Theorem wrd2ind 13477

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

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 13324	. . . . 5 |- ( A e. Word X -> ( # ` A ) e. NN0 )
2		eqeq2 2633	. . . . . . . . 9 |- ( n = 0 -> ( ( # ` x ) = n <-> ( # ` x ) = 0 ) )
3	2	anbi2d 740	. . . . . . . 8 |- ( n = 0 -> ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = n ) <-> ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = 0 ) ) )
4	3	imbi1d 331	. . . . . . 7 |- ( n = 0 -> ( ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = n ) -> ph ) <-> ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = 0 ) -> ph ) ) )
5	4	2ralbidv 2989	. . . . . 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 2633	. . . . . . . . 9 |- ( n = m -> ( ( # ` x ) = n <-> ( # ` x ) = m ) )
7	6	anbi2d 740	. . . . . . . 8 |- ( n = m -> ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = n ) <-> ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = m ) ) )
8	7	imbi1d 331	. . . . . . 7 |- ( n = m -> ( ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = n ) -> ph ) <-> ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = m ) -> ph ) ) )
9	8	2ralbidv 2989	. . . . . 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 2633	. . . . . . . . 9 |- ( n = ( m + 1 ) -> ( ( # ` x ) = n <-> ( # ` x ) = ( m + 1 ) ) )
11	10	anbi2d 740	. . . . . . . 8 |- ( n = ( m + 1 ) -> ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = n ) <-> ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) )
12	11	imbi1d 331	. . . . . . 7 |- ( n = ( m + 1 ) -> ( ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = n ) -> ph ) <-> ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) -> ph ) ) )
13	12	2ralbidv 2989	. . . . . 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 2633	. . . . . . . . 9 |- ( n = ( # ` A ) -> ( ( # ` x ) = n <-> ( # ` x ) = ( # ` A ) ) )
15	14	anbi2d 740	. . . . . . . 8 |- ( n = ( # ` A ) -> ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = n ) <-> ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( # ` A ) ) ) )
16	15	imbi1d 331	. . . . . . 7 |- ( n = ( # ` A ) -> ( ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = n ) -> ph ) <-> ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( # ` A ) ) -> ph ) ) )
17	16	2ralbidv 2989	. . . . . 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 2626	. . . . . . . . . . . 12 |- ( ( # ` x ) = 0 -> ( ( # ` x ) = ( # ` w ) <-> 0 = ( # ` w ) ) )
19	18	adantl 482	. . . . . . . . . . 11 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ ( # ` x ) = 0 ) -> ( ( # ` x ) = ( # ` w ) <-> 0 = ( # ` w ) ) )
20		eqcom 2629	. . . . . . . . . . . . . . 15 |- ( 0 = ( # ` w ) <-> ( # ` w ) = 0 )
21		hasheq0 13154	. . . . . . . . . . . . . . 15 |- ( w e. Word Y -> ( ( # ` w ) = 0 <-> w = (/) ) )
22	20, 21	syl5bb 272	. . . . . . . . . . . . . 14 |- ( w e. Word Y -> ( 0 = ( # ` w ) <-> w = (/) ) )
23		hasheq0 13154	. . . . . . . . . . . . . . . 16 |- ( x e. Word X -> ( ( # ` x ) = 0 <-> x = (/) ) )
24		wrd2ind.6	. . . . . . . . . . . . . . . . . 18 |- ps
25		wrd2ind.1	. . . . . . . . . . . . . . . . . 18 |- ( ( x = (/) /\ w = (/) ) -> ( ph <-> ps ) )
26	24, 25	mpbiri 248	. . . . . . . . . . . . . . . . 17 |- ( ( x = (/) /\ w = (/) ) -> ph )
27	26	ex 450	. . . . . . . . . . . . . . . 16 |- ( x = (/) -> ( w = (/) -> ph ) )
28	23, 27	syl6bi 243	. . . . . . . . . . . . . . 15 |- ( x e. Word X -> ( ( # ` x ) = 0 -> ( w = (/) -> ph ) ) )
29	28	com13 88	. . . . . . . . . . . . . 14 |- ( w = (/) -> ( ( # ` x ) = 0 -> ( x e. Word X -> ph ) ) )
30	22, 29	syl6bi 243	. . . . . . . . . . . . 13 |- ( w e. Word Y -> ( 0 = ( # ` w ) -> ( ( # ` x ) = 0 -> ( x e. Word X -> ph ) ) ) )
31	30	com24 95	. . . . . . . . . . . 12 |- ( w e. Word Y -> ( x e. Word X -> ( ( # ` x ) = 0 -> ( 0 = ( # ` w ) -> ph ) ) ) )
32	31	imp31 448	. . . . . . . . . . 11 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ ( # ` x ) = 0 ) -> ( 0 = ( # ` w ) -> ph ) )
33	19, 32	sylbid 230	. . . . . . . . . 10 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ ( # ` x ) = 0 ) -> ( ( # ` x ) = ( # ` w ) -> ph ) )
34	33	ex 450	. . . . . . . . 9 |- ( ( w e. Word Y /\ x e. Word X ) -> ( ( # ` x ) = 0 -> ( ( # ` x ) = ( # ` w ) -> ph ) ) )
35	34	com23 86	. . . . . . . 8 |- ( ( w e. Word Y /\ x e. Word X ) -> ( ( # ` x ) = ( # ` w ) -> ( ( # ` x ) = 0 -> ph ) ) )
36	35	impd 447	. . . . . . 7 |- ( ( w e. Word Y /\ x e. Word X ) -> ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = 0 ) -> ph ) )
37	36	rgen2 2975	. . . . . 6 |- A. w e. Word Y A. x e. Word X ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = 0 ) -> ph )
38		fveq2 6191	. . . . . . . . . . . . 13 |- ( x = y -> ( # ` x ) = ( # ` y ) )
