Theorem wwlkextinj 30362

Description: Lemma 2 for wwlkextbij 30365. (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 30361	. 2 |- ( N e. NN0 -> F : D --> R )
5		fvex 5701	. . . . . . 7 |- ( lastS ` d ) e. _V
6		fveq2 5691	. . . . . . . 8 |- ( t = d -> ( lastS ` t ) = ( lastS ` d ) )
7	6, 3	fvmptg 5772	. . . . . . 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 5701	. . . . . . 7 |- ( lastS ` x ) e. _V
10		fveq2 5691	. . . . . . . 8 |- ( t = x -> ( lastS ` t ) = ( lastS ` x ) )
11	10, 3	fvmptg 5772	. . . . . . 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 2458	. . . . 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 2507	. . . . . . 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 5691	. . . . . . . . . 10 |- ( w = d -> ( # ` w ) = ( # ` d ) )
17	16	eqeq1d 2451	. . . . . . . . 9 |- ( w = d -> ( ( # ` w ) = ( N + 2 ) <-> ( # ` d ) = ( N + 2 ) ) )
18		oveq1 6098	. . . . . . . . . 10 |- ( w = d -> ( w substr <. 0 , ( N + 1 ) >. ) = ( d substr <. 0 , ( N + 1 ) >. ) )
19	18	eqeq1d 2451	. . . . . . . . 9 |- ( w = d -> ( ( w substr <. 0 , ( N + 1 ) >. ) = W <-> ( d substr <. 0 , ( N + 1 ) >. ) = W ) )
20		fveq2 5691	. . . . . . . . . . 11 |- ( w = d -> ( lastS ` w ) = ( lastS ` d ) )
21	20	preq2d 3961	. . . . . . . . . 10 |- ( w = d -> { ( lastS ` W ) , ( lastS ` w ) } = { ( lastS ` W ) , ( lastS ` d ) } )
22	21	eleq1d 2509	. . . . . . . . 9 |- ( w = d -> ( { ( lastS ` W ) , ( lastS ` w ) } e. ran E <-> { ( lastS ` W ) , ( lastS ` d ) } e. ran E ) )
23	17, 19, 22	3anbi123d 1289	. . . . . . . 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 3117	. . . . . . 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 2507	. . . . . . 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 5691	. . . . . . . . . 10 |- ( w = x -> ( # ` w ) = ( # ` x ) )
28	27	eqeq1d 2451	. . . . . . . . 9 |- ( w = x -> ( ( # ` w ) = ( N + 2 ) <-> ( # ` x ) = ( N + 2 ) ) )
29		oveq1 6098	. . . . . . . . . 10 |- ( w = x -> ( w substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) )
30	29	eqeq1d 2451	. . . . . . . . 9 |- ( w = x -> ( ( w substr <. 0 , ( N + 1 ) >. ) = W <-> ( x substr <. 0 , ( N + 1 ) >. ) = W ) )
31		fveq2 5691	. . . . . . . . . . 11 |- ( w = x -> ( lastS ` w ) = ( lastS ` x ) )
32	31	preq2d 3961	. . . . . . . . . 10 |- ( w = x -> { ( lastS ` W ) , ( lastS ` w ) } = { ( lastS ` W ) , ( lastS ` x ) } )
33	32	eleq1d 2509	. . . . . . . . 9 |- ( w = x -> ( { ( lastS ` W ) , ( lastS ` w ) } e. ran E <-> { ( lastS ` W ) , ( lastS ` x ) } e. ran E ) )
34	28, 30, 33	3anbi123d 1289	. . . . . . . 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 3117	. . . . . . 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 2462	. . . . . . . . . . . . . . . . . 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 1009	. . . . . . . . . . . . . . . 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 1009	. . . . . . . . . . . . 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 2462	. . . . . . . . . . . . . . . . . . . . . 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 10437	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- 1 = ( 2 - 1 )
50	49	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( N e. NN0 -> 1 = ( 2 - 1 ) )
51	50	oveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( N e. NN0 -> ( N + 1 ) = ( N + ( 2 - 1 ) ) )
52		nn0cn 10589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( N e. NN0 -> N e. CC )
53		2cnd 10394	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( N e. NN0 -> 2 e. CC )
54		ax-1cn 9340	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- 1 e. CC
55	54	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( N e. NN0 -> 1 e. CC )
56	52, 53, 55	addsubassd 9739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( N e. NN0 -> ( ( N + 2 ) - 1 ) = ( N + ( 2 - 1 ) ) )
57	51, 56	eqtr4d 2478	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6098	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( # ` d ) = ( N + 2 ) -> ( ( # ` d ) - 1 ) = ( ( N + 2 ) - 1 ) )
60	59	eqeq2d 2454	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4060	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( N + 1 ) = ( ( # ` d ) - 1 ) -> <. 0 , ( N + 1 ) >. = <. 0 , ( ( # ` d ) - 1 ) >. )
64	63	oveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( N + 1 ) = ( ( # ` d ) - 1 ) -> ( d substr <. 0 , ( N + 1 ) >. ) = ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) )
65	63	oveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( N + 1 ) = ( ( # ` d ) - 1 ) -> ( x substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) )
66	64, 65	eqeq12d 2457	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1010	. . . . . . . . . . . . . . . 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 1008	. . . . . . . . . . . . 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 10588	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> N e. RR )
89		2re 10391	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 e. RR
90	89	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> 2 e. RR )
91		nn0ge0 10605	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> 0 <_ N )
92		2pos 10413	. . . . . . . . . . . . . . . . . . . . . . 23 |- 0 < 2
93	92	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> 0 < 2 )
94	88, 90, 91, 93	addgegt0d 9913	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> 0 < ( N + 2 ) )
95	94	adantl 466	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( # ` d ) = ( N + 2 ) /\ N e. NN0 ) -> 0 < ( N + 2 ) )
96		breq2 4296	. . . . . . . . . . . . . . . . . . . . 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 12133	. . . . . . . . . . . . . . . . . . 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 1009	. . . . . . . . . . . . . . 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 4296	. . . . . . . . . . . . . . . . . . . . 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 12133	. . . . . . . . . . . . . . . . . . 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 1009	. . . . . . . . . . . . . . 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 12343	. . . . . . . . . 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 1171	. . . . . . . 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 2808	. 2 |- ( N e. NN0 -> A. d e. D A. x e. D ( ( F ` d ) = ( F ` x ) -> d = x ) )
130		dff13 5971	. 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 965   = wceq 1369   e. wcel 1756   =/= wne 2606   A.wral 2715   {crab 2719   _Vcvv 2972   (/)c0 3637   {cpr 3879   <.cop 3883   class class class wbr 4292   e. cmpt 4350   ran crn 4841   -->wf 5414   -1-1->wf1 5415   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283   + caddc 9285   < clt 9418   - cmin 9595   2c2 10371   NN0cn0 10579   #chash 12103  Word cword 12221   lastS clsw 12222   substr csubstr 12225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-lsw 12230  df-substr 12233

This theorem is referenced by:  wwlkextbij0  30364