Theorem clwwlksf1 26917

Description: Lemma 3 for clwwlksbij 26920: F is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.)

Hypotheses

Ref	Expression
clwwlksbij.d	|- D = { w e. ( N WWalksN G ) | ( lastS ` w ) = ( w ` 0 ) }
clwwlksbij.f	|- F = ( t e. D |-> ( t substr <. 0 , N >. ) )

Assertion

Ref	Expression
clwwlksf1	|- ( N e. NN -> F : D -1-1-> ( N ClWWalksN G ) )

Distinct variable groups:   w, G   w, N   t, D   t, G, w   t, N

Allowed substitution hints:   D( w)   F( w, t)

Proof of Theorem clwwlksf1

Dummy variables i x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwwlksbij.d	. . 3 |- D = { w e. ( N WWalksN G ) | ( lastS ` w ) = ( w ` 0 ) }
2		clwwlksbij.f	. . 3 |- F = ( t e. D |-> ( t substr <. 0 , N >. ) )
3	1, 2	clwwlksf 26915	. 2 |- ( N e. NN -> F : D --> ( N ClWWalksN G ) )
4	1, 2	clwwlksfv 26916	. . . . . 6 |- ( x e. D -> ( F ` x ) = ( x substr <. 0 , N >. ) )
5	1, 2	clwwlksfv 26916	. . . . . 6 |- ( y e. D -> ( F ` y ) = ( y substr <. 0 , N >. ) )
6	4, 5	eqeqan12d 2638	. . . . 5 |- ( ( x e. D /\ y e. D ) -> ( ( F ` x ) = ( F ` y ) <-> ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) ) )
7	6	adantl 482	. . . 4 |- ( ( N e. NN /\ ( x e. D /\ y e. D ) ) -> ( ( F ` x ) = ( F ` y ) <-> ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) ) )
8		fveq2 6191	. . . . . . . . 9 |- ( w = x -> ( lastS ` w ) = ( lastS ` x ) )
9		fveq1 6190	. . . . . . . . 9 |- ( w = x -> ( w ` 0 ) = ( x ` 0 ) )
10	8, 9	eqeq12d 2637	. . . . . . . 8 |- ( w = x -> ( ( lastS ` w ) = ( w ` 0 ) <-> ( lastS ` x ) = ( x ` 0 ) ) )
11	10, 1	elrab2 3366	. . . . . . 7 |- ( x e. D <-> ( x e. ( N WWalksN G ) /\ ( lastS ` x ) = ( x ` 0 ) ) )
12		fveq2 6191	. . . . . . . . 9 |- ( w = y -> ( lastS ` w ) = ( lastS ` y ) )
13		fveq1 6190	. . . . . . . . 9 |- ( w = y -> ( w ` 0 ) = ( y ` 0 ) )
14	12, 13	eqeq12d 2637	. . . . . . . 8 |- ( w = y -> ( ( lastS ` w ) = ( w ` 0 ) <-> ( lastS ` y ) = ( y ` 0 ) ) )
15	14, 1	elrab2 3366	. . . . . . 7 |- ( y e. D <-> ( y e. ( N WWalksN G ) /\ ( lastS ` y ) = ( y ` 0 ) ) )
16	11, 15	anbi12i 733	. . . . . 6 |- ( ( x e. D /\ y e. D ) <-> ( ( x e. ( N WWalksN G ) /\ ( lastS ` x ) = ( x ` 0 ) ) /\ ( y e. ( N WWalksN G ) /\ ( lastS ` y ) = ( y ` 0 ) ) ) )
17		eqid 2622	. . . . . . . . . 10 |- (Vtx ` G ) = (Vtx ` G )
18		eqid 2622	. . . . . . . . . 10 |- (Edg ` G ) = (Edg ` G )
19	17, 18	wwlknp 26734	. . . . . . . . 9 |- ( x e. ( N WWalksN G ) -> ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) /\ A. i e. ( 0..^ N ) { ( x ` i ) , ( x ` ( i + 1 ) ) } e. (Edg ` G ) ) )
20	17, 18	wwlknp 26734	. . . . . . . . . . . . 13 |- ( y e. ( N WWalksN G ) -> ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) /\ A. i e. ( 0..^ N ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) ) )
21		simprlr 803	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) -> ( # ` x ) = ( N + 1 ) )
22		simpllr 799	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) -> ( # ` y ) = ( N + 1 ) )
23	21, 22	eqtr4d 2659	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) -> ( # ` x ) = ( # ` y ) )
24	23	ad2antlr 763	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) /\ ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) ) -> ( # ` x ) = ( # ` y ) )
25		nncn 11028	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( N e. NN -> N e. CC )
26		ax-1cn 9994	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 1 e. CC
27		pncan 10287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N + 1 ) - 1 ) = N )
28	27	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( N e. CC /\ 1 e. CC ) -> N = ( ( N + 1 ) - 1 ) )
29	25, 26, 28	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. NN -> N = ( ( N + 1 ) - 1 ) )
30		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( # ` x ) = ( N + 1 ) -> ( ( # ` x ) - 1 ) = ( ( N + 1 ) - 1 ) )
31	30	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( # ` x ) = ( N + 1 ) -> ( ( N + 1 ) - 1 ) = ( ( # ` x ) - 1 ) )
32	29, 31	sylan9eqr 2678	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( # ` x ) = ( N + 1 ) /\ N e. NN ) -> N = ( ( # ` x ) - 1 ) )
33	32	opeq2d 4409	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( # ` x ) = ( N + 1 ) /\ N e. NN ) -> <. 0 , N >. = <. 0 , ( ( # ` x ) - 1 ) >. )
34	33	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( # ` x ) = ( N + 1 ) /\ N e. NN ) -> ( x substr <. 0 , N >. ) = ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) )