39		fveq2 6191	. . . . . . . . . . . . 13 |- ( w = u -> ( # ` w ) = ( # ` u ) )
40	38, 39	eqeqan12d 2638	. . . . . . . . . . . 12 |- ( ( x = y /\ w = u ) -> ( ( # ` x ) = ( # ` w ) <-> ( # ` y ) = ( # ` u ) ) )
41	38	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( x = y -> ( ( # ` x ) = m <-> ( # ` y ) = m ) )
42	41	adantr 481	. . . . . . . . . . . 12 |- ( ( x = y /\ w = u ) -> ( ( # ` x ) = m <-> ( # ` y ) = m ) )
43	40, 42	anbi12d 747	. . . . . . . . . . 11 |- ( ( x = y /\ w = u ) -> ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = m ) <-> ( ( # ` y ) = ( # ` u ) /\ ( # ` y ) = m ) ) )
44		wrd2ind.2	. . . . . . . . . . 11 |- ( ( x = y /\ w = u ) -> ( ph <-> ch ) )
45	43, 44	imbi12d 334	. . . . . . . . . 10 |- ( ( x = y /\ w = u ) -> ( ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = m ) -> ph ) <-> ( ( ( # ` y ) = ( # ` u ) /\ ( # ` y ) = m ) -> ch ) ) )
46	45	ancoms 469	. . . . . . . . 9 |- ( ( w = u /\ x = y ) -> ( ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = m ) -> ph ) <-> ( ( ( # ` y ) = ( # ` u ) /\ ( # ` y ) = m ) -> ch ) ) )
47	46	cbvraldva 3177	. . . . . . . 8 |- ( w = u -> ( A. x e. Word X ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = m ) -> ph ) <-> A. y e. Word X ( ( ( # ` y ) = ( # ` u ) /\ ( # ` y ) = m ) -> ch ) ) )
48	47	cbvralv 3171	. . . . . . 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 ) )
49		swrdcl 13419	. . . . . . . . . . . . . . . . 17 |- ( w e. Word Y -> ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) e. Word Y )
50	49	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( w e. Word Y /\ x e. Word X ) -> ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) e. Word Y )
51	50	ad2antrl 764	. . . . . . . . . . . . . . 15 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) e. Word Y )
52		simprll 802	. . . . . . . . . . . . . . . 16 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> w e. Word Y )
53		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` x ) = ( m + 1 ) -> ( ( # ` x ) = ( # ` w ) <-> ( m + 1 ) = ( # ` w ) ) )
54		nn0p1nn 11332	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( m e. NN0 -> ( m + 1 ) e. NN )
55		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( # ` w ) = ( m + 1 ) -> ( ( # ` w ) e. NN <-> ( m + 1 ) e. NN ) )
56	55	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( m + 1 ) = ( # ` w ) -> ( ( # ` w ) e. NN <-> ( m + 1 ) e. NN ) )
57	54, 56	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( m + 1 ) = ( # ` w ) -> ( m e. NN0 -> ( # ` w ) e. NN ) )
58	53, 57	syl6bi 243	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` x ) = ( m + 1 ) -> ( ( # ` x ) = ( # ` w ) -> ( m e. NN0 -> ( # ` w ) e. NN ) ) )
59	58	impcom 446	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) -> ( m e. NN0 -> ( # ` w ) e. NN ) )
60	59	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) -> ( m e. NN0 -> ( # ` w ) e. NN ) )
61	60	impcom 446	. . . . . . . . . . . . . . . . . 18 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> ( # ` w ) e. NN )
62	61	nnge1d 11063	. . . . . . . . . . . . . . . . 17 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> 1 <_ ( # ` w ) )
63		wrdlenge1n0 13340	. . . . . . . . . . . . . . . . . 18 |- ( w e. Word Y -> ( w =/= (/) <-> 1 <_ ( # ` w ) ) )
64	52, 63	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> ( w =/= (/) <-> 1 <_ ( # ` w ) ) )
65	62, 64	mpbird 247	. . . . . . . . . . . . . . . 16 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> w =/= (/) )
66		lswcl 13355	. . . . . . . . . . . . . . . 16 |- ( ( w e. Word Y /\ w =/= (/) ) -> ( lastS ` w ) e. Y )
67	52, 65, 66	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> ( lastS ` w ) e. Y )
68	51, 67	jca 554	. . . . . . . . . . . . . 14 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) e. Word Y /\ ( lastS ` w ) e. Y ) )
69		swrdcl 13419	. . . . . . . . . . . . . . . 16 |- ( x e. Word X -> ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) e. Word X )
70	69	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( w e. Word Y /\ x e. Word X ) -> ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) e. Word X )
71	70	ad2antrl 764	. . . . . . . . . . . . . 14 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) e. Word X )
72		simprlr 803	. . . . . . . . . . . . . . 15 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> x e. Word X )
73		eleq1 2689	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` x ) = ( m + 1 ) -> ( ( # ` x ) e. NN <-> ( m + 1 ) e. NN ) )
74	54, 73	syl5ibr 236	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` x ) = ( m + 1 ) -> ( m e. NN0 -> ( # ` x ) e. NN ) )
