Theorem wrdind 11307

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 11121	. . 3 |- ( A e. Word B -> (♯ ` A ) e. NN0 )
2		eqeq2 2241	. . . . . 6 |- ( n = 0 -> ( (♯ ` x ) = n <-> (♯ ` x ) = 0 ) )
3	2	imbi1d 231	. . . . 5 |- ( n = 0 -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = 0 -> ph ) ) )
4	3	ralbidv 2532	. . . 4 |- ( n = 0 -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = 0 -> ph ) ) )
5		eqeq2 2241	. . . . . 6 |- ( n = m -> ( (♯ ` x ) = n <-> (♯ ` x ) = m ) )
6	5	imbi1d 231	. . . . 5 |- ( n = m -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = m -> ph ) ) )
7	6	ralbidv 2532	. . . 4 |- ( n = m -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = m -> ph ) ) )
8		eqeq2 2241	. . . . . 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 2532	. . . 4 |- ( n = ( m + 1 ) -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = ( m + 1 ) -> ph ) ) )
11		eqeq2 2241	. . . . . 6 |- ( n = (♯ ` A ) -> ( (♯ ` x ) = n <-> (♯ ` x ) = (♯ ` A ) ) )
12	11	imbi1d 231	. . . . 5 |- ( n = (♯ ` A ) -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = (♯ ` A ) -> ph ) ) )
13	12	ralbidv 2532	. . . 4 |- ( n = (♯ ` A ) -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = (♯ ` A ) -> ph ) ) )
14		wrdfin 11136	. . . . . . 7 |- ( x e. Word B -> x e. Fin )
15		fihasheq0 11056	. . . . . . 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 2585	. . . 4 |- A. x e. Word B ( (♯ ` x ) = 0 -> ph )
22		fveqeq2 5648	. . . . . . 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 2771	. . . . 5 |- ( A. x e. Word B ( (♯ ` x ) = m -> ph ) <-> A. y e. Word B ( (♯ ` y ) = m -> ch ) )
26		simprl 531	. . . . . . . . . . . . 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 10401	. . . . . . . . . . . . . 14 |- ( 0..^ (♯ ` x ) ) C_ ( 0 ... (♯ ` x ) )
28		simprr 533	. . . . . . . . . . . . . . . 16 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> (♯ ` x ) = ( m + 1 ) )
29		nn0p1nn 9441	. . . . . . . . . . . . . . . . 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 2308	. . . . . . . . . . . . . . 15 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> (♯ ` x ) e. NN )
32		fzo0end 10469	. . . . . . . . . . . . . . 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 3225	. . . . . . . . . . . . 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 11270	. . . . . . . . . . . . 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 6033	. . . . . . . . . . . 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 9412	. . . . . . . . . . . . . 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 8125	. . . . . . . . . . . . 13 |- 1 e. CC
41		pncan 8385	. . . . . . . . . . . . 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 2268	. . . . . . . . . . 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 5648	. . . . . . . . . . . . 13 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( (♯ ` y ) = m <-> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m ) )
45		vex 2805	. . . . . . . . . . . . . . 15 |- y e. _V
46	45, 23	sbcie 3066	. . . . . . . . . . . . . 14 |- ( [. y / x ]. ph <-> ch )
47		dfsbcq 3033	. . . . . . . . . . . . . 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 529	. . . . . . . . . . . 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 11121	. . . . . . . . . . . . . . . 16 |- ( x e. Word B -> (♯ ` x ) e. NN0 )
52	51	nn0zd 9600	. . . . . . . . . . . . . . 15 |- ( x e. Word B -> (♯ ` x ) e. ZZ )
53		1zzd 9506	. . . . . . . . . . . . . . 15 |- ( x e. Word B -> 1 e. ZZ )
54	52, 53	zsubcld 9607	. . . . . . . . . . . . . 14 |- ( x e. Word B -> ( (♯ ` x ) - 1 ) e. ZZ )
55		pfxclz 11264	. . . . . . . . . . . . . 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 2915	. . . . . . . . . . 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 9186	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> 1 <_ (♯ ` x ) )
61		wrdlenge1n0 11151	. . . . . . . . . . . . . 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 11168	. . . . . . . . . . . 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 6025	. . . . . . . . . . . . . 14 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( y ++ <" z "> ) = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) )
67	66	sbceq1d 3036	. . . . . . . . . . . . 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 11200	. . . . . . . . . . . . . . 15 |- ( z = (lastS ` x ) -> <" z "> = <" (lastS ` x ) "> )
70	69	oveq2d 6034	. . . . . . . . . . . . . 14 |- ( z = (lastS ` x ) -> ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) )
71	70	sbceq1d 3036	. . . . . . . . . . . . 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 11213	. . . . . . . . . . . . . 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 3068	. . . . . . . . . . . . 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 2872	. . . . . . . . . . 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 11058	. . . . . . . . . . . . 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 11298	. . . . . . . . . . . 12 |- ( ( x e. Word B /\ x =/= (/) ) -> ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) = x )
88	87	eqcomd 2237	. . . . . . . . . . 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 3041	. . . . . . . . . 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 2605	. . . . . 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 9594	. . 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 2232	. 2 |- ( A e. Word B -> (♯ ` A ) = (♯ ` A ) )
100		fveqeq2 5648	. . . 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 2906	. 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 1397   e. wcel 2202   =/= wne 2402   A.wral 2510   [.wsbc 3031   (/)c0 3494   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   Fincfn 6909   CCcc 8030   0cc0 8032   1c1 8033   + caddc 8035   <_ cle 8215   - cmin 8350   NNcn 9143   NN0cn0 9402   ZZcz 9479   ...cfz 10243  ..^cfzo 10377  ♯chash 11038  Word cword 11117  lastSclsw 11162   ++ cconcat 11171   <"cs1 11196   prefix cpfx 11257

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378  df-ihash 11039  df-word 11118  df-lsw 11163  df-concat 11172  df-s1 11197  df-substr 11231  df-pfx 11258

This theorem is referenced by: (None)