35	33	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( # ` x ) = ( N + 1 ) /\ N e. NN ) -> ( y substr <. 0 , N >. ) = ( y substr <. 0 , ( ( # ` x ) - 1 ) >. ) )
36	34, 35	eqeq12d 2637	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( # ` x ) = ( N + 1 ) /\ N e. NN ) -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) <-> ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) = ( y substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) )
37	36	ex 450	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( # ` x ) = ( N + 1 ) -> ( N e. NN -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) <-> ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) = ( y substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) ) )
38	37	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) -> ( N e. NN -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) <-> ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) = ( y substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) ) )
39	38	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) -> ( N e. NN -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) <-> ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) = ( y substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) ) )
40	39	impcom 446	. . . . . . . . . . . . . . . . . . . 20 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) <-> ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) = ( y substr <. 0 , ( ( # ` x ) - 1 ) >. ) ) )
41	40	biimpa 501	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) /\ ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) ) -> ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) = ( y substr <. 0 , ( ( # ` x ) - 1 ) >. ) )
42		simpll 790	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) -> y e. Word (Vtx ` G ) )
43		simpll 790	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) -> x e. Word (Vtx ` G ) )
44	42, 43	anim12ci 591	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) -> ( x e. Word (Vtx ` G ) /\ y e. Word (Vtx ` G ) ) )
45	44	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> ( x e. Word (Vtx ` G ) /\ y e. Word (Vtx ` G ) ) )
46		nnnn0 11299	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( N e. NN -> N e. NN0 )
47		0nn0 11307	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 0 e. NN0
48	46, 47	jctil 560	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( N e. NN -> ( 0 e. NN0 /\ N e. NN0 ) )
49	48	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> ( 0 e. NN0 /\ N e. NN0 ) )
50		nnre 11027	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. NN -> N e. RR )
51	50	lep1d 10955	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. NN -> N <_ ( N + 1 ) )
52		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( # ` x ) = ( N + 1 ) -> ( N <_ ( # ` x ) <-> N <_ ( N + 1 ) ) )
53	51, 52	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( # ` x ) = ( N + 1 ) -> ( N e. NN -> N <_ ( # ` x ) ) )
54	53	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) -> ( N e. NN -> N <_ ( # ` x ) ) )
55	54	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) -> ( N e. NN -> N <_ ( # ` x ) ) )
56	55	impcom 446	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> N <_ ( # ` x ) )
57		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( # ` y ) = ( N + 1 ) -> ( N <_ ( # ` y ) <-> N <_ ( N + 1 ) ) )
58	51, 57	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( # ` y ) = ( N + 1 ) -> ( N e. NN -> N <_ ( # ` y ) ) )
59	58	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) -> ( N e. NN -> N <_ ( # ` y ) ) )
60	59	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) -> ( N e. NN -> N <_ ( # ` y ) ) )
61	60	impcom 446	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> N <_ ( # ` y ) )
62		swrdspsleq 13449	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( x e. Word (Vtx ` G ) /\ y e. Word (Vtx ` G ) ) /\ ( 0 e. NN0 /\ N e. NN0 ) /\ ( N <_ ( # ` x ) /\ N <_ ( # ` y ) ) ) -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) <-> A. i e. ( 0..^ N ) ( x ` i ) = ( y ` i ) ) )
63	45, 49, 56, 61, 62	syl112anc 1330	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) <-> A. i e. ( 0..^ N ) ( x ` i ) = ( y ` i ) ) )
64		lbfzo0 12507	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( 0 e. ( 0..^ N ) <-> N e. NN )