75	74	ad2antll 765	. . . . . . . . . . . . . . . . . 18 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) -> ( m e. NN0 -> ( # ` x ) e. NN ) )
76	75	impcom 446	. . . . . . . . . . . . . . . . 17 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> ( # ` x ) e. NN )
77	76	nnge1d 11063	. . . . . . . . . . . . . . . 16 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> 1 <_ ( # ` x ) )
78		wrdlenge1n0 13340	. . . . . . . . . . . . . . . . 17 |- ( x e. Word X -> ( x =/= (/) <-> 1 <_ ( # ` x ) ) )
79	72, 78	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> ( x =/= (/) <-> 1 <_ ( # ` x ) ) )
80	77, 79	mpbird 247	. . . . . . . . . . . . . . 15 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> x =/= (/) )
81		lswcl 13355	. . . . . . . . . . . . . . 15 |- ( ( x e. Word X /\ x =/= (/) ) -> ( lastS ` x ) e. X )
82	72, 80, 81	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> ( lastS ` x ) e. X )
83	68, 71, 82	jca32 558	. . . . . . . . . . . . 13 |- ( ( m e. NN0 /\ ( ( w e. Word Y /\ x e. Word X ) /\ ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) ) ) -> ( ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) e. Word Y /\ ( lastS ` w ) e. Y ) /\ ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) e. Word X /\ ( lastS ` x ) e. X ) ) )
84	83	adantlr 751	. . . . . . . . . . . 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 substr <. 0 , ( ( # ` w ) - 1 ) >. ) e. Word Y /\ ( lastS ` w ) e. Y ) /\ ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) e. Word X /\ ( lastS ` x ) e. X ) ) )
85		simprl 794	. . . . . . . . . . . . . 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 ) )
86		simplr 792	. . . . . . . . . . . . . 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 ) )
87		simprrl 804	. . . . . . . . . . . . . . . . 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 ) )
88	87	oveq1d 6665	. . . . . . . . . . . . . . . 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 ) )
89		simprlr 803	. . . . . . . . . . . . . . . . 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 )
90		fzossfz 12488	. . . . . . . . . . . . . . . . . 18 |- ( 0..^ ( # ` x ) ) C_ ( 0 ... ( # ` x ) )
91		simprrr 805	. . . . . . . . . . . . . . . . . . . 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 ) )
92	54	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 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 )
93	91, 92	eqeltrd 2701	. . . . . . . . . . . . . . . . . . 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 )
94		fzo0end 12560	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` x ) e. NN -> ( ( # ` x ) - 1 ) e. ( 0..^ ( # ` x ) ) )
95	93, 94	syl 17	. . . . . . . . . . . . . . . . . 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 ) ) )
96	90, 95	sseldi 3601	. . . . . . . . . . . . . . . . 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 ) ) )
97		swrd0len 13422	. . . . . . . . . . . . . . . . 17 |- ( ( x e. Word X /\ ( ( # ` x ) - 1 ) e. ( 0 ... ( # ` x ) ) ) -> ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( ( # ` x ) - 1 ) )
98	89, 96, 97	syl2anc 693	. . . . . . . . . . . . . . . 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 substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( ( # ` x ) - 1 ) )
99		simprll 802	. . . . . . . . . . . . . . . . 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 )
100		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` w ) = ( # ` x ) -> ( ( # ` w ) - 1 ) = ( ( # ` x ) - 1 ) )
101		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` w ) = ( # ` x ) -> ( 0 ... ( # ` w ) ) = ( 0 ... ( # ` x ) ) )
102	100, 101	eleq12d 2695	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` w ) = ( # ` x ) -> ( ( ( # ` w ) - 1 ) e. ( 0 ... ( # ` w ) ) <-> ( ( # ` x ) - 1 ) e. ( 0 ... ( # ` x ) ) ) )
103	102	eqcoms 2630	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` x ) = ( # ` w ) -> ( ( ( # ` w ) - 1 ) e. ( 0 ... ( # ` w ) ) <-> ( ( # ` x ) - 1 ) e. ( 0 ... ( # ` x ) ) ) )
104	103	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) -> ( ( ( # ` w ) - 1 ) e. ( 0 ... ( # ` w ) ) <-> ( ( # ` x ) - 1 ) e. ( 0 ... ( # ` x ) ) ) )
105	104	ad2antll 765	. . . . . . . . . . . . . . . . . 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 ) ) ) )
106	96, 105	mpbird 247	. . . . . . . . . . . . . . . . 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 ) ) )
107		swrd0len 13422	. . . . . . . . . . . . . . . . 17 |- ( ( w e. Word Y /\ ( ( # ` w ) - 1 ) e. ( 0 ... ( # ` w ) ) ) -> ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) = ( ( # ` w ) - 1 ) )
108	99, 106, 107	syl2anc 693	. . . . . . . . . . . . . . . 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 substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) = ( ( # ` w ) - 1 ) )
