Theorem wrd2ind 11194

Description: Perform induction over the structure of two words of the same length. (Contributed by AV, 23-Jan-2019.) (Proof shortened by AV, 12-Oct-2022.)

Hypotheses

Ref	Expression
wrd2ind.1	|- ( ( x = (/) /\ w = (/) ) -> ( ph <-> ps ) )
wrd2ind.2	|- ( ( x = y /\ w = u ) -> ( ph <-> ch ) )
wrd2ind.3	|- ( ( x = ( y ++ <" z "> ) /\ w = ( u ++ <" s "> ) ) -> ( ph <-> th ) )
wrd2ind.4	|- ( x = A -> ( rh <-> ta ) )
wrd2ind.5	|- ( w = B -> ( ph <-> rh ) )
wrd2ind.6	|- ps
wrd2ind.7	|- ( ( ( y e. Word X /\ z e. X ) /\ ( u e. Word Y /\ s e. Y ) /\ (♯ ` y ) = (♯ ` u ) ) -> ( ch -> th ) )

Assertion

Ref	Expression
wrd2ind	|- ( ( A e. Word X /\ B e. Word Y /\ (♯ ` A ) = (♯ ` B ) ) -> ta )

Distinct variable groups:   x, w, A   y, w, z, B, x   u, s, w, x, y, z, X   Y, s, u, w, x, y, z   ch, w, x   ph, s, u, y, z   ta, x   th, w, x   rh, w

Allowed substitution hints:   ph( x, w)   ps( x, y, z, w, u, s)   ch( y, z, u, s)   th( y, z, u, s)   ta( y, z, w, u, s)   rh( x, y, z, u, s)   A( y, z, u, s)   B( u, s)

Proof of Theorem wrd2ind

Dummy variables n m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lencl 11015	. . . . 5 |- ( A e. Word X -> (♯ ` A ) e. NN0 )
2		eqeq2 2216	. . . . . . . . 9 |- ( n = 0 -> ( (♯ ` x ) = n <-> (♯ ` x ) = 0 ) )
3	2	anbi2d 464	. . . . . . . 8 |- ( n = 0 -> ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = n ) <-> ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = 0 ) ) )
4	3	imbi1d 231	. . . . . . 7 |- ( n = 0 -> ( ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = n ) -> ph ) <-> ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = 0 ) -> ph ) ) )
5	4	2ralbidv 2531	. . . . . 6 |- ( n = 0 -> ( A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = n ) -> ph ) <-> A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = 0 ) -> ph ) ) )
6		eqeq2 2216	. . . . . . . . 9 |- ( n = m -> ( (♯ ` x ) = n <-> (♯ ` x ) = m ) )
7	6	anbi2d 464	. . . . . . . 8 |- ( n = m -> ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = n ) <-> ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = m ) ) )
8	7	imbi1d 231	. . . . . . 7 |- ( n = m -> ( ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = n ) -> ph ) <-> ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = m ) -> ph ) ) )
9	8	2ralbidv 2531	. . . . . 6 |- ( n = m -> ( A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = n ) -> ph ) <-> A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = m ) -> ph ) ) )
10		eqeq2 2216	. . . . . . . . 9 |- ( n = ( m + 1 ) -> ( (♯ ` x ) = n <-> (♯ ` x ) = ( m + 1 ) ) )
11	10	anbi2d 464	. . . . . . . 8 |- ( n = ( m + 1 ) -> ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = n ) <-> ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) )
12	11	imbi1d 231	. . . . . . 7 |- ( n = ( m + 1 ) -> ( ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = n ) -> ph ) <-> ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) -> ph ) ) )
13	12	2ralbidv 2531	. . . . . 6 |- ( n = ( m + 1 ) -> ( A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = n ) -> ph ) <-> A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) -> ph ) ) )
14		eqeq2 2216	. . . . . . . . 9 |- ( n = (♯ ` A ) -> ( (♯ ` x ) = n <-> (♯ ` x ) = (♯ ` A ) ) )
15	14	anbi2d 464	. . . . . . . 8 |- ( n = (♯ ` A ) -> ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = n ) <-> ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = (♯ ` A ) ) ) )
16	15	imbi1d 231	. . . . . . 7 |- ( n = (♯ ` A ) -> ( ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = n ) -> ph ) <-> ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = (♯ ` A ) ) -> ph ) ) )
17	16	2ralbidv 2531	. . . . . 6 |- ( n = (♯ ` A ) -> ( A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = n ) -> ph ) <-> A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = (♯ ` A ) ) -> ph ) ) )
18		eqeq1 2213	. . . . . . . . . . 11 |- ( (♯ ` x ) = 0 -> ( (♯ ` x ) = (♯ ` w ) <-> 0 = (♯ ` w ) ) )
19	18	adantl 277	. . . . . . . . . 10 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ (♯ ` x ) = 0 ) -> ( (♯ ` x ) = (♯ ` w ) <-> 0 = (♯ ` w ) ) )
20		eqcom 2208	. . . . . . . . . . . . . 14 |- ( 0 = (♯ ` w ) <-> (♯ ` w ) = 0 )
21		wrdfin 11030	. . . . . . . . . . . . . . 15 |- ( w e. Word Y -> w e. Fin )
22		fihasheq0 10955	. . . . . . . . . . . . . . 15 |- ( w e. Fin -> ( (♯ ` w ) = 0 <-> w = (/) ) )
23	21, 22	syl 14	. . . . . . . . . . . . . 14 |- ( w e. Word Y -> ( (♯ ` w ) = 0 <-> w = (/) ) )
24	20, 23	bitrid 192	. . . . . . . . . . . . 13 |- ( w e. Word Y -> ( 0 = (♯ ` w ) <-> w = (/) ) )
25		wrdfin 11030	. . . . . . . . . . . . . . . 16 |- ( x e. Word X -> x e. Fin )
26		fihasheq0 10955	. . . . . . . . . . . . . . . 16 |- ( x e. Fin -> ( (♯ ` x ) = 0 <-> x = (/) ) )
27	25, 26	syl 14	. . . . . . . . . . . . . . 15 |- ( x e. Word X -> ( (♯ ` x ) = 0 <-> x = (/) ) )
28		wrd2ind.6	. . . . . . . . . . . . . . . . 17 |- ps
29		wrd2ind.1	. . . . . . . . . . . . . . . . 17 |- ( ( x = (/) /\ w = (/) ) -> ( ph <-> ps ) )
30	28, 29	mpbiri 168	. . . . . . . . . . . . . . . 16 |- ( ( x = (/) /\ w = (/) ) -> ph )
31	30	ex 115	. . . . . . . . . . . . . . 15 |- ( x = (/) -> ( w = (/) -> ph ) )
32	27, 31	biimtrdi 163	. . . . . . . . . . . . . 14 |- ( x e. Word X -> ( (♯ ` x ) = 0 -> ( w = (/) -> ph ) ) )
33	32	com13 80	. . . . . . . . . . . . 13 |- ( w = (/) -> ( (♯ ` x ) = 0 -> ( x e. Word X -> ph ) ) )
