Theorem wwlkextinj 24403

Description: Lemma 2 for wwlkextbij 24406. (Contributed by Alexander van der Vekens, 7-Aug-2018.)

Hypotheses

Ref	Expression
wwlkextbij.d	|- D = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) }
wwlkextbij.r	|- R = { n e. V | { ( lastS ` W ) , n } e. ran E }
wwlkextbij.f	|- F = ( t e. D |-> ( lastS ` t ) )

Assertion

Ref	Expression
wwlkextinj	|- ( N e. NN0 -> F : D -1-1-> R )

Distinct variable groups:   t, D   n, E, w   t, N, w   t, R   n, V, t, w   n, W, t, w

Allowed substitution hints:   D( w, n)   R( w, n)   E( t)   F( w, t, n)   N( n)

Proof of Theorem wwlkextinj

Dummy variables d x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlkextbij.d	. . 3 |- D = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) }
2		wwlkextbij.r	. . 3 |- R = { n e. V | { ( lastS ` W ) , n } e. ran E }
3		wwlkextbij.f	. . 3 |- F = ( t e. D |-> ( lastS ` t ) )
4	1, 2, 3	wwlkextfun 24402	. 2 |- ( N e. NN0 -> F : D --> R )
5		fvex 5874	. . . . . . 7 |- ( lastS ` d ) e. _V
6		fveq2 5864	. . . . . . . 8 |- ( t = d -> ( lastS ` t ) = ( lastS ` d ) )
7	6, 3	fvmptg 5946	. . . . . . 7 |- ( ( d e. D /\ ( lastS ` d ) e. _V ) -> ( F ` d ) = ( lastS ` d ) )
8	5, 7	mpan2 671	. . . . . 6 |- ( d e. D -> ( F ` d ) = ( lastS ` d ) )
9		fvex 5874	. . . . . . 7 |- ( lastS ` x ) e. _V
10		fveq2 5864	. . . . . . . 8 |- ( t = x -> ( lastS ` t ) = ( lastS ` x ) )
11	10, 3	fvmptg 5946	. . . . . . 7 |- ( ( x e. D /\ ( lastS ` x ) e. _V ) -> ( F ` x ) = ( lastS ` x ) )
12	9, 11	mpan2 671	. . . . . 6 |- ( x e. D -> ( F ` x ) = ( lastS ` x ) )
13	8, 12	eqeqan12d 2490	. . . . 5 |- ( ( d e. D /\ x e. D ) -> ( ( F ` d ) = ( F ` x ) <-> ( lastS ` d ) = ( lastS ` x ) ) )
14	13	adantl 466	. . . 4 |- ( ( N e. NN0 /\ ( d e. D /\ x e. D ) ) -> ( ( F ` d ) = ( F ` x ) <-> ( lastS ` d ) = ( lastS ` x ) ) )
15	1	eleq2i 2545	. . . . . . 7 |- ( d e. D <-> d e. { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } )
16		fveq2 5864	. . . . . . . . . 10 |- ( w = d -> ( # ` w ) = ( # ` d ) )
17	16	eqeq1d 2469	. . . . . . . . 9 |- ( w = d -> ( ( # ` w ) = ( N + 2 ) <-> ( # ` d ) = ( N + 2 ) ) )
18		oveq1 6289	. . . . . . . . . 10 |- ( w = d -> ( w substr <. 0 , ( N + 1 ) >. ) = ( d substr <. 0 , ( N + 1 ) >. ) )
19	18	eqeq1d 2469	. . . . . . . . 9 |- ( w = d -> ( ( w substr <. 0 , ( N + 1 ) >. ) = W <-> ( d substr <. 0 , ( N + 1 ) >. ) = W ) )
20		fveq2 5864	. . . . . . . . . . 11 |- ( w = d -> ( lastS ` w ) = ( lastS ` d ) )
21	20	preq2d 4113	. . . . . . . . . 10 |- ( w = d -> { ( lastS ` W ) , ( lastS ` w ) } = { ( lastS ` W ) , ( lastS ` d ) } )
22	21	eleq1d 2536	. . . . . . . . 9 |- ( w = d -> ( { ( lastS ` W ) , ( lastS ` w ) } e. ran E <-> { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) )
23	17, 19, 22	3anbi123d 1299	. . . . . . . 8 |- ( w = d -> ( ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) <-> ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) )
24	23	elrab 3261	. . . . . . 7 |- ( d e. { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } <-> ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) )
25	15, 24	bitri 249	. . . . . 6 |- ( d e. D <-> ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) )
26	1	eleq2i 2545	. . . . . . 7 |- ( x e. D <-> x e. { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } )
27		fveq2 5864	. . . . . . . . . 10 |- ( w = x -> ( # ` w ) = ( # ` x ) )
28	27	eqeq1d 2469	. . . . . . . . 9 |- ( w = x -> ( ( # ` w ) = ( N + 2 ) <-> ( # ` x ) = ( N + 2 ) ) )
29		oveq1 6289	. . . . . . . . . 10 |- ( w = x -> ( w substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) )
30	29	eqeq1d 2469	. . . . . . . . 9 |- ( w = x -> ( ( w substr <. 0 , ( N + 1 ) >. ) = W <-> ( x substr <. 0 , ( N + 1 ) >. ) = W ) )