109	88, 98, 108	3eqtr4d 2666	. . . . . . . . . . . . . . 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 substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) )
110	91	oveq1d 6665	. . . . . . . . . . . . . . . 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 ) )
111		nn0cn 11302	. . . . . . . . . . . . . . . . . 18 |- ( m e. NN0 -> m e. CC )
112	111	ad2antrr 762	. . . . . . . . . . . . . . . . 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 )
113		ax-1cn 9994	. . . . . . . . . . . . . . . . 17 |- 1 e. CC
114		pncan 10287	. . . . . . . . . . . . . . . . 17 |- ( ( m e. CC /\ 1 e. CC ) -> ( ( m + 1 ) - 1 ) = m )
115	112, 113, 114	sylancl 694	. . . . . . . . . . . . . . . 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 )
116	98, 110, 115	3eqtrd 2660	. . . . . . . . . . . . . . 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 substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = m )
117	109, 116	jca 554	. . . . . . . . . . . . . 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 substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = m ) )
118	70	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) -> ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) e. Word X )
119		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( # ` y ) = ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) )
120		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 |- ( u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) -> ( # ` u ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) )
121	119, 120	eqeqan12d 2638	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) -> ( ( # ` y ) = ( # ` u ) <-> ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) )
122	121	expcom 451	. . . . . . . . . . . . . . . . . . . 20 |- ( u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) -> ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( ( # ` y ) = ( # ` u ) <-> ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) ) )
123	122	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) -> ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( ( # ` y ) = ( # ` u ) <-> ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) ) )
124	123	imp 445	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) -> ( ( # ` y ) = ( # ` u ) <-> ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) )
125	119	eqeq1d 2624	. . . . . . . . . . . . . . . . . . 19 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( ( # ` y ) = m <-> ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = m ) )
126	125	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) -> ( ( # ` y ) = m <-> ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = m ) )
127	124, 126	anbi12d 747	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) -> ( ( ( # ` y ) = ( # ` u ) /\ ( # ` y ) = m ) <-> ( ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = m ) ) )
128		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 |- y e. _V
129		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 |- u e. _V
130	128, 129, 44	sbc2ie 3505	. . . . . . . . . . . . . . . . . . . 20 |- ( [. y / x ]. [. u / w ]. ph <-> ch )
131	130	bicomi 214	. . . . . . . . . . . . . . . . . . 19 |- ( ch <-> [. y / x ]. [. u / w ]. ph )
132	131	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) -> ( ch <-> [. y / x ]. [. u / w ]. ph ) )
133		simpr 477	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) -> y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) )
134	133	sbceq1d 3440	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) -> ( [. y / x ]. [. u / w ]. ph <-> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. u / w ]. ph ) )
135		dfsbcq 3437	. . . . . . . . . . . . . . . . . . . . 21 |- ( u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) -> ( [. u / w ]. ph <-> [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph ) )
136	135	sbcbidv 3490	. . . . . . . . . . . . . . . . . . . 20 |- ( u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) -> ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. u / w ]. ph <-> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph ) )
137	136	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) -> ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. u / w ]. ph <-> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph ) )
138	137	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) -> ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. u / w ]. ph <-> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph ) )
139	132, 134, 138	3bitrd 294	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) -> ( ch <-> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph ) )