34	24, 33	biimtrdi 163	. . . . . . . . . . . 12 |- ( w e. Word Y -> ( 0 = (♯ ` w ) -> ( (♯ ` x ) = 0 -> ( x e. Word X -> ph ) ) ) )
35	34	com24 87	. . . . . . . . . . 11 |- ( w e. Word Y -> ( x e. Word X -> ( (♯ ` x ) = 0 -> ( 0 = (♯ ` w ) -> ph ) ) ) )
36	35	imp31 256	. . . . . . . . . 10 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ (♯ ` x ) = 0 ) -> ( 0 = (♯ ` w ) -> ph ) )
37	19, 36	sylbid 150	. . . . . . . . 9 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ (♯ ` x ) = 0 ) -> ( (♯ ` x ) = (♯ ` w ) -> ph ) )
38	37	ex 115	. . . . . . . 8 |- ( ( w e. Word Y /\ x e. Word X ) -> ( (♯ ` x ) = 0 -> ( (♯ ` x ) = (♯ ` w ) -> ph ) ) )
39	38	impcomd 255	. . . . . . 7 |- ( ( w e. Word Y /\ x e. Word X ) -> ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = 0 ) -> ph ) )
40	39	rgen2 2593	. . . . . 6 |- A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = 0 ) -> ph )
41		fveq2 5588	. . . . . . . . . . . . 13 |- ( x = y -> (♯ ` x ) = (♯ ` y ) )
42		fveq2 5588	. . . . . . . . . . . . 13 |- ( w = u -> (♯ ` w ) = (♯ ` u ) )
43	41, 42	eqeqan12d 2222	. . . . . . . . . . . 12 |- ( ( x = y /\ w = u ) -> ( (♯ ` x ) = (♯ ` w ) <-> (♯ ` y ) = (♯ ` u ) ) )
44		fveqeq2 5597	. . . . . . . . . . . . 13 |- ( x = y -> ( (♯ ` x ) = m <-> (♯ ` y ) = m ) )
45	44	adantr 276	. . . . . . . . . . . 12 |- ( ( x = y /\ w = u ) -> ( (♯ ` x ) = m <-> (♯ ` y ) = m ) )
46	43, 45	anbi12d 473	. . . . . . . . . . 11 |- ( ( x = y /\ w = u ) -> ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = m ) <-> ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) ) )
47		wrd2ind.2	. . . . . . . . . . 11 |- ( ( x = y /\ w = u ) -> ( ph <-> ch ) )
48	46, 47	imbi12d 234	. . . . . . . . . 10 |- ( ( x = y /\ w = u ) -> ( ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = m ) -> ph ) <-> ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) )
49	48	ancoms 268	. . . . . . . . 9 |- ( ( w = u /\ x = y ) -> ( ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = m ) -> ph ) <-> ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) )
50	49	cbvraldva 2748	. . . . . . . 8 |- ( w = u -> ( A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = m ) -> ph ) <-> A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) )
51	50	cbvralvw 2743	. . . . . . 7 |- ( A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = m ) -> ph ) <-> A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) )
52		lencl 11015	. . . . . . . . . . . . . . . . . . . 20 |- ( w e. Word Y -> (♯ ` w ) e. NN0 )
53	52	nn0zd 9508	. . . . . . . . . . . . . . . . . . 19 |- ( w e. Word Y -> (♯ ` w ) e. ZZ )
54		1zzd 9414	. . . . . . . . . . . . . . . . . . 19 |- ( w e. Word Y -> 1 e. ZZ )
55	53, 54	zsubcld 9515	. . . . . . . . . . . . . . . . . 18 |- ( w e. Word Y -> ( (♯ ` w ) - 1 ) e. ZZ )
56		pfxclz 11150	. . . . . . . . . . . . . . . . . 18 |- ( ( w e. Word Y /\ ( (♯ ` w ) - 1 ) e. ZZ ) -> ( w prefix ( (♯ ` w ) - 1 ) ) e. Word Y )
57	55, 56	mpdan 421	. . . . . . . . . . . . . . . . 17 |- ( w e. Word Y -> ( w prefix ( (♯ ` w ) - 1 ) ) e. Word Y )
58	57	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( w e. Word Y /\ x e. Word X ) -> ( w prefix ( (♯ ` w ) - 1 ) ) e. Word Y )
59	58	ad2antrl 490	. . . . . . . . . . . . . . 15 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( w prefix ( (♯ ` w ) - 1 ) ) e. Word Y )
60		simprll 537	. . . . . . . . . . . . . . . 16 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> w e. Word Y )
61		eqeq1 2213	. . . . . . . . . . . . . . . . . . . . . 22 |- ( (♯ ` x ) = ( m + 1 ) -> ( (♯ ` x ) = (♯ ` w ) <-> ( m + 1 ) = (♯ ` w ) ) )
62		nn0p1nn 9349	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( m e. NN0 -> ( m + 1 ) e. NN )
63		eleq1 2269	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( (♯ ` w ) = ( m + 1 ) -> ( (♯ ` w ) e. NN <-> ( m + 1 ) e. NN ) )
64	63	eqcoms 2209	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( m + 1 ) = (♯ ` w ) -> ( (♯ ` w ) e. NN <-> ( m + 1 ) e. NN ) )
65	62, 64	imbitrrid 156	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( m + 1 ) = (♯ ` w ) -> ( m e. NN0 -> (♯ ` w ) e. NN ) )
66	61, 65	biimtrdi 163	. . . . . . . . . . . . . . . . . . . . 21 |- ( (♯ ` x ) = ( m + 1 ) -> ( (♯ ` x ) = (♯ ` w ) -> ( m e. NN0 -> (♯ ` w ) e. NN ) ) )
67	66	impcom 125	. . . . . . . . . . . . . . . . . . . 20 |- ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) -> ( m e. NN0 -> (♯ ` w ) e. NN ) )
68	67	adantl 277	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( m e. NN0 -> (♯ ` w ) e. NN ) )
69	68	impcom 125	. . . . . . . . . . . . . . . . . 18 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (♯ ` w ) e. NN )
70	69	nnge1d 9094	. . . . . . . . . . . . . . . . 17 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> 1 <_ (♯ ` w ) )
71		wrdlenge1n0 11044	. . . . . . . . . . . . . . . . . 18 |- ( w e. Word Y -> ( w =/= (/) <-> 1 <_ (♯ ` w ) ) )
72	60, 71	syl 14	. . . . . . . . . . . . . . . . 17 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( w =/= (/) <-> 1 <_ (♯ ` w ) ) )
73	70, 72	mpbird 167	. . . . . . . . . . . . . . . 16 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> w =/= (/) )
74		lswcl 11061	. . . . . . . . . . . . . . . 16 |- ( ( w e. Word Y /\ w =/= (/) ) -> (lastS ` w ) e. Y )
