Theorem wrdind 11193

Description: Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 12-Oct-2022.)

Hypotheses

Ref	Expression
wrdind.1	|- ( x = (/) -> ( ph <-> ps ) )
wrdind.2	|- ( x = y -> ( ph <-> ch ) )
wrdind.3	|- ( x = ( y ++ <" z "> ) -> ( ph <-> th ) )
wrdind.4	|- ( x = A -> ( ph <-> ta ) )
wrdind.5	|- ps
wrdind.6	|- ( ( y e. Word B /\ z e. B ) -> ( ch -> th ) )

Assertion

Ref	Expression
wrdind	|- ( A e. Word B -> ta )

Distinct variable groups:   x, A   x, y, z, B   ch, x   ph, y, z   ta, x   th, x

Allowed substitution hints:   ph( x)   ps( x, y, z)   ch( y, z)   th( y, z)   ta( y, z)   A( y, z)

Proof of Theorem wrdind

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

Step	Hyp	Ref	Expression
1		lencl 11015	. . 3 |- ( A e. Word B -> (♯ ` A ) e. NN0 )
2		eqeq2 2216	. . . . . 6 |- ( n = 0 -> ( (♯ ` x ) = n <-> (♯ ` x ) = 0 ) )
3	2	imbi1d 231	. . . . 5 |- ( n = 0 -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = 0 -> ph ) ) )
4	3	ralbidv 2507	. . . 4 |- ( n = 0 -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = 0 -> ph ) ) )
5		eqeq2 2216	. . . . . 6 |- ( n = m -> ( (♯ ` x ) = n <-> (♯ ` x ) = m ) )
6	5	imbi1d 231	. . . . 5 |- ( n = m -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = m -> ph ) ) )
7	6	ralbidv 2507	. . . 4 |- ( n = m -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = m -> ph ) ) )
8		eqeq2 2216	. . . . . 6 |- ( n = ( m + 1 ) -> ( (♯ ` x ) = n <-> (♯ ` x ) = ( m + 1 ) ) )
9	8	imbi1d 231	. . . . 5 |- ( n = ( m + 1 ) -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = ( m + 1 ) -> ph ) ) )
10	9	ralbidv 2507	. . . 4 |- ( n = ( m + 1 ) -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = ( m + 1 ) -> ph ) ) )
11		eqeq2 2216	. . . . . 6 |- ( n = (♯ ` A ) -> ( (♯ ` x ) = n <-> (♯ ` x ) = (♯ ` A ) ) )
12	11	imbi1d 231	. . . . 5 |- ( n = (♯ ` A ) -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = (♯ ` A ) -> ph ) ) )
13	12	ralbidv 2507	. . . 4 |- ( n = (♯ ` A ) -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = (♯ ` A ) -> ph ) ) )
14		wrdfin 11030	. . . . . . 7 |- ( x e. Word B -> x e. Fin )
15		fihasheq0 10955	. . . . . . 7 |- ( x e. Fin -> ( (♯ ` x ) = 0 <-> x = (/) ) )
16	14, 15	syl 14	. . . . . 6 |- ( x e. Word B -> ( (♯ ` x ) = 0 <-> x = (/) ) )
17		wrdind.5	. . . . . . 7 |- ps
18		wrdind.1	. . . . . . 7 |- ( x = (/) -> ( ph <-> ps ) )
19	17, 18	mpbiri 168	. . . . . 6 |- ( x = (/) -> ph )
20	16, 19	biimtrdi 163	. . . . 5 |- ( x e. Word B -> ( (♯ ` x ) = 0 -> ph ) )
21	20	rgen 2560	. . . 4 |- A. x e. Word B ( (♯ ` x ) = 0 -> ph )
22		fveqeq2 5597	. . . . . . 7 |- ( x = y -> ( (♯ ` x ) = m <-> (♯ ` y ) = m ) )
23		wrdind.2	. . . . . . 7 |- ( x = y -> ( ph <-> ch ) )
24	22, 23	imbi12d 234	. . . . . 6 |- ( x = y -> ( ( (♯ ` x ) = m -> ph ) <-> ( (♯ ` y ) = m -> ch ) ) )
25	24	cbvralvw 2743	. . . . 5 |- ( A. x e. Word B ( (♯ ` x ) = m -> ph ) <-> A. y e. Word B ( (♯ ` y ) = m -> ch ) )
26		simprl 529	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> x e. Word B )
27		fzossfz 10303	. . . . . . . . . . . . . 14 |- ( 0..^ (♯ ` x ) ) C_ ( 0 ... (♯ ` x ) )
28		simprr 531	. . . . . . . . . . . . . . . 16 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> (♯ ` x ) = ( m + 1 ) )