31		fveq2 5864	. . . . . . . . . . 11 |- ( w = x -> ( lastS ` w ) = ( lastS ` x ) )
32	31	preq2d 4113	. . . . . . . . . 10 |- ( w = x -> { ( lastS ` W ) , ( lastS ` w ) } = { ( lastS ` W ) , ( lastS ` x ) } )
33	32	eleq1d 2536	. . . . . . . . 9 |- ( w = x -> ( { ( lastS ` W ) , ( lastS ` w ) } e. ran E <-> { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) )
34	28, 30, 33	3anbi123d 1299	. . . . . . . 8 |- ( w = x -> ( ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) <-> ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) )
35	34	elrab 3261	. . . . . . 7 |- ( x e. { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } <-> ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) )
36	26, 35	bitri 249	. . . . . 6 |- ( x e. D <-> ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) )
37		eqtr3 2495	. . . . . . . . . . . . . . . . . 18 |- ( ( ( # ` d ) = ( N + 2 ) /\ ( # ` x ) = ( N + 2 ) ) -> ( # ` d ) = ( # ` x ) )
38	37	expcom 435	. . . . . . . . . . . . . . . . 17 |- ( ( # ` x ) = ( N + 2 ) -> ( ( # ` d ) = ( N + 2 ) -> ( # ` d ) = ( # ` x ) ) )
39	38	3ad2ant1 1017	. . . . . . . . . . . . . . . 16 |- ( ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) -> ( ( # ` d ) = ( N + 2 ) -> ( # ` d ) = ( # ` x ) ) )
40	39	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) -> ( ( # ` d ) = ( N + 2 ) -> ( # ` d ) = ( # ` x ) ) )
41	40	com12 31	. . . . . . . . . . . . . 14 |- ( ( # ` d ) = ( N + 2 ) -> ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) -> ( # ` d ) = ( # ` x ) ) )
42	41	3ad2ant1 1017	. . . . . . . . . . . . 13 |- ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) -> ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) -> ( # ` d ) = ( # ` x ) ) )
43	42	adantl 466	. . . . . . . . . . . 12 |- ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) -> ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) -> ( # ` d ) = ( # ` x ) ) )
44	43	imp 429	. . . . . . . . . . 11 |- ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) -> ( # ` d ) = ( # ` x ) )
45	44	adantr 465	. . . . . . . . . 10 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) -> ( # ` d ) = ( # ` x ) )
46	45	adantr 465	. . . . . . . . 9 |- ( ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) /\ ( lastS ` d ) = ( lastS ` x ) ) -> ( # ` d ) = ( # ` x ) )
47		simpr 461	. . . . . . . . 9 |- ( ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) /\ ( lastS ` d ) = ( lastS ` x ) ) -> ( lastS ` d ) = ( lastS ` x ) )
48		eqtr3 2495	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ ( x substr <. 0 , ( N + 1 ) >. ) = W ) -> ( d substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) )
49		1e2m1 10647	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- 1 = ( 2 - 1 )
50	49	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( N e. NN0 -> 1 = ( 2 - 1 ) )
51	50	oveq2d 6298	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( N e. NN0 -> ( N + 1 ) = ( N + ( 2 - 1 ) ) )
52		nn0cn 10801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( N e. NN0 -> N e. CC )
53		2cnd 10604	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( N e. NN0 -> 2 e. CC )
54		ax-1cn 9546	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- 1 e. CC
55	54	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( N e. NN0 -> 1 e. CC )
56	52, 53, 55	addsubassd 9946	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( N e. NN0 -> ( ( N + 2 ) - 1 ) = ( N + ( 2 - 1 ) ) )
57	51, 56	eqtr4d 2511	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. NN0 -> ( N + 1 ) = ( ( N + 2 ) - 1 ) )
58	57	adantr 465	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( N e. NN0 /\ ( # ` d ) = ( N + 2 ) ) -> ( N + 1 ) = ( ( N + 2 ) - 1 ) )