140	127, 139	imbi12d 334	. . . . . . . . . . . . . . . 16 |- ( ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) -> ( ( ( ( # ` y ) = ( # ` u ) /\ ( # ` y ) = m ) -> ch ) <-> ( ( ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = m ) -> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph ) ) )
141	118, 140	rspcdv 3312	. . . . . . . . . . . . . . 15 |- ( ( ( w e. Word Y /\ x e. Word X ) /\ u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) -> ( A. y e. Word X ( ( ( # ` y ) = ( # ` u ) /\ ( # ` y ) = m ) -> ch ) -> ( ( ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = m ) -> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph ) ) )
142	50, 141	rspcimdv 3310	. . . . . . . . . . . . . 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 substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) /\ ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = m ) -> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph ) ) )
143	85, 86, 117, 142	syl3c 66	. . . . . . . . . . . . 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 substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph )
144	143, 109	jca 554	. . . . . . . . . . . 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 substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) )
145		dfsbcq 3437	. . . . . . . . . . . . . . . 16 |- ( u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) -> ( [. u / w ]. [. y / x ]. ph <-> [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. [. y / x ]. ph ) )
146		sbccom 3509	. . . . . . . . . . . . . . . 16 |- ( [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. [. y / x ]. ph <-> [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph )
147	145, 146	syl6bb 276	. . . . . . . . . . . . . . 15 |- ( u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) -> ( [. u / w ]. [. y / x ]. ph <-> [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph ) )
148	120	eqeq2d 2632	. . . . . . . . . . . . . . 15 |- ( u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) -> ( ( # ` y ) = ( # ` u ) <-> ( # ` y ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) )
149	147, 148	anbi12d 747	. . . . . . . . . . . . . 14 |- ( u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) -> ( ( [. u / w ]. [. y / x ]. ph /\ ( # ` y ) = ( # ` u ) ) <-> ( [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` y ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) ) )
150		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) -> ( u ++ <" s "> ) = ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" s "> ) )
151	150	sbceq1d 3440	. . . . . . . . . . . . . 14 |- ( u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) -> ( [. ( u ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph <-> [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) )
152	149, 151	imbi12d 334	. . . . . . . . . . . . 13 |- ( u = ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) -> ( ( ( [. u / w ]. [. y / x ]. ph /\ ( # ` y ) = ( # ` u ) ) -> [. ( u ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) <-> ( ( [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` y ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) -> [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) ) )
153		s1eq 13380	. . . . . . . . . . . . . . . . 17 |- ( s = ( lastS ` w ) -> <" s "> = <" ( lastS ` w ) "> )
154	153	oveq2d 6666	. . . . . . . . . . . . . . . 16 |- ( s = ( lastS ` w ) -> ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" s "> ) = ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) )
155	154	sbceq1d 3440	. . . . . . . . . . . . . . 15 |- ( s = ( lastS ` w ) -> ( [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph <-> [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) )
156	155	imbi2d 330	. . . . . . . . . . . . . 14 |- ( s = ( lastS ` w ) -> ( ( ( [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` y ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) -> [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) <-> ( ( [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` y ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) -> [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) ) )
157		sbccom 3509	. . . . . . . . . . . . . . . 16 |- ( [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph <-> [. ( y ++ <" z "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph )
158	157	a1i 11	. . . . . . . . . . . . . . 15 |- ( s = ( lastS ` w ) -> ( [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph <-> [. ( y ++ <" z "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) )