75	60, 73, 74	syl2anc 411	. . . . . . . . . . . . . . 15 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (lastS ` w ) e. Y )
76	59, 75	jca 306	. . . . . . . . . . . . . 14 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( ( w prefix ( (♯ ` w ) - 1 ) ) e. Word Y /\ (lastS ` w ) e. Y ) )
77		lencl 11015	. . . . . . . . . . . . . . . . . . 19 |- ( x e. Word X -> (♯ ` x ) e. NN0 )
78	77	nn0zd 9508	. . . . . . . . . . . . . . . . . 18 |- ( x e. Word X -> (♯ ` x ) e. ZZ )
79		1zzd 9414	. . . . . . . . . . . . . . . . . 18 |- ( x e. Word X -> 1 e. ZZ )
80	78, 79	zsubcld 9515	. . . . . . . . . . . . . . . . 17 |- ( x e. Word X -> ( (♯ ` x ) - 1 ) e. ZZ )
81		pfxclz 11150	. . . . . . . . . . . . . . . . 17 |- ( ( x e. Word X /\ ( (♯ ` x ) - 1 ) e. ZZ ) -> ( x prefix ( (♯ ` x ) - 1 ) ) e. Word X )
82	80, 81	mpdan 421	. . . . . . . . . . . . . . . 16 |- ( x e. Word X -> ( x prefix ( (♯ ` x ) - 1 ) ) e. Word X )
83	82	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( w e. Word Y /\ x e. Word X ) -> ( x prefix ( (♯ ` x ) - 1 ) ) e. Word X )
84	83	ad2antrl 490	. . . . . . . . . . . . . 14 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( x prefix ( (♯ ` x ) - 1 ) ) e. Word X )
85		simprlr 538	. . . . . . . . . . . . . . 15 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> x e. Word X )
86		eleq1 2269	. . . . . . . . . . . . . . . . . . . 20 |- ( (♯ ` x ) = ( m + 1 ) -> ( (♯ ` x ) e. NN <-> ( m + 1 ) e. NN ) )
87	62, 86	imbitrrid 156	. . . . . . . . . . . . . . . . . . 19 |- ( (♯ ` x ) = ( m + 1 ) -> ( m e. NN0 -> (♯ ` x ) e. NN ) )
88	87	ad2antll 491	. . . . . . . . . . . . . . . . . 18 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( m e. NN0 -> (♯ ` x ) e. NN ) )
89	88	impcom 125	. . . . . . . . . . . . . . . . 17 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (♯ ` x ) e. NN )
90	89	nnge1d 9094	. . . . . . . . . . . . . . . 16 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> 1 <_ (♯ ` x ) )
91		wrdlenge1n0 11044	. . . . . . . . . . . . . . . . 17 |- ( x e. Word X -> ( x =/= (/) <-> 1 <_ (♯ ` x ) ) )
92	85, 91	syl 14	. . . . . . . . . . . . . . . 16 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( x =/= (/) <-> 1 <_ (♯ ` x ) ) )
93	90, 92	mpbird 167	. . . . . . . . . . . . . . 15 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> x =/= (/) )
94		lswcl 11061	. . . . . . . . . . . . . . 15 |- ( ( x e. Word X /\ x =/= (/) ) -> (lastS ` x ) e. X )
95	85, 93, 94	syl2anc 411	. . . . . . . . . . . . . 14 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (lastS ` x ) e. X )
96	76, 84, 95	jca32 310	. . . . . . . . . . . . 13 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( ( ( w prefix ( (♯ ` w ) - 1 ) ) e. Word Y /\ (lastS ` w ) e. Y ) /\ ( ( x prefix ( (♯ ` x ) - 1 ) ) e. Word X /\ (lastS ` x ) e. X ) ) )
97	96	adantlr 477	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( ( ( w prefix ( (♯ ` w ) - 1 ) ) e. Word Y /\ (lastS ` w ) e. Y ) /\ ( ( x prefix ( (♯ ` x ) - 1 ) ) e. Word X /\ (lastS ` x ) e. X ) ) )
98		simprl 529	. . . . . . . . . . . . . 14 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( w e. Word Y /\ x e. Word X ) )
99		simplr 528	. . . . . . . . . . . . . 14 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) )
100		simprrl 539	. . . . . . . . . . . . . . . . 17 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (♯ ` x ) = (♯ ` w ) )
101	100	oveq1d 5971	. . . . . . . . . . . . . . . 16 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( (♯ ` x ) - 1 ) = ( (♯ ` w ) - 1 ) )
102		simprlr 538	. . . . . . . . . . . . . . . . 17 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> x e. Word X )
103		fzossfz 10303	. . . . . . . . . . . . . . . . . 18 |- ( 0..^ (♯ ` x ) ) C_ ( 0 ... (♯ ` x ) )
104		simprrr 540	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (♯ ` x ) = ( m + 1 ) )
105	62	ad2antrr 488	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( m + 1 ) e. NN )
106	104, 105	eqeltrd 2283	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (♯ ` x ) e. NN )
107		fzo0end 10369	. . . . . . . . . . . . . . . . . . 19 |- ( (♯ ` x ) e. NN -> ( (♯ ` x ) - 1 ) e. ( 0..^ (♯ ` x ) ) )
108	106, 107	syl 14	. . . . . . . . . . . . . . . . . 18 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( (♯ ` x ) - 1 ) e. ( 0..^ (♯ ` x ) ) )
109	103, 108	sselid 3195	. . . . . . . . . . . . . . . . 17 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( (♯ ` x ) - 1 ) e. ( 0 ... (♯ ` x ) ) )
110		pfxlen 11156	. . . . . . . . . . . . . . . . 17 |- ( ( x e. Word X /\ ( (♯ ` x ) - 1 ) e. ( 0 ... (♯ ` x ) ) ) -> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = ( (♯ ` x ) - 1 ) )
111	102, 109, 110	syl2anc 411	. . . . . . . . . . . . . . . 16 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = ( (♯ ` x ) - 1 ) )
112		simprll 537	. . . . . . . . . . . . . . . . 17 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> w e. Word Y )
113		oveq1 5963	. . . . . . . . . . . . . . . . . . . . . 22 |- ( (♯ ` w ) = (♯ ` x ) -> ( (♯ ` w ) - 1 ) = ( (♯ ` x ) - 1 ) )
114		oveq2 5964	. . . . . . . . . . . . . . . . . . . . . 22 |- ( (♯ ` w ) = (♯ ` x ) -> ( 0 ... (♯ ` w ) ) = ( 0 ... (♯ ` x ) ) )
115	113, 114	eleq12d 2277	. . . . . . . . . . . . . . . . . . . . 21 |- ( (♯ ` w ) = (♯ ` x ) -> ( ( (♯ ` w ) - 1 ) e. ( 0 ... (♯ ` w ) ) <-> ( (♯ ` x ) - 1 ) e. ( 0 ... (♯ ` x ) ) ) )
