Theorem wrdind 13476

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

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 13324	. . 3 |- ( A e. Word B -> ( # ` A ) e. NN0 )
2		eqeq2 2633	. . . . . 6 |- ( n = 0 -> ( ( # ` x ) = n <-> ( # ` x ) = 0 ) )
3	2	imbi1d 331	. . . . 5 |- ( n = 0 -> ( ( ( # ` x ) = n -> ph ) <-> ( ( # ` x ) = 0 -> ph ) ) )
4	3	ralbidv 2986	. . . 4 |- ( n = 0 -> ( A. x e. Word B ( ( # ` x ) = n -> ph ) <-> A. x e. Word B ( ( # ` x ) = 0 -> ph ) ) )
5		eqeq2 2633	. . . . . 6 |- ( n = m -> ( ( # ` x ) = n <-> ( # ` x ) = m ) )
6	5	imbi1d 331	. . . . 5 |- ( n = m -> ( ( ( # ` x ) = n -> ph ) <-> ( ( # ` x ) = m -> ph ) ) )
7	6	ralbidv 2986	. . . 4 |- ( n = m -> ( A. x e. Word B ( ( # ` x ) = n -> ph ) <-> A. x e. Word B ( ( # ` x ) = m -> ph ) ) )
8		eqeq2 2633	. . . . . 6 |- ( n = ( m + 1 ) -> ( ( # ` x ) = n <-> ( # ` x ) = ( m + 1 ) ) )
9	8	imbi1d 331	. . . . 5 |- ( n = ( m + 1 ) -> ( ( ( # ` x ) = n -> ph ) <-> ( ( # ` x ) = ( m + 1 ) -> ph ) ) )
10	9	ralbidv 2986	. . . 4 |- ( n = ( m + 1 ) -> ( A. x e. Word B ( ( # ` x ) = n -> ph ) <-> A. x e. Word B ( ( # ` x ) = ( m + 1 ) -> ph ) ) )
11		eqeq2 2633	. . . . . 6 |- ( n = ( # ` A ) -> ( ( # ` x ) = n <-> ( # ` x ) = ( # ` A ) ) )
12	11	imbi1d 331	. . . . 5 |- ( n = ( # ` A ) -> ( ( ( # ` x ) = n -> ph ) <-> ( ( # ` x ) = ( # ` A ) -> ph ) ) )
13	12	ralbidv 2986	. . . 4 |- ( n = ( # ` A ) -> ( A. x e. Word B ( ( # ` x ) = n -> ph ) <-> A. x e. Word B ( ( # ` x ) = ( # ` A ) -> ph ) ) )
14		hasheq0 13154	. . . . . 6 |- ( x e. Word B -> ( ( # ` x ) = 0 <-> x = (/) ) )
15		wrdind.5	. . . . . . 7 |- ps
16		wrdind.1	. . . . . . 7 |- ( x = (/) -> ( ph <-> ps ) )
17	15, 16	mpbiri 248	. . . . . 6 |- ( x = (/) -> ph )
18	14, 17	syl6bi 243	. . . . 5 |- ( x e. Word B -> ( ( # ` x ) = 0 -> ph ) )
19	18	rgen 2922	. . . 4 |- A. x e. Word B ( ( # ` x ) = 0 -> ph )
20		fveq2 6191	. . . . . . . 8 |- ( x = y -> ( # ` x ) = ( # ` y ) )
21	20	eqeq1d 2624	. . . . . . 7 |- ( x = y -> ( ( # ` x ) = m <-> ( # ` y ) = m ) )
22		wrdind.2	. . . . . . 7 |- ( x = y -> ( ph <-> ch ) )
23	21, 22	imbi12d 334	. . . . . 6 |- ( x = y -> ( ( ( # ` x ) = m -> ph ) <-> ( ( # ` y ) = m -> ch ) ) )
24	23	cbvralv 3171	. . . . 5 |- ( A. x e. Word B ( ( # ` x ) = m -> ph ) <-> A. y e. Word B ( ( # ` y ) = m -> ch ) )
25		swrdcl 13419	. . . . . . . . . . . 12 |- ( x e. Word B -> ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) e. Word B )
26	25	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) e. Word B )
27		simplr 792	. . . . . . . . . . 11 |- ( ( ( 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 ) )
28		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> x e. Word B )
29		fzossfz 12488	. . . . . . . . . . . . . 14 |- ( 0..^ ( # ` x ) ) C_ ( 0 ... ( # ` x ) )
30		simprr 796	. . . . . . . . . . . . . . . 16 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( # ` x ) = ( m + 1 ) )
31		nn0p1nn 11332	. . . . . . . . . . . . . . . . 17 |- ( m e. NN0 -> ( m + 1 ) e. NN )
32	31	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( m + 1 ) e. NN )
33	30, 32	eqeltrd 2701	. . . . . . . . . . . . . . 15 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( # ` x ) e. NN )
34		fzo0end 12560	. . . . . . . . . . . . . . 15 |- ( ( # ` x ) e. NN -> ( ( # ` x ) - 1 ) e. ( 0..^ ( # ` x ) ) )
35	33, 34	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( ( # ` x ) - 1 ) e. ( 0..^ ( # ` x ) ) )
36	29, 35	sseldi 3601	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( ( # ` x ) - 1 ) e. ( 0 ... ( # ` x ) ) )
37		swrd0len 13422	. . . . . . . . . . . . 13 |- ( ( x e. Word B /\ ( ( # ` x ) - 1 ) e. ( 0 ... ( # ` x ) ) ) -> ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( ( # ` x ) - 1 ) )
38	28, 36, 37	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = ( ( # ` x ) - 1 ) )
39	30	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( ( # ` x ) - 1 ) = ( ( m + 1 ) - 1 ) )
40		nn0cn 11302	. . . . . . . . . . . . . 14 |- ( m e. NN0 -> m e. CC )
41	40	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> m e. CC )