59		oveq1 6289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( # ` d ) = ( N + 2 ) -> ( ( # ` d ) - 1 ) = ( ( N + 2 ) - 1 ) )
60	59	eqeq2d 2481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( # ` d ) = ( N + 2 ) -> ( ( N + 1 ) = ( ( # ` d ) - 1 ) <-> ( N + 1 ) = ( ( N + 2 ) - 1 ) ) )
61	60	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( N e. NN0 /\ ( # ` d ) = ( N + 2 ) ) -> ( ( N + 1 ) = ( ( # ` d ) - 1 ) <-> ( N + 1 ) = ( ( N + 2 ) - 1 ) ) )
62	58, 61	mpbird 232	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( N e. NN0 /\ ( # ` d ) = ( N + 2 ) ) -> ( N + 1 ) = ( ( # ` d ) - 1 ) )
63		opeq2 4214	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( N + 1 ) = ( ( # ` d ) - 1 ) -> <. 0 , ( N + 1 ) >. = <. 0 , ( ( # ` d ) - 1 ) >. )
64	63	oveq2d 6298	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( N + 1 ) = ( ( # ` d ) - 1 ) -> ( d substr <. 0 , ( N + 1 ) >. ) = ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) )
65	63	oveq2d 6298	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( N + 1 ) = ( ( # ` d ) - 1 ) -> ( x substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) )
66	64, 65	eqeq12d 2489	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( N + 1 ) = ( ( # ` d ) - 1 ) -> ( ( d substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) <-> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) )
67	62, 66	syl 16	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. NN0 /\ ( # ` d ) = ( N + 2 ) ) -> ( ( d substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) <-> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) )
68	67	biimpd 207	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N e. NN0 /\ ( # ` d ) = ( N + 2 ) ) -> ( ( d substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) )
69	68	ex 434	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. NN0 -> ( ( # ` d ) = ( N + 2 ) -> ( ( d substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
70	69	com13 80	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( d substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) -> ( ( # ` d ) = ( N + 2 ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
71	48, 70	syl 16	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ ( x substr <. 0 , ( N + 1 ) >. ) = W ) -> ( ( # ` d ) = ( N + 2 ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
72	71	ex 434	. . . . . . . . . . . . . . . . . . . 20 |- ( ( d substr <. 0 , ( N + 1 ) >. ) = W -> ( ( x substr <. 0 , ( N + 1 ) >. ) = W -> ( ( # ` d ) = ( N + 2 ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) ) )
73	72	com23 78	. . . . . . . . . . . . . . . . . . 19 |- ( ( d substr <. 0 , ( N + 1 ) >. ) = W -> ( ( # ` d ) = ( N + 2 ) -> ( ( x substr <. 0 , ( N + 1 ) >. ) = W -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) ) )
74	73	impcom 430	. . . . . . . . . . . . . . . . . 18 |- ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W ) -> ( ( x substr <. 0 , ( N + 1 ) >. ) = W -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
75	74	com12 31	. . . . . . . . . . . . . . . . 17 |- ( ( x substr <. 0 , ( N + 1 ) >. ) = W -> ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
76	75	3ad2ant2 1018	. . . . . . . . . . . . . . . 16 |- ( ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) -> ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
77	76	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) -> ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
78	77	com12 31	. . . . . . . . . . . . . 14 |- ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W ) -> ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