65	64	biimpri 218	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( N e. NN -> 0 e. ( 0..^ N ) )
66	65	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> 0 e. ( 0..^ N ) )
67		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( i = 0 -> ( x ` i ) = ( x ` 0 ) )
68		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( i = 0 -> ( y ` i ) = ( y ` 0 ) )
69	67, 68	eqeq12d 2637	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( i = 0 -> ( ( x ` i ) = ( y ` i ) <-> ( x ` 0 ) = ( y ` 0 ) ) )
70	69	rspcv 3305	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( 0 e. ( 0..^ N ) -> ( A. i e. ( 0..^ N ) ( x ` i ) = ( y ` i ) -> ( x ` 0 ) = ( y ` 0 ) ) )
71	66, 70	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> ( A. i e. ( 0..^ N ) ( x ` i ) = ( y ` i ) -> ( x ` 0 ) = ( y ` 0 ) ) )
72	63, 71	sylbid 230	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> ( x ` 0 ) = ( y ` 0 ) ) )
73	72	imp 445	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) /\ ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) ) -> ( x ` 0 ) = ( y ` 0 ) )
74		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) -> ( lastS ` x ) = ( x ` 0 ) )
75		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) -> ( lastS ` y ) = ( y ` 0 ) )
76	74, 75	eqeqan12rd 2640	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) -> ( ( lastS ` x ) = ( lastS ` y ) <-> ( x ` 0 ) = ( y ` 0 ) ) )
77	76	ad2antlr 763	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) /\ ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) ) -> ( ( lastS ` x ) = ( lastS ` y ) <-> ( x ` 0 ) = ( y ` 0 ) ) )
78	73, 77	mpbird 247	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) /\ ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) ) -> ( lastS ` x ) = ( lastS ` y ) )
79	24, 41, 78	jca32 558	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) /\ ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) ) -> ( ( # ` x ) = ( # ` y ) /\ ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) = ( y substr <. 0 , ( ( # ` x ) - 1 ) >. ) /\ ( lastS ` x ) = ( lastS ` y ) ) ) )
80	43	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) -> x e. Word (Vtx ` G ) )
81	80	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> x e. Word (Vtx ` G ) )
82	42	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) -> y e. Word (Vtx ` G ) )
83	82	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> y e. Word (Vtx ` G ) )
84		1red 10055	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N e. NN -> 1 e. RR )
85		nngt0 11049	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N e. NN -> 0 < N )
86		0lt1 10550	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- 0 < 1
87	86	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N e. NN -> 0 < 1 )
88	50, 84, 85, 87	addgt0d 10602	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( N e. NN -> 0 < ( N + 1 ) )
89		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` x ) = ( N + 1 ) -> ( 0 < ( # ` x ) <-> 0 < ( N + 1 ) ) )
90	88, 89	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( # ` x ) = ( N + 1 ) -> ( N e. NN -> 0 < ( # ` x ) ) )
91	90	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) -> ( N e. NN -> 0 < ( # ` x ) ) )
92	91	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) -> ( N e. NN -> 0 < ( # ` x ) ) )
93	92	impcom 446	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> 0 < ( # ` x ) )
94	81, 83, 93	3jca 1242	. . . . . . . . . . . . . . . . . . . 20 |- ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) -> ( x e. Word (Vtx ` G ) /\ y e. Word (Vtx ` G ) /\ 0 < ( # ` x ) ) )
95	94	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) /\ ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) ) -> ( x e. Word (Vtx ` G ) /\ y e. Word (Vtx ` G ) /\ 0 < ( # ` x ) ) )
96		2swrd1eqwrdeq 13454	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. Word (Vtx ` G ) /\ y e. Word (Vtx ` G ) /\ 0 < ( # ` x ) ) -> ( x = y <-> ( ( # ` x ) = ( # ` y ) /\ ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) = ( y substr <. 0 , ( ( # ` x ) - 1 ) >. ) /\ ( lastS ` x ) = ( lastS ` y ) ) ) ) )
97	95, 96	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) /\ ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) ) -> ( x = y <-> ( ( # ` x ) = ( # ` y ) /\ ( ( x substr <. 0 , ( ( # ` x ) - 1 ) >. ) = ( y substr <. 0 , ( ( # ` x ) - 1 ) >. ) /\ ( lastS ` x ) = ( lastS ` y ) ) ) ) )