42		ax-1cn 9994	. . . . . . . . . . . . 13 |- 1 e. CC
43		pncan 10287	. . . . . . . . . . . . 13 |- ( ( m e. CC /\ 1 e. CC ) -> ( ( m + 1 ) - 1 ) = m )
44	41, 42, 43	sylancl 694	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( ( m + 1 ) - 1 ) = m )
45	38, 39, 44	3eqtrd 2660	. . . . . . . . . . 11 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = m )
46		fveq2 6191	. . . . . . . . . . . . . 14 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( # ` y ) = ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) )
47	46	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( ( # ` y ) = m <-> ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = m ) )
48		vex 3203	. . . . . . . . . . . . . . 15 |- y e. _V
49	48, 22	sbcie 3470	. . . . . . . . . . . . . 14 |- ( [. y / x ]. ph <-> ch )
50		dfsbcq 3437	. . . . . . . . . . . . . 14 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( [. y / x ]. ph <-> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. ph ) )
51	49, 50	syl5bbr 274	. . . . . . . . . . . . 13 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( ch <-> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. ph ) )
52	47, 51	imbi12d 334	. . . . . . . . . . . 12 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( ( ( # ` y ) = m -> ch ) <-> ( ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = m -> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. ph ) ) )
53	52	rspcv 3305	. . . . . . . . . . 11 |- ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) e. Word B -> ( A. y e. Word B ( ( # ` y ) = m -> ch ) -> ( ( # ` ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) = m -> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. ph ) ) )
54	26, 27, 45, 53	syl3c 66	. . . . . . . . . 10 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. ph )
55	33	nnge1d 11063	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> 1 <_ ( # ` x ) )
56		wrdlenge1n0 13340	. . . . . . . . . . . . . 14 |- ( x e. Word B -> ( x =/= (/) <-> 1 <_ ( # ` x ) ) )
57	56	ad2antrl 764	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( x =/= (/) <-> 1 <_ ( # ` x ) ) )
58	55, 57	mpbird 247	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> x =/= (/) )
59		lswcl 13355	. . . . . . . . . . . 12 |- ( ( x e. Word B /\ x =/= (/) ) -> ( lastS ` x ) e. B )
60	28, 58, 59	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( lastS ` x ) e. B )
61		oveq1 6657	. . . . . . . . . . . . . 14 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( y ++ <" z "> ) = ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" z "> ) )
62	61	sbceq1d 3440	. . . . . . . . . . . . 13 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( [. ( y ++ <" z "> ) / x ]. ph <-> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" z "> ) / x ]. ph ) )
63	50, 62	imbi12d 334	. . . . . . . . . . . 12 |- ( y = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) -> ( ( [. y / x ]. ph -> [. ( y ++ <" z "> ) / x ]. ph ) <-> ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. ph -> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" z "> ) / x ]. ph ) ) )
64		s1eq 13380	. . . . . . . . . . . . . . 15 |- ( z = ( lastS ` x ) -> <" z "> = <" ( lastS ` x ) "> )
65	64	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( z = ( lastS ` x ) -> ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" z "> ) = ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) )
66	65	sbceq1d 3440	. . . . . . . . . . . . 13 |- ( z = ( lastS ` x ) -> ( [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" z "> ) / x ]. ph <-> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. ph ) )
67	66	imbi2d 330	. . . . . . . . . . . 12 |- ( z = ( lastS ` x ) -> ( ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. ph -> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" z "> ) / x ]. ph ) <-> ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. ph -> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. ph ) ) )
68		wrdind.6	. . . . . . . . . . . . 13 |- ( ( y e. Word B /\ z e. B ) -> ( ch -> th ) )