79	78	3adant3 1016	. . . . . . . . . . . . 13 |- ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) -> ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
80	79	adantl 466	. . . . . . . . . . . 12 |- ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) -> ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
81	80	imp 429	. . . . . . . . . . 11 |- ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) )
82	81	imp 429	. . . . . . . . . 10 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) )
83	82	adantr 465	. . . . . . . . 9 |- ( ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) /\ ( lastS ` d ) = ( lastS ` x ) ) -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) )
84		simpl 457	. . . . . . . . . . . . . 14 |- ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) -> d e. Word V )
85		simpl 457	. . . . . . . . . . . . . 14 |- ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) -> x e. Word V )
86	84, 85	anim12i 566	. . . . . . . . . . . . 13 |- ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) -> ( d e. Word V /\ x e. Word V ) )
87	86	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) -> ( d e. Word V /\ x e. Word V ) )
88		nn0re 10800	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> N e. RR )
89		2re 10601	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 e. RR
90	89	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> 2 e. RR )
91		nn0ge0 10817	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> 0 <_ N )
92		2pos 10623	. . . . . . . . . . . . . . . . . . . . . . 23 |- 0 < 2
93	92	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> 0 < 2 )
94	88, 90, 91, 93	addgegt0d 10122	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> 0 < ( N + 2 ) )
95	94	adantl 466	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( # ` d ) = ( N + 2 ) /\ N e. NN0 ) -> 0 < ( N + 2 ) )
96		breq2 4451	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` d ) = ( N + 2 ) -> ( 0 < ( # ` d ) <-> 0 < ( N + 2 ) ) )
97	96	adantr 465	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( # ` d ) = ( N + 2 ) /\ N e. NN0 ) -> ( 0 < ( # ` d ) <-> 0 < ( N + 2 ) ) )
98	95, 97	mpbird 232	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( # ` d ) = ( N + 2 ) /\ N e. NN0 ) -> 0 < ( # ` d ) )
99		hashgt0n0 12397	. . . . . . . . . . . . . . . . . . 19 |- ( ( d e. Word V /\ 0 < ( # ` d ) ) -> d =/= (/) )
100	98, 99	sylan2 474	. . . . . . . . . . . . . . . . . 18 |- ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ N e. NN0 ) ) -> d =/= (/) )
101	100	exp32 605	. . . . . . . . . . . . . . . . 17 |- ( d e. Word V -> ( ( # ` d ) = ( N + 2 ) -> ( N e. NN0 -> d =/= (/) ) ) )
102	101	com12 31	. . . . . . . . . . . . . . . 16 |- ( ( # ` d ) = ( N + 2 ) -> ( d e. Word V -> ( N e. NN0 -> d =/= (/) ) ) )
103	102	3ad2ant1 1017	. . . . . . . . . . . . . . 15 |- ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) -> ( d e. Word V -> ( N e. NN0 -> d =/= (/) ) ) )
104	103	impcom 430	. . . . . . . . . . . . . 14 |- ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) -> ( N e. NN0 -> d =/= (/) ) )
105	104	adantr 465	. . . . . . . . . . . . 13 |- ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) -> ( N e. NN0 -> d =/= (/) ) )
106	105	imp 429	. . . . . . . . . . . 12 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) -> d =/= (/) )
107	94	adantl 466	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( # ` x ) = ( N + 2 ) /\ N e. NN0 ) -> 0 < ( N + 2 ) )
108		breq2 4451	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` x ) = ( N + 2 ) -> ( 0 < ( # ` x ) <-> 0 < ( N + 2 ) ) )
109	108	adantr 465	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( # ` x ) = ( N + 2 ) /\ N e. NN0 ) -> ( 0 < ( # ` x ) <-> 0 < ( N + 2 ) ) )