29		nn0p1nn 9349	. . . . . . . . . . . . . . . . 17 |- ( m e. NN0 -> ( m + 1 ) e. NN )
30	29	ad2antrr 488	. . . . . . . . . . . . . . . 16 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( m + 1 ) e. NN )
31	28, 30	eqeltrd 2283	. . . . . . . . . . . . . . 15 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> (♯ ` x ) e. NN )
32		fzo0end 10369	. . . . . . . . . . . . . . 15 |- ( (♯ ` x ) e. NN -> ( (♯ ` x ) - 1 ) e. ( 0..^ (♯ ` x ) ) )
33	31, 32	syl 14	. . . . . . . . . . . . . 14 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( (♯ ` x ) - 1 ) e. ( 0..^ (♯ ` x ) ) )
34	27, 33	sselid 3195	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( (♯ ` x ) - 1 ) e. ( 0 ... (♯ ` x ) ) )
35		pfxlen 11156	. . . . . . . . . . . . 13 |- ( ( x e. Word B /\ ( (♯ ` x ) - 1 ) e. ( 0 ... (♯ ` x ) ) ) -> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = ( (♯ ` x ) - 1 ) )
36	26, 34, 35	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = ( (♯ ` x ) - 1 ) )
37	28	oveq1d 5971	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( (♯ ` x ) - 1 ) = ( ( m + 1 ) - 1 ) )
38		nn0cn 9320	. . . . . . . . . . . . . 14 |- ( m e. NN0 -> m e. CC )
39	38	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> m e. CC )
40		ax-1cn 8033	. . . . . . . . . . . . 13 |- 1 e. CC
41		pncan 8293	. . . . . . . . . . . . 13 |- ( ( m e. CC /\ 1 e. CC ) -> ( ( m + 1 ) - 1 ) = m )
42	39, 40, 41	sylancl 413	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( ( m + 1 ) - 1 ) = m )
43	36, 37, 42	3eqtrd 2243	. . . . . . . . . . 11 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m )
44		fveqeq2 5597	. . . . . . . . . . . . 13 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( (♯ ` y ) = m <-> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m ) )
45		vex 2776	. . . . . . . . . . . . . . 15 |- y e. _V
46	45, 23	sbcie 3037	. . . . . . . . . . . . . 14 |- ( [. y / x ]. ph <-> ch )
47		dfsbcq 3004	. . . . . . . . . . . . . 14 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( [. y / x ]. ph <-> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. ph ) )
48	46, 47	bitr3id 194	. . . . . . . . . . . . 13 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( ch <-> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. ph ) )
49	44, 48	imbi12d 234	. . . . . . . . . . . 12 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( ( (♯ ` y ) = m -> ch ) <-> ( (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m -> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. ph ) ) )
50		simplr 528	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> A. y e. Word B ( (♯ ` y ) = m -> ch ) )
51		lencl 11015	. . . . . . . . . . . . . . . 16 |- ( x e. Word B -> (♯ ` x ) e. NN0 )
52	51	nn0zd 9508	. . . . . . . . . . . . . . 15 |- ( x e. Word B -> (♯ ` x ) e. ZZ )
53		1zzd 9414	. . . . . . . . . . . . . . 15 |- ( x e. Word B -> 1 e. ZZ )
54	52, 53	zsubcld 9515	. . . . . . . . . . . . . 14 |- ( x e. Word B -> ( (♯ ` x ) - 1 ) e. ZZ )
55		pfxclz 11150	. . . . . . . . . . . . . 14 |- ( ( x e. Word B /\ ( (♯ ` x ) - 1 ) e. ZZ ) -> ( x prefix ( (♯ ` x ) - 1 ) ) e. Word B )
56	54, 55	mpdan 421	. . . . . . . . . . . . 13 |- ( x e. Word B -> ( x prefix ( (♯ ` x ) - 1 ) ) e. Word B )