69		ovex 6678	. . . . . . . . . . . . . 14 |- ( y ++ <" z "> ) e. _V
70		wrdind.3	. . . . . . . . . . . . . 14 |- ( x = ( y ++ <" z "> ) -> ( ph <-> th ) )
71	69, 70	sbcie 3470	. . . . . . . . . . . . 13 |- ( [. ( y ++ <" z "> ) / x ]. ph <-> th )
72	68, 49, 71	3imtr4g 285	. . . . . . . . . . . 12 |- ( ( y e. Word B /\ z e. B ) -> ( [. y / x ]. ph -> [. ( y ++ <" z "> ) / x ]. ph ) )
73	63, 67, 72	vtocl2ga 3274	. . . . . . . . . . 11 |- ( ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) e. Word B /\ ( lastS ` x ) e. B ) -> ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. ph -> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. ph ) )
74	26, 60, 73	syl2anc 693	. . . . . . . . . 10 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( [. ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) / x ]. ph -> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. ph ) )
75	54, 74	mpd 15	. . . . . . . . 9 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. ph )
76		wrdfin 13323	. . . . . . . . . . . . . 14 |- ( x e. Word B -> x e. Fin )
77	76	ad2antrl 764	. . . . . . . . . . . . 13 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> x e. Fin )
78		hashnncl 13157	. . . . . . . . . . . . 13 |- ( x e. Fin -> ( ( # ` x ) e. NN <-> x =/= (/) ) )
79	77, 78	syl 17	. . . . . . . . . . . 12 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( ( # ` x ) e. NN <-> x =/= (/) ) )
80	33, 79	mpbid 222	. . . . . . . . . . 11 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> x =/= (/) )
81		swrdccatwrd 13468	. . . . . . . . . . . 12 |- ( ( x e. Word B /\ x =/= (/) ) -> ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) = x )
82	81	eqcomd 2628	. . . . . . . . . . 11 |- ( ( x e. Word B /\ x =/= (/) ) -> x = ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) )
83	28, 80, 82	syl2anc 693	. . . . . . . . . 10 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> x = ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) )
84		sbceq1a 3446	. . . . . . . . . 10 |- ( x = ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) -> ( ph <-> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. ph ) )
85	83, 84	syl 17	. . . . . . . . 9 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ( ph <-> [. ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) ++ <" ( lastS ` x ) "> ) / x ]. ph ) )
86	75, 85	mpbird 247	. . . . . . . 8 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ ( x e. Word B /\ ( # ` x ) = ( m + 1 ) ) ) -> ph )
87	86	expr 643	. . . . . . 7 |- ( ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) /\ x e. Word B ) -> ( ( # ` x ) = ( m + 1 ) -> ph ) )
88	87	ralrimiva 2966	. . . . . 6 |- ( ( m e. NN0 /\ A. y e. Word B ( ( # ` y ) = m -> ch ) ) -> A. x e. Word B ( ( # ` x ) = ( m + 1 ) -> ph ) )
89	88	ex 450	. . . . 5 |- ( m e. NN0 -> ( A. y e. Word B ( ( # ` y ) = m -> ch ) -> A. x e. Word B ( ( # ` x ) = ( m + 1 ) -> ph ) ) )
90	24, 89	syl5bi 232	. . . 4 |- ( m e. NN0 -> ( A. x e. Word B ( ( # ` x ) = m -> ph ) -> A. x e. Word B ( ( # ` x ) = ( m + 1 ) -> ph ) ) )
91	4, 7, 10, 13, 19, 90	nn0ind 11472	. . 3 |- ( ( # ` A ) e. NN0 -> A. x e. Word B ( ( # ` x ) = ( # ` A ) -> ph ) )
92	1, 91	syl 17	. 2 |- ( A e. Word B -> A. x e. Word B ( ( # ` x ) = ( # ` A ) -> ph ) )
93		eqidd 2623	. 2 |- ( A e. Word B -> ( # ` A ) = ( # ` A ) )
94		fveq2 6191	. . . . 5 |- ( x = A -> ( # ` x ) = ( # ` A ) )
95	94	eqeq1d 2624	. . . 4 |- ( x = A -> ( ( # ` x ) = ( # ` A ) <-> ( # ` A ) = ( # ` A ) ) )
96		wrdind.4	. . . 4 |- ( x = A -> ( ph <-> ta ) )
97	95, 96	imbi12d 334	. . 3 |- ( x = A -> ( ( ( # ` x ) = ( # ` A ) -> ph ) <-> ( ( # ` A ) = ( # ` A ) -> ta ) ) )
98	97	rspcv 3305	. 2 |- ( A e. Word B -> ( A. x e. Word B ( ( # ` x ) = ( # ` A ) -> ph ) -> ( ( # ` A ) = ( # ` A ) -> ta ) ) )
99	92, 93, 98	mp2d 49	1 |- ( A e. Word B -> ta )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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:  frmdgsum  17399  gsumwrev  17796  gsmsymgrfix  17848  efginvrel2  18140  signstfvneq0  30649  signstfvc  30651  mrsubvrs  31419