Theorem wrdind 11240

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 11062	. . 3 |- ( A e. Word B -> (♯ ` A ) e. NN0 )
2		eqeq2 2239	. . . . . 6 |- ( n = 0 -> ( (♯ ` x ) = n <-> (♯ ` x ) = 0 ) )
3	2	imbi1d 231	. . . . 5 |- ( n = 0 -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = 0 -> ph ) ) )
4	3	ralbidv 2530	. . . 4 |- ( n = 0 -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = 0 -> ph ) ) )
5		eqeq2 2239	. . . . . 6 |- ( n = m -> ( (♯ ` x ) = n <-> (♯ ` x ) = m ) )
6	5	imbi1d 231	. . . . 5 |- ( n = m -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = m -> ph ) ) )
7	6	ralbidv 2530	. . . 4 |- ( n = m -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = m -> ph ) ) )
8		eqeq2 2239	. . . . . 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 2530	. . . 4 |- ( n = ( m + 1 ) -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = ( m + 1 ) -> ph ) ) )
11		eqeq2 2239	. . . . . 6 |- ( n = (♯ ` A ) -> ( (♯ ` x ) = n <-> (♯ ` x ) = (♯ ` A ) ) )
12	11	imbi1d 231	. . . . 5 |- ( n = (♯ ` A ) -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = (♯ ` A ) -> ph ) ) )
13	12	ralbidv 2530	. . . 4 |- ( n = (♯ ` A ) -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = (♯ ` A ) -> ph ) ) )
14		wrdfin 11077	. . . . . . 7 |- ( x e. Word B -> x e. Fin )
15		fihasheq0 11002	. . . . . . 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 2583	. . . 4 |- A. x e. Word B ( (♯ ` x ) = 0 -> ph )
22		fveqeq2 5632	. . . . . . 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 2769	. . . . 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 10350	. . . . . . . . . . . . . 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 9396	. . . . . . . . . . . . . . . . 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 2306	. . . . . . . . . . . . . . 15 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> (♯ ` x ) e. NN )
32		fzo0end 10416	. . . . . . . . . . . . . . 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 3222	. . . . . . . . . . . . 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 11203	. . . . . . . . . . . . 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 6009	. . . . . . . . . . . 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 9367	. . . . . . . . . . . . . 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 8080	. . . . . . . . . . . . 13 |- 1 e. CC
41		pncan 8340	. . . . . . . . . . . . 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 2266	. . . . . . . . . . 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 5632	. . . . . . . . . . . . 13 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( (♯ ` y ) = m <-> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m ) )
45		vex 2802	. . . . . . . . . . . . . . 15 |- y e. _V
46	45, 23	sbcie 3063	. . . . . . . . . . . . . 14 |- ( [. y / x ]. ph <-> ch )
47		dfsbcq 3030	. . . . . . . . . . . . . 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 11062	. . . . . . . . . . . . . . . 16 |- ( x e. Word B -> (♯ ` x ) e. NN0 )
52	51	nn0zd 9555	. . . . . . . . . . . . . . 15 |- ( x e. Word B -> (♯ ` x ) e. ZZ )
53		1zzd 9461	. . . . . . . . . . . . . . 15 |- ( x e. Word B -> 1 e. ZZ )
54	52, 53	zsubcld 9562	. . . . . . . . . . . . . 14 |- ( x e. Word B -> ( (♯ ` x ) - 1 ) e. ZZ )
55		pfxclz 11197	. . . . . . . . . . . . . 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 2912	. . . . . . . . . . 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 9141	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> 1 <_ (♯ ` x ) )
61		wrdlenge1n0 11091	. . . . . . . . . . . . . 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 11108	. . . . . . . . . . . 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 6001	. . . . . . . . . . . . . 14 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( y ++ <" z "> ) = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) )
67	66	sbceq1d 3033	. . . . . . . . . . . . 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 11138	. . . . . . . . . . . . . . 15 |- ( z = (lastS ` x ) -> <" z "> = <" (lastS ` x ) "> )
70	69	oveq2d 6010	. . . . . . . . . . . . . 14 |- ( z = (lastS ` x ) -> ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) )
71	70	sbceq1d 3033	. . . . . . . . . . . . 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 11151	. . . . . . . . . . . . . 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 3065	. . . . . . . . . . . . 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 2869	. . . . . . . . . . 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 11004	. . . . . . . . . . . . 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 11231	. . . . . . . . . . . 12 |- ( ( x e. Word B /\ x =/= (/) ) -> ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) = x )
88	87	eqcomd 2235	. . . . . . . . . . 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 3038	. . . . . . . . . 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 2603	. . . . . 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 9549	. . 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 2230	. 2 |- ( A e. Word B -> (♯ ` A ) = (♯ ` A ) )
100		fveqeq2 5632	. . . 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 2903	. 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 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   [.wsbc 3028   (/)c0 3491   class class class wbr 4082   ` cfv 5314  (class class class)co 5994   Fincfn 6877   CCcc 7985   0cc0 7987   1c1 7988   + caddc 7990   <_ cle 8170   - cmin 8305   NNcn 9098   NN0cn0 9357   ZZcz 9434   ...cfz 10192  ..^cfzo 10326  ♯chash 10984  Word cword 11058  lastSclsw 11102   ++ cconcat 11111   <"cs1 11134   prefix cpfx 11190

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-1o 6552  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-inn 9099  df-n0 9358  df-z 9435  df-uz 9711  df-fz 10193  df-fzo 10327  df-ihash 10985  df-word 11059  df-lsw 11103  df-concat 11112  df-s1 11135  df-substr 11164  df-pfx 11191

This theorem is referenced by: (None)