159	158	imbi2d 330	. . . . . . . . . . . . . 14 |- ( s = ( lastS ` w ) -> ( ( ( [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` y ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) -> [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) <-> ( ( [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` y ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) -> [. ( y ++ <" z "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) ) )
160	156, 159	bitrd 268	. . . . . . . . . . . . 13 |- ( s = ( lastS ` w ) -> ( ( ( [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` y ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) -> [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph ) <-> ( ( [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` y ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) -> [. ( y ++ <" z "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) ) )
161		dfsbcq 3437	. . . . . . . . . . . . . . 15 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph <-> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph ) )
162	119	eqeq1d 2624	. . . . . . . . . . . . . . 15 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( ( # ` y ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) <-> ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) )
163	161, 162	anbi12d 747	. . . . . . . . . . . . . 14 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( ( [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` y ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) <-> ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) ) )
164		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( y ++ <" z "> ) = ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" z "> ) )
165	164	sbceq1d 3440	. . . . . . . . . . . . . 14 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( [. ( y ++ <" z "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph <-> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" z "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) )
166	163, 165	imbi12d 334	. . . . . . . . . . . . 13 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( ( ( [. y / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` y ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) -> [. ( y ++ <" z "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) <-> ( ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) -> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" z "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) ) )
167		s1eq 13380	. . . . . . . . . . . . . . . 16 |- ( z = ( lastS ` x ) -> <" z "> = <" ( lastS ` x ) "> )
168	167	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( z = ( lastS ` x ) -> ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" z "> ) = ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) )
169	168	sbceq1d 3440	. . . . . . . . . . . . . 14 |- ( z = ( lastS ` x ) -> ( [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" z "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph <-> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) )
170	169	imbi2d 330	. . . . . . . . . . . . 13 |- ( z = ( lastS ` x ) -> ( ( ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) -> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" z "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) <-> ( ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) -> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) ) )
171		simplr 792	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ ( # ` y ) = ( # ` u ) ) -> ( y e. Word X /\ z e. X ) )
172		simpll 790	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ ( # ` y ) = ( # ` u ) ) -> ( u e. Word Y /\ s e. Y ) )
173		simpr 477	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ ( # ` y ) = ( # ` u ) ) -> ( # ` y ) = ( # ` u ) )
174		wrd2ind.7	. . . . . . . . . . . . . . . . . 18 |- ( ( ( y e. Word X /\ z e. X ) /\ ( u e. Word Y /\ s e. Y ) /\ ( # ` y ) = ( # ` u ) ) -> ( ch -> th ) )
175	171, 172, 173, 174	syl3anc 1326	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( u e. Word Y /\ s e. Y ) /\ ( y e. Word X /\ z e. X ) ) /\ ( # ` y ) = ( # ` u ) ) -> ( ch -> th ) )
176	44	ancoms 469	. . . . . . . . . . . . . . . . . 18 |- ( ( w = u /\ x = y ) -> ( ph <-> ch ) )
177	129, 128, 176	sbc2ie 3505	. . . . . . . . . . . . . . . . 17 |- ( [. u / w ]. [. y / x ]. ph <-> ch )
178		ovex 6678	. . . . . . . . . . . . . . . . . 18 |- ( u ++ <" s "> ) e. _V
179		ovex 6678	. . . . . . . . . . . . . . . . . 18 |- ( y ++ <" z "> ) e. _V
180		wrd2ind.3	. . . . . . . . . . . . . . . . . . 19 |- ( ( x = ( y ++ <" z "> ) /\ w = ( u ++ <" s "> ) ) -> ( ph <-> th ) )
181	180	ancoms 469	. . . . . . . . . . . . . . . . . 18 |- ( ( w = ( u ++ <" s "> ) /\ x = ( y ++ <" z "> ) ) -> ( ph <-> th ) )