110	107, 109	mpbird 232	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( # ` x ) = ( N + 2 ) /\ N e. NN0 ) -> 0 < ( # ` x ) )
111		hashgt0n0 12397	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. Word V /\ 0 < ( # ` x ) ) -> x =/= (/) )
112	110, 111	sylan2 474	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ N e. NN0 ) ) -> x =/= (/) )
113	112	exp32 605	. . . . . . . . . . . . . . . . 17 |- ( x e. Word V -> ( ( # ` x ) = ( N + 2 ) -> ( N e. NN0 -> x =/= (/) ) ) )
114	113	com12 31	. . . . . . . . . . . . . . . 16 |- ( ( # ` x ) = ( N + 2 ) -> ( x e. Word V -> ( N e. NN0 -> x =/= (/) ) ) )
115	114	3ad2ant1 1017	. . . . . . . . . . . . . . 15 |- ( ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) -> ( x e. Word V -> ( N e. NN0 -> x =/= (/) ) ) )
116	115	impcom 430	. . . . . . . . . . . . . 14 |- ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) -> ( N e. NN0 -> x =/= (/) ) )
117	116	adantl 466	. . . . . . . . . . . . 13 |- ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) -> ( N e. NN0 -> x =/= (/) ) )
118	117	imp 429	. . . . . . . . . . . 12 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) -> x =/= (/) )
119	87, 106, 118	jca32 535	. . . . . . . . . . 11 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) -> ( ( d e. Word V /\ x e. Word V ) /\ ( d =/= (/) /\ x =/= (/) ) ) )
120	119	adantr 465	. . . . . . . . . 10 |- ( ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) /\ ( lastS ` d ) = ( lastS ` x ) ) -> ( ( d e. Word V /\ x e. Word V ) /\ ( d =/= (/) /\ x =/= (/) ) ) )
121		wrdeqswrdlsw 12631	. . . . . . . . . 10 |- ( ( ( d e. Word V /\ x e. Word V ) /\ ( d =/= (/) /\ x =/= (/) ) ) -> ( d = x <-> ( ( # ` d ) = ( # ` x ) /\ ( lastS ` d ) = ( lastS ` x ) /\ ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
122	120, 121	syl 16	. . . . . . . . 9 |- ( ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) /\ ( lastS ` d ) = ( lastS ` x ) ) -> ( d = x <-> ( ( # ` d ) = ( # ` x ) /\ ( lastS ` d ) = ( lastS ` x ) /\ ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
123	46, 47, 83, 122	mpbir3and 1179	. . . . . . . 8 |- ( ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) /\ ( lastS ` d ) = ( lastS ` x ) ) -> d = x )
124	123	ex 434	. . . . . . 7 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) /\ N e. NN0 ) -> ( ( lastS ` d ) = ( lastS ` x ) -> d = x ) )
125	124	ex 434	. . . . . 6 |- ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) ) ) -> ( N e. NN0 -> ( ( lastS ` d ) = ( lastS ` x ) -> d = x ) ) )
126	25, 36, 125	syl2anb 479	. . . . 5 |- ( ( d e. D /\ x e. D ) -> ( N e. NN0 -> ( ( lastS ` d ) = ( lastS ` x ) -> d = x ) ) )
127	126	impcom 430	. . . 4 |- ( ( N e. NN0 /\ ( d e. D /\ x e. D ) ) -> ( ( lastS ` d ) = ( lastS ` x ) -> d = x ) )
128	14, 127	sylbid 215	. . 3 |- ( ( N e. NN0 /\ ( d e. D /\ x e. D ) ) -> ( ( F ` d ) = ( F ` x ) -> d = x ) )
129	128	ralrimivva 2885	. 2 |- ( N e. NN0 -> A. d e. D A. x e. D ( ( F ` d ) = ( F ` x ) -> d = x ) )
130		dff13 6152	. 2 |- ( F : D -1-1-> R <-> ( F : D --> R /\ A. d e. D A. x e. D ( ( F ` d ) = ( F ` x ) -> d = x ) ) )
131	4, 129, 130	sylanbrc 664	1 |- ( N e. NN0 -> F : D -1-1-> R )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113   (/)c0 3785   {cpr 4029   <.cop 4033   class class class wbr 4447   |-> cmpt 4505   ran crn 5000   -->wf 5582   -1-1->wf1 5583   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   + caddc 9491   < clt 9624   - cmin 9801   2c2 10581   NN0cn0 10791   #chash 12367  Word cword 12494   lastS clsw 12495   substr csubstr 12498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-lsw 12503  df-substr 12506

This theorem is referenced by:  wwlkextbij0  24405