116	115	eqcoms 2209	. . . . . . . . . . . . . . . . . . . 20 |- ( (♯ ` x ) = (♯ ` w ) -> ( ( (♯ ` w ) - 1 ) e. ( 0 ... (♯ ` w ) ) <-> ( (♯ ` x ) - 1 ) e. ( 0 ... (♯ ` x ) ) ) )
117	116	adantr 276	. . . . . . . . . . . . . . . . . . 19 |- ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) -> ( ( (♯ ` w ) - 1 ) e. ( 0 ... (♯ ` w ) ) <-> ( (♯ ` x ) - 1 ) e. ( 0 ... (♯ ` x ) ) ) )
118	117	ad2antll 491	. . . . . . . . . . . . . . . . . 18 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( ( (♯ ` w ) - 1 ) e. ( 0 ... (♯ ` w ) ) <-> ( (♯ ` x ) - 1 ) e. ( 0 ... (♯ ` x ) ) ) )
119	109, 118	mpbird 167	. . . . . . . . . . . . . . . . 17 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( (♯ ` w ) - 1 ) e. ( 0 ... (♯ ` w ) ) )
120		pfxlen 11156	. . . . . . . . . . . . . . . . 17 |- ( ( w e. Word Y /\ ( (♯ ` w ) - 1 ) e. ( 0 ... (♯ ` w ) ) ) -> (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) = ( (♯ ` w ) - 1 ) )
121	112, 119, 120	syl2anc 411	. . . . . . . . . . . . . . . 16 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) = ( (♯ ` w ) - 1 ) )
122	101, 111, 121	3eqtr4d 2249	. . . . . . . . . . . . . . 15 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) )
123	104	oveq1d 5971	. . . . . . . . . . . . . . . 16 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( (♯ ` x ) - 1 ) = ( ( m + 1 ) - 1 ) )
124		nn0cn 9320	. . . . . . . . . . . . . . . . . 18 |- ( m e. NN0 -> m e. CC )
125	124	ad2antrr 488	. . . . . . . . . . . . . . . . 17 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> m e. CC )
126		ax-1cn 8033	. . . . . . . . . . . . . . . . 17 |- 1 e. CC
127		pncan 8293	. . . . . . . . . . . . . . . . 17 |- ( ( m e. CC /\ 1 e. CC ) -> ( ( m + 1 ) - 1 ) = m )
128	125, 126, 127	sylancl 413	. . . . . . . . . . . . . . . 16 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( ( m + 1 ) - 1 ) = m )
129	111, 123, 128	3eqtrd 2243	. . . . . . . . . . . . . . 15 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m )
130	122, 129	jca 306	. . . . . . . . . . . . . 14 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m ) )
131	83	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) -> ( x prefix ( (♯ ` x ) - 1 ) ) e. Word X )
132		fveq2 5588	. . . . . . . . . . . . . . . . . . . . . 22 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> (♯ ` y ) = (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) )
133		fveq2 5588	. . . . . . . . . . . . . . . . . . . . . 22 |- ( u = ( w prefix ( (♯ ` w ) - 1 ) ) -> (♯ ` u ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) )
134	132, 133	eqeqan12d 2222	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( y = ( x prefix ( (♯ ` x ) - 1 ) ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) -> ( (♯ ` y ) = (♯ ` u ) <-> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) )
135	134	expcom 116	. . . . . . . . . . . . . . . . . . . 20 |- ( u = ( w prefix ( (♯ ` w ) - 1 ) ) -> ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( (♯ ` y ) = (♯ ` u ) <-> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) ) )
136	135	adantl 277	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) -> ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( (♯ ` y ) = (♯ ` u ) <-> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) ) )
137	136	imp 124	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ y = ( x prefix ( (♯ ` x ) - 1 ) ) ) -> ( (♯ ` y ) = (♯ ` u ) <-> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) )
138		fveqeq2 5597	. . . . . . . . . . . . . . . . . . 19 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( (♯ ` y ) = m <-> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m ) )
139	138	adantl 277	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ y = ( x prefix ( (♯ ` x ) - 1 ) ) ) -> ( (♯ ` y ) = m <-> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m ) )
140	137, 139	anbi12d 473	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ y = ( x prefix ( (♯ ` x ) - 1 ) ) ) -> ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) <-> ( (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m ) ) )
141		vex 2776	. . . . . . . . . . . . . . . . . . . . 21 |- y e. _V
142		vex 2776	. . . . . . . . . . . . . . . . . . . . 21 |- u e. _V
143	141, 142, 47	sbc2ie 3074	. . . . . . . . . . . . . . . . . . . 20 |- ( [. y / x ]. [. u / w ]. ph <-> ch )
144	143	bicomi 132	. . . . . . . . . . . . . . . . . . 19 |- ( ch <-> [. y / x ]. [. u / w ]. ph )
145	144	a1i 9	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ y = ( x prefix ( (♯ ` x ) - 1 ) ) ) -> ( ch <-> [. y / x ]. [. u / w ]. ph ) )
146		simpr 110	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ y = ( x prefix ( (♯ ` x ) - 1 ) ) ) -> y = ( x prefix ( (♯ ` x ) - 1 ) ) )
147	146	sbceq1d 3007	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ y = ( x prefix ( (♯ ` x ) - 1 ) ) ) -> ( [. y / x ]. [. u / w ]. ph <-> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. u / w ]. ph ) )
148		dfsbcq 3004	. . . . . . . . . . . . . . . . . . . . 21 |- ( u = ( w prefix ( (♯ ` w ) - 1 ) ) -> ( [. u / w ]. ph <-> [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph ) )
149	148	sbcbidv 3061	. . . . . . . . . . . . . . . . . . . 20 |- ( u = ( w prefix ( (♯ ` w ) - 1 ) ) -> ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. u / w ]. ph <-> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph ) )