57	56	ad2antrl 490	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( x prefix ( (♯ ` x ) - 1 ) ) e. Word B )
58	49, 50, 57	rspcdva 2886	. . . . . . . . . . 11 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m -> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. ph ) )
59	43, 58	mpd 13	. . . . . . . . . 10 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. ph )
60	31	nnge1d 9094	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> 1 <_ (♯ ` x ) )
61		wrdlenge1n0 11044	. . . . . . . . . . . . . 14 |- ( x e. Word B -> ( x =/= (/) <-> 1 <_ (♯ ` x ) ) )
62	61	ad2antrl 490	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( x =/= (/) <-> 1 <_ (♯ ` x ) ) )
63	60, 62	mpbird 167	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> x =/= (/) )
64		lswcl 11061	. . . . . . . . . . . 12 |- ( ( x e. Word B /\ x =/= (/) ) -> (lastS ` x ) e. B )
65	26, 63, 64	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> (lastS ` x ) e. B )
66		oveq1 5963	. . . . . . . . . . . . . 14 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( y ++ <" z "> ) = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) )
67	66	sbceq1d 3007	. . . . . . . . . . . . 13 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( [. ( y ++ <" z "> ) / x ]. ph <-> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) / x ]. ph ) )
68	47, 67	imbi12d 234	. . . . . . . . . . . 12 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( ( [. y / x ]. ph -> [. ( y ++ <" z "> ) / x ]. ph ) <-> ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. ph -> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) / x ]. ph ) ) )
69		s1eq 11091	. . . . . . . . . . . . . . 15 |- ( z = (lastS ` x ) -> <" z "> = <" (lastS ` x ) "> )
70	69	oveq2d 5972	. . . . . . . . . . . . . 14 |- ( z = (lastS ` x ) -> ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) )
71	70	sbceq1d 3007	. . . . . . . . . . . . 13 |- ( z = (lastS ` x ) -> ( [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) / x ]. ph <-> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. ph ) )
72	71	imbi2d 230	. . . . . . . . . . . 12 |- ( z = (lastS ` x ) -> ( ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. ph -> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) / x ]. ph ) <-> ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. ph -> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. ph ) ) )
73		wrdind.6	. . . . . . . . . . . . 13 |- ( ( y e. Word B /\ z e. B ) -> ( ch -> th ) )
74	46	a1i 9	. . . . . . . . . . . . 13 |- ( ( y e. Word B /\ z e. B ) -> ( [. y / x ]. ph <-> ch ) )
75		ccatws1cl 11104	. . . . . . . . . . . . . 14 |- ( ( y e. Word B /\ z e. B ) -> ( y ++ <" z "> ) e. Word B )
76		wrdind.3	. . . . . . . . . . . . . . 15 |- ( x = ( y ++ <" z "> ) -> ( ph <-> th ) )
77	76	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ( y e. Word B /\ z e. B ) /\ x = ( y ++ <" z "> ) ) -> ( ph <-> th ) )
78	75, 77	sbcied 3039	. . . . . . . . . . . . 13 |- ( ( y e. Word B /\ z e. B ) -> ( [. ( y ++ <" z "> ) / x ]. ph <-> th ) )
79	73, 74, 78	3imtr4d 203	. . . . . . . . . . . 12 |- ( ( y e. Word B /\ z e. B ) -> ( [. y / x ]. ph -> [. ( y ++ <" z "> ) / x ]. ph ) )
