Theorem wrdind 11477

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 11291	. . 3 |- ( A e. Word B -> (♯ ` A ) e. NN0 )
2		eqeq2 2248	. . . . . 6 |- ( n = 0 -> ( (♯ ` x ) = n <-> (♯ ` x ) = 0 ) )
3	2	imbi1d 231	. . . . 5 |- ( n = 0 -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = 0 -> ph ) ) )
4	3	ralbidv 2550	. . . 4 |- ( n = 0 -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = 0 -> ph ) ) )
5		eqeq2 2248	. . . . . 6 |- ( n = m -> ( (♯ ` x ) = n <-> (♯ ` x ) = m ) )
6	5	imbi1d 231	. . . . 5 |- ( n = m -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = m -> ph ) ) )
7	6	ralbidv 2550	. . . 4 |- ( n = m -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = m -> ph ) ) )
8		eqeq2 2248	. . . . . 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 2550	. . . 4 |- ( n = ( m + 1 ) -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = ( m + 1 ) -> ph ) ) )
11		eqeq2 2248	. . . . . 6 |- ( n = (♯ ` A ) -> ( (♯ ` x ) = n <-> (♯ ` x ) = (♯ ` A ) ) )
12	11	imbi1d 231	. . . . 5 |- ( n = (♯ ` A ) -> ( ( (♯ ` x ) = n -> ph ) <-> ( (♯ ` x ) = (♯ ` A ) -> ph ) ) )
13	12	ralbidv 2550	. . . 4 |- ( n = (♯ ` A ) -> ( A. x e. Word B ( (♯ ` x ) = n -> ph ) <-> A. x e. Word B ( (♯ ` x ) = (♯ ` A ) -> ph ) ) )
14		wrdfin 11306	. . . . . . 7 |- ( x e. Word B -> x e. Fin )
15		fihasheq0 11215	. . . . . . 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 2603	. . . 4 |- A. x e. Word B ( (♯ ` x ) = 0 -> ph )
22		fveqeq2 5702	. . . . . . 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 2790	. . . . 5 |- ( A. x e. Word B ( (♯ ` x ) = m -> ph ) <-> A. y e. Word B ( (♯ ` y ) = m -> ch ) )
26		simprl 535	. . . . . . . . . . . . 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 10556	. . . . . . . . . . . . . 14 |- ( 0..^ (♯ ` x ) ) C_ ( 0 ... (♯ ` x ) )
28		simprr 537	. . . . . . . . . . . . . . . 16 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> (♯ ` x ) = ( m + 1 ) )
29		nn0p1nn 9585	. . . . . . . . . . . . . . . . 17 |- ( m e. NN0 -> ( m + 1 ) e. NN )
30	29	ad2antrr 492	. . . . . . . . . . . . . . . 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 2315	. . . . . . . . . . . . . . 15 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> (♯ ` x ) e. NN )
32		fzo0end 10624	. . . . . . . . . . . . . . 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 3246	. . . . . . . . . . . . 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 11440	. . . . . . . . . . . . 13 |- ( ( x e. Word B /\ ( (♯ ` x ) - 1 ) e. ( 0 ... (♯ ` x ) ) ) -> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = ( (♯ ` x ) - 1 ) )
36	26, 34, 35	syl2anc 415	. . . . . . . . . . . 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 6094	. . . . . . . . . . . 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 9556	. . . . . . . . . . . . . 14 |- ( m e. NN0 -> m e. CC )
39	38	ad2antrr 492	. . . . . . . . . . . . 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 8266	. . . . . . . . . . . . 13 |- 1 e. CC
41		pncan 8526	. . . . . . . . . . . . 13 |- ( ( m e. CC /\ 1 e. CC ) -> ( ( m + 1 ) - 1 ) = m )
42	39, 40, 41	sylancl 417	. . . . . . . . . . . 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 2275	. . . . . . . . . . 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 5702	. . . . . . . . . . . . 13 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( (♯ ` y ) = m <-> (♯ ` ( x prefix ( (♯ ` x ) - 1 ) ) ) = m ) )
45		vex 2824	. . . . . . . . . . . . . . 15 |- y e. _V
46	45, 23	sbcie 3086	. . . . . . . . . . . . . 14 |- ( [. y / x ]. ph <-> ch )
47		dfsbcq 3053	. . . . . . . . . . . . . 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 533	. . . . . . . . . . . 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 11291	. . . . . . . . . . . . . . . 16 |- ( x e. Word B -> (♯ ` x ) e. NN0 )