150	149	adantl 277	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) -> ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. u / w ]. ph <-> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph ) )
151	150	adantr 276	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ y = ( x prefix ( (♯ ` x ) - 1 ) ) ) -> ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. u / w ]. ph <-> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph ) )
152	145, 147, 151	3bitrd 214	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ y = ( x prefix ( (♯ ` x ) - 1 ) ) ) -> ( ch <-> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph ) )
153	140, 152	imbi12d 234	. . . . . . . . . . . . . . . 16 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ y = ( x prefix ( (♯ ` x ) - 1 ) ) ) -> ( ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) <-> ( ( (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m ) -> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph ) ) )
154	131, 153	rspcdv 2884	. . . . . . . . . . . . . . 15 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w prefix ( (♯ ` w ) - 1 ) ) ) -> ( A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) -> ( ( (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m ) -> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph ) ) )
155	58, 154	rspcimdv 2882	. . . . . . . . . . . . . 14 |- ( ( w e. Word Y /\ x e. Word X ) -> ( A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) -> ( ( (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) /\ (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m ) -> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph ) ) )
156	98, 99, 130, 155	syl3c 63	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph )
157	156, 122	jca 306	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) )
158		dfsbcq 3004	. . . . . . . . . . . . . . . 16 |- ( u = ( w prefix ( (♯ ` w ) - 1 ) ) -> ( [. u / w ]. [. y / x ]. ph <-> [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. [. y / x ]. ph ) )
159		sbccom 3078	. . . . . . . . . . . . . . . 16 |- ( [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. [. y / x ]. ph <-> [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph )
160	158, 159	bitrdi 196	. . . . . . . . . . . . . . 15 |- ( u = ( w prefix ( (♯ ` w ) - 1 ) ) -> ( [. u / w ]. [. y / x ]. ph <-> [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph ) )
161	133	eqeq2d 2218	. . . . . . . . . . . . . . 15 |- ( u = ( w prefix ( (♯ ` w ) - 1 ) ) -> ( (♯ ` y ) = (♯ ` u ) <-> (♯ ` y ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) )
162	160, 161	anbi12d 473	. . . . . . . . . . . . . 14 |- ( u = ( w prefix ( (♯ ` w ) - 1 ) ) -> ( ( [. u / w ]. [. y / x ]. ph /\ (♯ ` y ) = (♯ ` u ) ) <-> ( [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` y ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) ) )
163		oveq1 5963	. . . . . . . . . . . . . . 15 |- ( u = ( w prefix ( (♯ ` w ) - 1 ) ) -> ( u ++ <" s "> ) = ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" s "> ) )
164	163	sbceq1d 3007	. . . . . . . . . . . . . 14 |- ( u = ( w prefix ( (♯ ` w ) - 1 ) ) -> ( [. ( u ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph <-> [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) )
165	162, 164	imbi12d 234	. . . . . . . . . . . . 13 |- ( u = ( w prefix ( (♯ ` w ) - 1 ) ) -> ( ( ( [. u / w ]. [. y / x ]. ph /\ (♯ ` y ) = (♯ ` u ) ) -> [. ( u ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) <-> ( ( [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` y ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) -> [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) ) )
166		s1eq 11091	. . . . . . . . . . . . . . . . 17 |- ( s = (lastS ` w ) -> <" s "> = <" (lastS ` w ) "> )
167	166	oveq2d 5972	. . . . . . . . . . . . . . . 16 |- ( s = (lastS ` w ) -> ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" s "> ) = ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) )
168	167	sbceq1d 3007	. . . . . . . . . . . . . . 15 |- ( s = (lastS ` w ) -> ( [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph <-> [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) )
169	168	imbi2d 230	. . . . . . . . . . . . . 14 |- ( s = (lastS ` w ) -> ( ( ( [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` y ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) -> [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) <-> ( ( [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` y ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) -> [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) ) )
170		sbccom 3078	. . . . . . . . . . . . . . . 16 |- ( [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph <-> [. ( y ++ <" z "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph )
171	170	a1i 9	. . . . . . . . . . . . . . 15 |- ( s = (lastS ` w ) -> ( [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph <-> [. ( y ++ <" z "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) )
172	171	imbi2d 230	. . . . . . . . . . . . . 14 |- ( s = (lastS ` w ) -> ( ( ( [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` y ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) -> [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) <-> ( ( [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` y ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) -> [. ( y ++ <" z "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) ) )