182	178, 179, 181	sbc2ie 3505	. . . . . . . . . . . . . . . . 17 |- ( [. ( u ++ <" s "> ) / w ]. [. ( y ++ <" z "> ) / x ]. ph <-> th )
183	175, 177, 182	3imtr4g 285	. . . . . . . . . . . . . . . 16 |- ( ( ( ( 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 ) )
184	183	ex 450	. . . . . . . . . . . . . . 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 ) ) )
185	184	com23 86	. . . . . . . . . . . . . 14 |- ( ( ( 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 ) ) )
186	185	impd 447	. . . . . . . . . . . . 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 ) )
187	152, 160, 166, 170, 186	vtocl4ga 3278	. . . . . . . . . . . 12 |- ( ( ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) e. Word Y /\ ( lastS ` w ) e. Y ) /\ ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) e. Word X /\ ( lastS ` x ) e. X ) ) -> ( ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. [. ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) / w ]. ph /\ ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( # ` ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ) ) -> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) )
188	84, 144, 187	sylc 65	. . . . . . . . . . 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 substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph )
189		eqtr2 2642	. . . . . . . . . . . . . . . 16 |- ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( m + 1 ) ) -> ( # ` w ) = ( m + 1 ) )
190	189	ad2antll 765	. . . . . . . . . . . . . . 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 ) )
191	190, 92	eqeltrd 2701	. . . . . . . . . . . . . 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 )
192		wrdfin 13323	. . . . . . . . . . . . . . . . 17 |- ( w e. Word Y -> w e. Fin )
193	192	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( w e. Word Y /\ x e. Word X ) -> w e. Fin )
194	193	ad2antrl 764	. . . . . . . . . . . . . . 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 )
195		hashnncl 13157	. . . . . . . . . . . . . . 15 |- ( w e. Fin -> ( ( # ` w ) e. NN <-> w =/= (/) ) )
196	194, 195	syl 17	. . . . . . . . . . . . . 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 =/= (/) ) )
197	191, 196	mpbid 222	. . . . . . . . . . . . 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 =/= (/) )
198		swrdccatwrd 13468	. . . . . . . . . . . . . 14 |- ( ( w e. Word Y /\ w =/= (/) ) -> ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) = w )
199	198	eqcomd 2628	. . . . . . . . . . . . 13 |- ( ( w e. Word Y /\ w =/= (/) ) -> w = ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) )
200	99, 197, 199	syl2anc 693	. . . . . . . . . . . 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 substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) )
201		wrdfin 13323	. . . . . . . . . . . . . . . . 17 |- ( x e. Word X -> x e. Fin )
202	201	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( w e. Word Y /\ x e. Word X ) -> x e. Fin )
203	202	ad2antrl 764	. . . . . . . . . . . . . . 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 )
204		hashnncl 13157	. . . . . . . . . . . . . . 15 |- ( x e. Fin -> ( ( # ` x ) e. NN <-> x =/= (/) ) )
205	203, 204	syl 17	. . . . . . . . . . . . . 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 =/= (/) ) )
206	93, 205	mpbid 222	. . . . . . . . . . . . 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 =/= (/) )
207		swrdccatwrd 13468	. . . . . . . . . . . . . 14 |- ( ( x e. Word X /\ x =/= (/) ) -> ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) = x )
208	207	eqcomd 2628	. . . . . . . . . . . . 13 |- ( ( x e. Word X /\ x =/= (/) ) -> x = ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) )
209	89, 206, 208	syl2anc 693	. . . . . . . . . . . 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 substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) )
210		sbceq1a 3446	. . . . . . . . . . . . 13 |- ( w = ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) -> ( ph <-> [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) )
211		sbceq1a 3446	. . . . . . . . . . . . 13 |- ( x = ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) -> ( [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph <-> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) )
212	210, 211	sylan9bb 736	. . . . . . . . . . . 12 |- ( ( w = ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) /\ x = ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) ) -> ( ph <-> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) )
213	200, 209, 212	syl2anc 693	. . . . . . . . . . 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 substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. [. ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) / w ]. ph ) )
