is referenced by:  efgrelexlemb  18163