173	169, 172	bitrd 188	. . . . . . . . . . . . 13 |- ( s = (lastS ` w ) -> ( ( ( [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` y ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) -> [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) <-> ( ( [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` y ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) -> [. ( y ++ <" z "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) ) )
174		dfsbcq 3004	. . . . . . . . . . . . . . 15 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph <-> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph ) )
175		fveqeq2 5597	. . . . . . . . . . . . . . 15 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( (♯ ` y ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) <-> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) )
176	174, 175	anbi12d 473	. . . . . . . . . . . . . 14 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( ( [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` y ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) <-> ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) ) )
177		oveq1 5963	. . . . . . . . . . . . . . 15 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( y ++ <" z "> ) = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) )
178	177	sbceq1d 3007	. . . . . . . . . . . . . 14 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( [. ( y ++ <" z "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph <-> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) )
179	176, 178	imbi12d 234	. . . . . . . . . . . . 13 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( ( ( [. y / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` y ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) -> [. ( y ++ <" z "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) <-> ( ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) -> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) ) )
180		s1eq 11091	. . . . . . . . . . . . . . . 16 |- ( z = (lastS ` x ) -> <" z "> = <" (lastS ` x ) "> )
181	180	oveq2d 5972	. . . . . . . . . . . . . . 15 |- ( z = (lastS ` x ) -> ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) )
182	181	sbceq1d 3007	. . . . . . . . . . . . . 14 |- ( z = (lastS ` x ) -> ( [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph <-> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) )
183	182	imbi2d 230	. . . . . . . . . . . . 13 |- ( z = (lastS ` x ) -> ( ( ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) -> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) <-> ( ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) -> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) ) )
184		simplr 528	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ (♯ ` y ) = (♯ ` u ) ) -> ( y e. Word X /\ z e. X ) )
185		simpll 527	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ (♯ ` y ) = (♯ ` u ) ) -> ( u e. Word Y /\ s e. Y ) )
186		simpr 110	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ (♯ ` y ) = (♯ ` u ) ) -> (♯ ` y ) = (♯ ` u ) )
187		wrd2ind.7	. . . . . . . . . . . . . . . . 17 |- ( ( ( y e. Word X /\ z e. X ) /\ ( u e. Word Y /\ s e. Y ) /\ (♯ ` y ) = (♯ ` u ) ) -> ( ch -> th ) )
188	184, 185, 186, 187	syl3anc 1250	. . . . . . . . . . . . . . . 16 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ (♯ ` y ) = (♯ ` u ) ) -> ( ch -> th ) )
189	47	ancoms 268	. . . . . . . . . . . . . . . . . 18 |- ( ( w = u /\ x = y ) -> ( ph <-> ch ) )
190	142, 141, 189	sbc2ie 3074	. . . . . . . . . . . . . . . . 17 |- ( [. u / w ]. [. y / x ]. ph <-> ch )
191	190	a1i 9	. . . . . . . . . . . . . . . 16 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ (♯ ` y ) = (♯ ` u ) ) -> ( [. u / w ]. [. y / x ]. ph <-> ch ) )
192		ccatws1cl 11104	. . . . . . . . . . . . . . . . . 18 |- ( ( u e. Word Y /\ s e. Y ) -> ( u ++ <" s "> ) e. Word Y )
193	192	ad2antrr 488	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ (♯ ` y ) = (♯ ` u ) ) -> ( u ++ <" s "> ) e. Word Y )
194		ccatws1cl 11104	. . . . . . . . . . . . . . . . . 18 |- ( ( y e. Word X /\ z e. X ) -> ( y ++ <" z "> ) e. Word X )
195	184, 194	syl 14	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ (♯ ` y ) = (♯ ` u ) ) -> ( y ++ <" z "> ) e. Word X )
196		nfv 1552	. . . . . . . . . . . . . . . . . 18 |- F/ w th
197		nfv 1552	. . . . . . . . . . . . . . . . . 18 |- F/ x th
198		nfv 1552	. . . . . . . . . . . . . . . . . 18 |- F/ w ( y ++ <" z "> ) e. Word X
199		wrd2ind.3	. . . . . . . . . . . . . . . . . . 19 |- ( ( x = ( y ++ <" z "> ) /\ w = ( u ++ <" s "> ) ) -> ( ph <-> th ) )
200	199	ancoms 268	. . . . . . . . . . . . . . . . . 18 |- ( ( w = ( u ++ <" s "> ) /\ x = ( y ++ <" z "> ) ) -> ( ph <-> th ) )
201	196, 197, 198, 200	sbc2iegf 3073	. . . . . . . . . . . . . . . . 17 |- ( ( ( u ++ <" s "> ) e. Word Y /\ ( y ++ <" z "> ) e. Word X ) -> ( [. ( u ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph <-> th ) )
202	193, 195, 201	syl2anc 411	. . . . . . . . . . . . . . . 16 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ (♯ ` y ) = (♯ ` u ) ) -> ( [. ( u ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph <-> th ) )