52	51	nn0zd 9749	. . . . . . . . . . . . . . 15 |- ( x e. Word B -> (♯ ` x ) e. ZZ )
53		1zzd 9654	. . . . . . . . . . . . . . 15 |- ( x e. Word B -> 1 e. ZZ )
54	52, 53	zsubcld 9756	. . . . . . . . . . . . . 14 |- ( x e. Word B -> ( (♯ ` x ) - 1 ) e. ZZ )
55		pfxclz 11434	. . . . . . . . . . . . . 14 |- ( ( x e. Word B /\ ( (♯ ` x ) - 1 ) e. ZZ ) -> ( x prefix ( (♯ ` x ) - 1 ) ) e. Word B )
56	54, 55	mpdan 425	. . . . . . . . . . . . 13 |- ( x e. Word B -> ( x prefix ( (♯ ` x ) - 1 ) ) e. Word B )
57	56	ad2antrl 494	. . . . . . . . . . . 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 2934	. . . . . . . . . . 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 9330	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> 1 <_ (♯ ` x ) )
61		wrdlenge1n0 11321	. . . . . . . . . . . . . 14 |- ( x e. Word B -> ( x =/= (/) <-> 1 <_ (♯ ` x ) ) )
62	61	ad2antrl 494	. . . . . . . . . . . . 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 11338	. . . . . . . . . . . 12 |- ( ( x e. Word B /\ x =/= (/) ) -> (lastS ` x ) e. B )
65	26, 63, 64	syl2anc 415	. . . . . . . . . . 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 6086	. . . . . . . . . . . . . 14 |- ( y = ( x prefix ( (♯ ` x ) - 1 ) ) -> ( y ++ <" z "> ) = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) )
67	66	sbceq1d 3056	. . . . . . . . . . . . 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 11370	. . . . . . . . . . . . . . 15 |- ( z = (lastS ` x ) -> <" z "> = <" (lastS ` x ) "> )
70	69	oveq2d 6095	. . . . . . . . . . . . . 14 |- ( z = (lastS ` x ) -> ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" z "> ) = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) )
71	70	sbceq1d 3056	. . . . . . . . . . . . 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 11383	. . . . . . . . . . . . . 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 3088	. . . . . . . . . . . . 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 2891	. . . . . . . . . . 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 415	. . . . . . . . . 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 494	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( (♯ ` y ) = m -> ch ) ) /\ ( x e. Word B /\ (♯ ` x ) = ( m + 1 ) ) ) -> x e. Fin )
84		hashnncl 11217	. . . . . . . . . . . . 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 11468	. . . . . . . . . . . 12 |- ( ( x e. Word B /\ x =/= (/) ) -> ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) = x )
88	87	eqcomd 2244	. . . . . . . . . . 11 |- ( ( x e. Word B /\ x =/= (/) ) -> x = ( ( x prefix ( (♯ ` x ) - 1 ) ) ++ <" (lastS ` x ) "> ) )
89	26, 86, 88	syl2anc 415	. . . . . . . . . 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 3061	. . . . . . . . . 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 2623	. . . . . 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 9743	. . 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 2239	. 2 |- ( A e. Word B -> (♯ ` A ) = (♯ ` A ) )
100		fveqeq2 5702	. . . 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 2925	. 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 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   [.wsbc 3051   (/)c0 3520   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   Fincfn 7016   CCcc 8171   0cc0 8173   1c1 8174   + caddc 8176   <_ cle 8355   - cmin 8491   NNcn 9287   NN0cn0 9546   ZZcz 9627   ...cfz 10394  ..^cfzo 10532  ♯chash 11197  Word cword 11287  lastSclsw 11332   ++ cconcat 11341   <"cs1 11366   prefix cpfx 11427

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-fz 10395  df-fzo 10533  df-ihash 11198  df-word 11288  df-lsw 11333  df-concat 11342  df-s1 11367  df-substr 11401  df-pfx 11428

This theorem is referenced by: (None)