98	79, 97	mpbird 247	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) ) /\ ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) ) -> x = y )
99	98	exp31 630	. . . . . . . . . . . . . . . 16 |- ( N e. NN -> ( ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) /\ ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) ) -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) ) )
100	99	expdcom 455	. . . . . . . . . . . . . . 15 |- ( ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) /\ ( lastS ` y ) = ( y ` 0 ) ) -> ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) -> ( N e. NN -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) ) ) )
101	100	ex 450	. . . . . . . . . . . . . 14 |- ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) ) -> ( ( lastS ` y ) = ( y ` 0 ) -> ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) -> ( N e. NN -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) ) ) ) )
102	101	3adant3 1081	. . . . . . . . . . . . 13 |- ( ( y e. Word (Vtx ` G ) /\ ( # ` y ) = ( N + 1 ) /\ A. i e. ( 0..^ N ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) ) -> ( ( lastS ` y ) = ( y ` 0 ) -> ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) -> ( N e. NN -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) ) ) ) )
103	20, 102	syl 17	. . . . . . . . . . . 12 |- ( y e. ( N WWalksN G ) -> ( ( lastS ` y ) = ( y ` 0 ) -> ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) -> ( N e. NN -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) ) ) ) )
104	103	imp 445	. . . . . . . . . . 11 |- ( ( y e. ( N WWalksN G ) /\ ( lastS ` y ) = ( y ` 0 ) ) -> ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) /\ ( lastS ` x ) = ( x ` 0 ) ) -> ( N e. NN -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) ) ) )
105	104	expdcom 455	. . . . . . . . . 10 |- ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) ) -> ( ( lastS ` x ) = ( x ` 0 ) -> ( ( y e. ( N WWalksN G ) /\ ( lastS ` y ) = ( y ` 0 ) ) -> ( N e. NN -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) ) ) ) )
106	105	3adant3 1081	. . . . . . . . 9 |- ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = ( N + 1 ) /\ A. i e. ( 0..^ N ) { ( x ` i ) , ( x ` ( i + 1 ) ) } e. (Edg ` G ) ) -> ( ( lastS ` x ) = ( x ` 0 ) -> ( ( y e. ( N WWalksN G ) /\ ( lastS ` y ) = ( y ` 0 ) ) -> ( N e. NN -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) ) ) ) )
107	19, 106	syl 17	. . . . . . . 8 |- ( x e. ( N WWalksN G ) -> ( ( lastS ` x ) = ( x ` 0 ) -> ( ( y e. ( N WWalksN G ) /\ ( lastS ` y ) = ( y ` 0 ) ) -> ( N e. NN -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) ) ) ) )
108	107	imp31 448	. . . . . . 7 |- ( ( ( x e. ( N WWalksN G ) /\ ( lastS ` x ) = ( x ` 0 ) ) /\ ( y e. ( N WWalksN G ) /\ ( lastS ` y ) = ( y ` 0 ) ) ) -> ( N e. NN -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) ) )
109	108	com12 32	. . . . . 6 |- ( N e. NN -> ( ( ( x e. ( N WWalksN G ) /\ ( lastS ` x ) = ( x ` 0 ) ) /\ ( y e. ( N WWalksN G ) /\ ( lastS ` y ) = ( y ` 0 ) ) ) -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) ) )
110	16, 109	syl5bi 232	. . . . 5 |- ( N e. NN -> ( ( x e. D /\ y e. D ) -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) ) )
111	110	imp 445	. . . 4 |- ( ( N e. NN /\ ( x e. D /\ y e. D ) ) -> ( ( x substr <. 0 , N >. ) = ( y substr <. 0 , N >. ) -> x = y ) )
112	7, 111	sylbid 230	. . 3 |- ( ( N e. NN /\ ( x e. D /\ y e. D ) ) -> ( ( F ` x ) = ( F ` y ) -> x = y ) )
113	112	ralrimivva 2971	. 2 |- ( N e. NN -> A. x e. D A. y e. D ( ( F ` x ) = ( F ` y ) -> x = y ) )
114		dff13 6512	. 2 |- ( F : D -1-1-> ( N ClWWalksN G ) <-> ( F : D --> ( N ClWWalksN G ) /\ A. x e. D A. y e. D ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
115	3, 113, 114	sylanbrc 698	1 |- ( N e. NN -> F : D -1-1-> ( N ClWWalksN G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   {cpr 4179   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   substr csubstr 13295  Vtxcvtx 25874  Edgcedg 25939   WWalksN cwwlksn 26718   ClWWalksN cclwwlksn 26876

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-fal 1489  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-map 7859  df-pm 7860  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-s1 13302  df-substr 13303  df-wwlks 26722  df-wwlksn 26723  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  clwwlksf1o  26919