203	188, 191, 202	3imtr4d 203	. . . . . . . . . . . . . . 15 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ (♯ ` y ) = (♯ ` u ) ) -> ( [. u / w ]. [. y / x ]. ph -> [. ( u ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) )
204	203	ex 115	. . . . . . . . . . . . . 14 |- ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) -> ( (♯ ` y ) = (♯ ` u ) -> ( [. u / w ]. [. y / x ]. ph -> [. ( u ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) ) )
205	204	impcomd 255	. . . . . . . . . . . . 13 |- ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) -> ( ( [. u / w ]. [. y / x ]. ph /\ (♯ ` y ) = (♯ ` u ) ) -> [. ( u ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) )
206	165, 173, 179, 183, 205	vtocl4ga 2847	. . . . . . . . . . . 12 |- ( ( ( ( w prefix ( (♯ ` w ) - 1 ) ) e. Word Y /\ (lastS ` w ) e. Y ) /\ ( ( x prefix ( (♯ ` x ) - 1 ) ) e. Word X /\ (lastS ` x ) e. X ) ) -> ( ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. [. ( w prefix ( (♯ ` w ) - 1 ) ) / w ]. ph /\ (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = (♯ ` ( w prefix ( (♯ ` w ) - 1 ) ) ) ) -> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) )
207	97, 157, 206	sylc 62	. . . . . . . . . . 11 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph )
208		eqtr2 2225	. . . . . . . . . . . . . . . 16 |- ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) -> (♯ ` w ) = ( m + 1 ) )
209	208	ad2antll 491	. . . . . . . . . . . . . . 15 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (♯ ` w ) = ( m + 1 ) )
210	209, 105	eqeltrd 2283	. . . . . . . . . . . . . 14 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> (♯ ` w ) e. NN )
211	21	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( w e. Word Y /\ x e. Word X ) -> w e. Fin )
212	211	ad2antrl 490	. . . . . . . . . . . . . . 15 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> w e. Fin )
213		hashnncl 10957	. . . . . . . . . . . . . . 15 |- ( w e. Fin -> ( (♯ ` w ) e. NN <-> w =/= (/) ) )
214	212, 213	syl 14	. . . . . . . . . . . . . 14 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( (♯ ` w ) e. NN <-> w =/= (/) ) )
215	210, 214	mpbid 147	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> w =/= (/) )
216		pfxlswccat 11184	. . . . . . . . . . . . . 14 |- ( ( w e. Word Y /\ w =/= (/) ) -> ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) = w )
217	216	eqcomd 2212	. . . . . . . . . . . . 13 |- ( ( w e. Word Y /\ w =/= (/) ) -> w = ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) )
218	112, 215, 217	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> w = ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) )
219	25	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( w e. Word Y /\ x e. Word X ) -> x e. Fin )
220	219	ad2antrl 490	. . . . . . . . . . . . . . 15 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> x e. Fin )
221		hashnncl 10957	. . . . . . . . . . . . . . 15 |- ( x e. Fin -> ( (♯ ` x ) e. NN <-> x =/= (/) ) )
222	220, 221	syl 14	. . . . . . . . . . . . . 14 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( (♯ ` x ) e. NN <-> x =/= (/) ) )
223	106, 222	mpbid 147	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> x =/= (/) )
224		pfxlswccat 11184	. . . . . . . . . . . . . 14 |- ( ( x e. Word X /\ x =/= (/) ) -> ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) = x )
225	224	eqcomd 2212	. . . . . . . . . . . . 13 |- ( ( x e. Word X /\ x =/= (/) ) -> x = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) )
226	102, 223, 225	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> x = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) )
227		sbceq1a 3012	. . . . . . . . . . . . 13 |- ( w = ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) -> ( ph <-> [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) )
228		sbceq1a 3012	. . . . . . . . . . . . 13 |- ( x = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) -> ( [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph <-> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) )
229	227, 228	sylan9bb 462	. . . . . . . . . . . 12 |- ( ( w = ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) /\ x = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) ) -> ( ph <-> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) )
230	218, 226, 229	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ( ph <-> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. [. ( ( w prefix ( (♯ ` w ) - 1 ) ) ++ <" (lastS ` w ) "> ) / w ]. ph ) )
231	207, 230	mpbird 167	. . . . . . . . . 10 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) ) ) -> ph )
232	231	expr 375	. . . . . . . . 9 |- ( ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) /\ ( w e. Word Y /\ x e. Word X ) ) -> ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) -> ph ) )
233	232	ralrimivva 2589	. . . . . . . 8 |- ( ( m e. NN0 /\ A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) ) -> A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) -> ph ) )
234	233	ex 115	. . . . . . 7 |- ( m e. NN0 -> ( A. u e. Word Y A. y e. Word X ( ( (♯ ` y ) = (♯ ` u ) /\ (♯ ` y ) = m ) -> ch ) -> A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) -> ph ) ) )
235	51, 234	biimtrid 152	. . . . . 6 |- ( m e. NN0 -> ( A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = m ) -> ph ) -> A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = ( m + 1 ) ) -> ph ) ) )