80	68, 72, 79	vtocl2ga 2843	. . . . . . . . . . 11 |- ( ( ( x prefix ( (♯ ` x ) - 1 ) ) e. Word B /\ (lastS ` x ) e. B ) -> ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. ph -> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. ph ) )
81	57, 65, 80	syl2anc 411	. . . . . . . . . 10 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( [. ( x prefix ( (♯ ` x ) - 1 ) ) / x ]. ph -> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. ph ) )
82	59, 81	mpd 13	. . . . . . . . 9 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. ph )
83	14	ad2antrl 490	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> x e. Fin )
84		hashnncl 10957	. . . . . . . . . . . . 13 |- ( x e. Fin -> ( (♯ ` x ) e. NN <-> x =/= (/) ) )
85	83, 84	syl 14	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( (♯ ` x ) e. NN <-> x =/= (/) ) )
86	31, 85	mpbid 147	. . . . . . . . . . 11 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> x =/= (/) )
87		pfxlswccat 11184	. . . . . . . . . . . 12 |- ( ( x e. Word B /\ x =/= (/) ) -> ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) = x )
88	87	eqcomd 2212	. . . . . . . . . . 11 |- ( ( x e. Word B /\ x =/= (/) ) -> x = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) )
89	26, 86, 88	syl2anc 411	. . . . . . . . . 10 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> x = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) )
90		sbceq1a 3012	. . . . . . . . . 10 |- ( x = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) -> ( ph <-> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. ph ) )
91	89, 90	syl 14	. . . . . . . . 9 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> ( ph <-> [. ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) / x ]. ph ) )
92	82, 91	mpbird 167	. . . . . . . 8 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> ph )
93	92	expr 375	. . . . . . 7 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ x e. Word B ) -> ( (♯ ` x ) = ( m + 1 ) -> ph ) )
94	93	ralrimiva 2580	. . . . . 6 |- ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) -> A. x e. Word B ( (♯ ` x ) = ( m + 1 ) -> ph ) )
95	94	ex 115	. . . . 5 |- ( m e. NN0 -> ( A. y e. Word B ( (♯ ` y ) = m -> ch ) -> A. x e. Word B ( (♯ ` x ) = ( m + 1 ) -> ph ) ) )
96	25, 95	biimtrid 152	. . . 4 |- ( m e. NN0 -> ( A. x e. Word B ( (♯ ` x ) = m -> ph ) -> A. x e. Word B ( (♯ ` x ) = ( m + 1 ) -> ph ) ) )
97	4, 7, 10, 13, 21, 96	nn0ind 9502	. . 3 |- ( (♯ ` A ) e. NN0 -> A. x e. Word B ( (♯ ` x ) = (♯ ` A ) -> ph ) )
98	1, 97	syl 14	. 2 |- ( A e. Word B -> A. x e. Word B ( (♯ ` x ) = (♯ ` A ) -> ph ) )
99		eqidd 2207	. 2 |- ( A e. Word B -> (♯ ` A ) = (♯ ` A ) )
100		fveqeq2 5597	. . . 4 |- ( x = A -> ( (♯ ` x ) = (♯ ` A ) <-> (♯ ` A ) = (♯ ` A ) ) )
101		wrdind.4	. . . 4 |- ( x = A -> ( ph <-> ta ) )
102	100, 101	imbi12d 234	. . 3 |- ( x = A -> ( ( (♯ ` x ) = (♯ ` A ) -> ph ) <-> ( (♯ ` A ) = (♯ ` A ) -> ta ) ) )
103	102	rspcv 2877	. 2 |- ( A e. Word B -> ( A. x e. Word B ( (♯ ` x ) = (♯ ` A ) -> ph ) -> ( (♯ ` A ) = (♯ ` A ) -> ta ) ) )
104	98, 99, 103	mp2d 47	1 |- ( A e. Word B -> ta )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = 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)