214	188, 213	mpbird 247	. . . . . . . . . 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 )
215	214	expr 643	. . . . . . . . 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 ) )
216	215	ralrimivva 2971	. . . . . . . 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 ) )
217	216	ex 450	. . . . . . 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 ) ) )
218	48, 217	syl5bi 232	. . . . . 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 ) ) )
219	5, 9, 13, 17, 37, 218	nn0ind 11472	. . . . 5 |- ( ( # ` A ) e. NN0 -> A. w e. Word Y A. x e. Word X ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( # ` A ) ) -> ph ) )
220	1, 219	syl 17	. . . 4 |- ( A e. Word X -> A. w e. Word Y A. x e. Word X ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( # ` A ) ) -> ph ) )
221	220	3ad2ant1 1082	. . 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 ) )
222		fveq2 6191	. . . . . . . . 9 |- ( w = B -> ( # ` w ) = ( # ` B ) )
223	222	eqeq2d 2632	. . . . . . . 8 |- ( w = B -> ( ( # ` x ) = ( # ` w ) <-> ( # ` x ) = ( # ` B ) ) )
224	223	anbi1d 741	. . . . . . 7 |- ( w = B -> ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( # ` A ) ) <-> ( ( # ` x ) = ( # ` B ) /\ ( # ` x ) = ( # ` A ) ) ) )
225		wrd2ind.5	. . . . . . 7 |- ( w = B -> ( ph <-> rh ) )
226	224, 225	imbi12d 334	. . . . . 6 |- ( w = B -> ( ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( # ` A ) ) -> ph ) <-> ( ( ( # ` x ) = ( # ` B ) /\ ( # ` x ) = ( # ` A ) ) -> rh ) ) )
227	226	ralbidv 2986	. . . . 5 |- ( w = B -> ( A. x e. Word X ( ( ( # ` x ) = ( # ` w ) /\ ( # ` x ) = ( # ` A ) ) -> ph ) <-> A. x e. Word X ( ( ( # ` x ) = ( # ` B ) /\ ( # ` x ) = ( # ` A ) ) -> rh ) ) )
228	227	rspcv 3305	. . . 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 ) ) )
229	228	3ad2ant2 1083	. . 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 ) ) )
230	221, 229	mpd 15	. 2 |- ( ( A e. Word X /\ B e. Word Y /\ ( # ` A ) = ( # ` B ) ) -> A. x e. Word X ( ( ( # ` x ) = ( # ` B ) /\ ( # ` x ) = ( # ` A ) ) -> rh ) )
231		eqidd 2623	. 2 |- ( ( A e. Word X /\ B e. Word Y /\ ( # ` A ) = ( # ` B ) ) -> ( # ` A ) = ( # ` A ) )
232		fveq2 6191	. . . . . . . . . . 11 |- ( x = A -> ( # ` x ) = ( # ` A ) )
233	232	eqeq1d 2624	. . . . . . . . . 10 |- ( x = A -> ( ( # ` x ) = ( # ` B ) <-> ( # ` A ) = ( # ` B ) ) )
234	232	eqeq1d 2624	. . . . . . . . . 10 |- ( x = A -> ( ( # ` x ) = ( # ` A ) <-> ( # ` A ) = ( # ` A ) ) )
235	233, 234	anbi12d 747	. . . . . . . . 9 |- ( x = A -> ( ( ( # ` x ) = ( # ` B ) /\ ( # ` x ) = ( # ` A ) ) <-> ( ( # ` A ) = ( # ` B ) /\ ( # ` A ) = ( # ` A ) ) ) )
236		wrd2ind.4	. . . . . . . . 9 |- ( x = A -> ( rh <-> ta ) )
237	235, 236	imbi12d 334	. . . . . . . 8 |- ( x = A -> ( ( ( ( # ` x ) = ( # ` B ) /\ ( # ` x ) = ( # ` A ) ) -> rh ) <-> ( ( ( # ` A ) = ( # ` B ) /\ ( # ` A ) = ( # ` A ) ) -> ta ) ) )
238	237	rspcv 3305	. . . . . . 7 |- ( A e. Word X -> ( A. x e. Word X ( ( ( # ` x ) = ( # ` B ) /\ ( # ` x ) = ( # ` A ) ) -> rh ) -> ( ( ( # ` A ) = ( # ` B ) /\ ( # ` A ) = ( # ` A ) ) -> ta ) ) )
239	238	com23 86	. . . . . 6 |- ( A e. Word X -> ( ( ( # ` A ) = ( # ` B ) /\ ( # ` A ) = ( # ` A ) ) -> ( A. x e. Word X ( ( ( # ` x ) = ( # ` B ) /\ ( # ` x ) = ( # ` A ) ) -> rh ) -> ta ) ) )
240	239	expd 452	. . . . 5 |- ( A e. Word X -> ( ( # ` A ) = ( # ` B ) -> ( ( # ` A ) = ( # ` A ) -> ( A. x e. Word X ( ( ( # ` x ) = ( # ` B ) /\ ( # ` x ) = ( # ` A ) ) -> rh ) -> ta ) ) ) )
241	240	com34 91	. . . 4 |- ( A e. Word X -> ( ( # ` A ) = ( # ` B ) -> ( A. x e. Word X ( ( ( # ` x ) = ( # ` B ) /\ ( # ` x ) = ( # ` A ) ) -> rh ) -> ( ( # ` A ) = ( # ` A ) -> ta ) ) ) )
242	241	imp 445	. . 3 |- ( ( A e. Word X /\ ( # ` A ) = ( # ` B ) ) -> ( A. x e. Word X ( ( ( # ` x ) = ( # ` B ) /\ ( # ` x ) = ( # ` A ) ) -> rh ) -> ( ( # ` A ) = ( # ` A ) -> ta ) ) )
243	242	3adant2 1080	. 2 |- ( ( A e. Word X /\ B e. Word Y /\ ( # ` A ) = ( # ` B ) ) -> ( A. x e. Word X ( ( ( # ` x ) = ( # ` B ) /\ ( # ` x ) = ( # ` A ) ) -> rh ) -> ( ( # ` A ) = ( # ` A ) -> ta ) ) )
244	230, 231, 243	mp2d 49	1 |- ( ( A e. Word X /\ B e. Word Y /\ ( # ` A ) = ( # ` B ) ) -> ta )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [.wsbc 3435   (/)c0 3915   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294   substr csubstr 13295

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-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-oadd 7564  df-er 7742  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-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303

This theorem is referenced by:  gsmsymgreq  17852