236	5, 9, 13, 17, 40, 235	nn0ind 9502	. . . . 5 |- ( (♯ ` A ) e. NN0 -> A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = (♯ ` A ) ) -> ph ) )
237	1, 236	syl 14	. . . 4 |- ( A e. Word X -> A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = (♯ ` A ) ) -> ph ) )
238	237	3ad2ant1 1021	. . 3 |- ( ( A e. Word X /\ B e. Word Y /\ (♯ ` A ) = (♯ ` B ) ) -> A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = (♯ ` A ) ) -> ph ) )
239		fveq2 5588	. . . . . . . . 9 |- ( w = B -> (♯ ` w ) = (♯ ` B ) )
240	239	eqeq2d 2218	. . . . . . . 8 |- ( w = B -> ( (♯ ` x ) = (♯ ` w ) <-> (♯ ` x ) = (♯ ` B ) ) )
241	240	anbi1d 465	. . . . . . 7 |- ( w = B -> ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = (♯ ` A ) ) <-> ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) ) )
242		wrd2ind.5	. . . . . . 7 |- ( w = B -> ( ph <-> rh ) )
243	241, 242	imbi12d 234	. . . . . 6 |- ( w = B -> ( ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = (♯ ` A ) ) -> ph ) <-> ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) -> rh ) ) )
244	243	ralbidv 2507	. . . . 5 |- ( w = B -> ( A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = (♯ ` A ) ) -> ph ) <-> A. x e. Word X ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) -> rh ) ) )
245	244	rspcv 2877	. . . 4 |- ( B e. Word Y -> ( A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = (♯ ` A ) ) -> ph ) -> A. x e. Word X ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) -> rh ) ) )
246	245	3ad2ant2 1022	. . 3 |- ( ( A e. Word X /\ B e. Word Y /\ (♯ ` A ) = (♯ ` B ) ) -> ( A. w e. Word Y A. x e. Word X ( ( (♯ ` x ) = (♯ ` w ) /\ (♯ ` x ) = (♯ ` A ) ) -> ph ) -> A. x e. Word X ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) -> rh ) ) )
247	238, 246	mpd 13	. 2 |- ( ( A e. Word X /\ B e. Word Y /\ (♯ ` A ) = (♯ ` B ) ) -> A. x e. Word X ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) -> rh ) )
248		eqidd 2207	. 2 |- ( ( A e. Word X /\ B e. Word Y /\ (♯ ` A ) = (♯ ` B ) ) -> (♯ ` A ) = (♯ ` A ) )
249		fveqeq2 5597	. . . . . . . . . 10 |- ( x = A -> ( (♯ ` x ) = (♯ ` B ) <-> (♯ ` A ) = (♯ ` B ) ) )
250		fveqeq2 5597	. . . . . . . . . 10 |- ( x = A -> ( (♯ ` x ) = (♯ ` A ) <-> (♯ ` A ) = (♯ ` A ) ) )
251	249, 250	anbi12d 473	. . . . . . . . 9 |- ( x = A -> ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) <-> ( (♯ ` A ) = (♯ ` B ) /\ (♯ ` A ) = (♯ ` A ) ) ) )
252		wrd2ind.4	. . . . . . . . 9 |- ( x = A -> ( rh <-> ta ) )
253	251, 252	imbi12d 234	. . . . . . . 8 |- ( x = A -> ( ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) -> rh ) <-> ( ( (♯ ` A ) = (♯ ` B ) /\ (♯ ` A ) = (♯ ` A ) ) -> ta ) ) )
254	253	rspcv 2877	. . . . . . 7 |- ( A e. Word X -> ( A. x e. Word X ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) -> rh ) -> ( ( (♯ ` A ) = (♯ ` B ) /\ (♯ ` A ) = (♯ ` A ) ) -> ta ) ) )
255	254	com23 78	. . . . . 6 |- ( A e. Word X -> ( ( (♯ ` A ) = (♯ ` B ) /\ (♯ ` A ) = (♯ ` A ) ) -> ( A. x e. Word X ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) -> rh ) -> ta ) ) )
256	255	expd 258	. . . . 5 |- ( A e. Word X -> ( (♯ ` A ) = (♯ ` B ) -> ( (♯ ` A ) = (♯ ` A ) -> ( A. x e. Word X ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) -> rh ) -> ta ) ) ) )
257	256	com34 83	. . . 4 |- ( A e. Word X -> ( (♯ ` A ) = (♯ ` B ) -> ( A. x e. Word X ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) -> rh ) -> ( (♯ ` A ) = (♯ ` A ) -> ta ) ) ) )
258	257	imp 124	. . 3 |- ( ( A e. Word X /\ (♯ ` A ) = (♯ ` B ) ) -> ( A. x e. Word X ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) -> rh ) -> ( (♯ ` A ) = (♯ ` A ) -> ta ) ) )
259	258	3adant2 1019	. 2 |- ( ( A e. Word X /\ B e. Word Y /\ (♯ ` A ) = (♯ ` B ) ) -> ( A. x e. Word X ( ( (♯ ` x ) = (♯ ` B ) /\ (♯ ` x ) = (♯ ` A ) ) -> rh ) -> ( (♯ ` A ) = (♯ ` A ) -> ta ) ) )
260	247, 248, 259	mp2d 47	1 |- ( ( A e. Word X /\ B e. Word Y /\ (♯ ` A ) = (♯ ` B ) ) -> ta )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2177   =/= wne 2377   A.wral 2485   [.wsbc 3002   (/)c0 3464   class class class wbr 4050   ` cfv 5279  (class class class)co 5956   Fincfn 6839   CCcc 7938   0cc0 7940   1c1 7941   + caddc 7943   <_ cle 8123   - cmin 8258   NNcn 9051   NN0cn0 9310   ZZcz 9387   ...cfz 10145  ..^cfzo 10279  ♯chash 10937  Word cword 11011  lastSclsw 11055   ++ cconcat 11064   <"cs1 11087   prefix cpfx 11143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-mulrcl 8039  ax-addcom 8040  ax-mulcom 8041  ax-addass 8042  ax-mulass 8043  ax-distr 8044  ax-i2m1 8045  ax-0lt1 8046  ax-1rid 8047  ax-0id 8048  ax-rnegex 8049  ax-precex 8050  ax-cnre 8051  ax-pre-ltirr 8052  ax-pre-ltwlin 8053  ax-pre-lttrn 8054  ax-pre-apti 8055  ax-pre-ltadd 8056  ax-pre-mulgt0 8057

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-iord 4420  df-on 4422  df-ilim 4423  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-frec 6489  df-1o 6514  df-er 6632  df-en 6840  df-dom 6841  df-fin 6842  df-pnf 8124  df-mnf 8125  df-xr 8126  df-ltxr 8127  df-le 8128  df-sub 8260  df-neg 8261  df-reap 8663  df-ap 8670  df-inn 9052  df-n0 9311  df-z 9388  df-uz 9664  df-fz 10146  df-fzo 10280  df-ihash 10938  df-word 11012  df-lsw 11056  df-concat 11065  df-s1 11088  df-substr 11117  df-pfx 11144

This theorem is referenced by: (None)