Theorem numclwlk1lem2f1 27227

Description: T is a 1-1 function. (Contributed by AV, 26-Sep-2018.) (Revised by AV, 29-May-2021.)

Hypotheses

Ref	Expression
extwwlkfab.v	|- V = (Vtx ` G )
extwwlkfab.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
extwwlkfab.c	|- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
numclwwlk.t	|- T = ( u e. ( X C N ) |-> <. ( u substr <. 0 , ( N - 2 ) >. ) , ( u ` ( N - 1 ) ) >. )

Assertion

Ref	Expression
numclwlk1lem2f1	|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> T : ( X C N ) -1-1-> ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) )

Distinct variable groups:   n, G, u, v, w   n, N, u, v, w   n, V, v, w   n, X, u, v, w   w, F   u, C   u, F   u, V   u, T

Allowed substitution hints:   C( w, v, n)   T( w, v, n)   F( v, n)

Proof of Theorem numclwlk1lem2f1

Dummy variables a p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		extwwlkfab.v	. . 3 |- V = (Vtx ` G )
2		extwwlkfab.f	. . 3 |- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
3		extwwlkfab.c	. . 3 |- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
4		numclwwlk.t	. . 3 |- T = ( u e. ( X C N ) |-> <. ( u substr <. 0 , ( N - 2 ) >. ) , ( u ` ( N - 1 ) ) >. )
5	1, 2, 3, 4	numclwlk1lem2f 27225	. 2 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> T : ( X C N ) --> ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) )
6	1, 2, 3, 4	numclwlk1lem2fv 27226	. . . . . 6 |- ( p e. ( X C N ) -> ( T ` p ) = <. ( p substr <. 0 , ( N - 2 ) >. ) , ( p ` ( N - 1 ) ) >. )
7	6	ad2antrl 764	. . . . 5 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( p e. ( X C N ) /\ a e. ( X C N ) ) ) -> ( T ` p ) = <. ( p substr <. 0 , ( N - 2 ) >. ) , ( p ` ( N - 1 ) ) >. )
8	1, 2, 3, 4	numclwlk1lem2fv 27226	. . . . . 6 |- ( a e. ( X C N ) -> ( T ` a ) = <. ( a substr <. 0 , ( N - 2 ) >. ) , ( a ` ( N - 1 ) ) >. )
9	8	ad2antll 765	. . . . 5 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( p e. ( X C N ) /\ a e. ( X C N ) ) ) -> ( T ` a ) = <. ( a substr <. 0 , ( N - 2 ) >. ) , ( a ` ( N - 1 ) ) >. )
10	7, 9	eqeq12d 2637	. . . 4 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( p e. ( X C N ) /\ a e. ( X C N ) ) ) -> ( ( T ` p ) = ( T ` a ) <-> <. ( p substr <. 0 , ( N - 2 ) >. ) , ( p ` ( N - 1 ) ) >. = <. ( a substr <. 0 , ( N - 2 ) >. ) , ( a ` ( N - 1 ) ) >. ) )
11		ovex 6678	. . . . . 6 |- ( p substr <. 0 , ( N - 2 ) >. ) e. _V
12		fvex 6201	. . . . . 6 |- ( p ` ( N - 1 ) ) e. _V
13	11, 12	opth 4945	. . . . 5 |- ( <. ( p substr <. 0 , ( N - 2 ) >. ) , ( p ` ( N - 1 ) ) >. = <. ( a substr <. 0 , ( N - 2 ) >. ) , ( a ` ( N - 1 ) ) >. <-> ( ( p substr <. 0 , ( N - 2 ) >. ) = ( a substr <. 0 , ( N - 2 ) >. ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) )
14		uzuzle23 11729	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 3 ) -> N e. ( ZZ>= ` 2 ) )
15	3	numclwwlkovgel 27221	. . . . . . . . . 10 |- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( p e. ( X C N ) <-> ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) )
16	14, 15	sylan2 491	. . . . . . . . 9 |- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( p e. ( X C N ) <-> ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) )
17	1	clwwlknbp 26885	. . . . . . . . . . 11 |- ( p e. ( N ClWWalksN G ) -> ( p e. Word V /\ ( # ` p ) = N ) )
18	17	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) -> ( p e. Word V /\ ( # ` p ) = N ) )
19		3simpc 1060	. . . . . . . . . 10 |- ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) -> ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) )
20	18, 19	jca 554	. . . . . . . . 9 |- ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) -> ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) )
21	16, 20	syl6bi 243	. . . . . . . 8 |- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( p e. ( X C N ) -> ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) ) )
22	21	3adant1 1079	. . . . . . 7 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( p e. ( X C N ) -> ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) ) )
23	3	numclwwlkovgel 27221	. . . . . . . . . 10 |- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( a e. ( X C N ) <-> ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) )
24	14, 23	sylan2 491	. . . . . . . . 9 |- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( a e. ( X C N ) <-> ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) )
25	1	clwwlknbp 26885	. . . . . . . . . . 11 |- ( a e. ( N ClWWalksN G ) -> ( a e. Word V /\ ( # ` a ) = N ) )
26	25	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) -> ( a e. Word V /\ ( # ` a ) = N ) )
27		3simpc 1060	. . . . . . . . . 10 |- ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) -> ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) )
28	26, 27	jca 554	. . . . . . . . 9 |- ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) -> ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) )
29	24, 28	syl6bi 243	. . . . . . . 8 |- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( a e. ( X C N ) -> ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) )
30	29	3adant1 1079	. . . . . . 7 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( a e. ( X C N ) -> ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) )
31		simpll 790	. . . . . . . . . . 11 |- ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) -> p e. Word V )
32	31	ad2antrl 764	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> p e. Word V )
33		simprll 802	. . . . . . . . . . 11 |- ( ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) -> a e. Word V )
34	33	adantl 482	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> a e. Word V )
35		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( N = ( # ` p ) -> ( N e. ( ZZ>= ` 3 ) <-> ( # ` p ) e. ( ZZ>= ` 3 ) ) )
36	35	eqcoms 2630	. . . . . . . . . . . . . . . 16 |- ( ( # ` p ) = N -> ( N e. ( ZZ>= ` 3 ) <-> ( # ` p ) e. ( ZZ>= ` 3 ) ) )
37		eluz2 11693	. . . . . . . . . . . . . . . . 17 |- ( ( # ` p ) e. ( ZZ>= ` 3 ) <-> ( 3 e. ZZ /\ ( # ` p ) e. ZZ /\ 3 <_ ( # ` p ) ) )
38		1red 10055	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` p ) e. ZZ -> 1 e. RR )
39		3re 11094	. . . . . . . . . . . . . . . . . . . . . 22 |- 3 e. RR
40	39	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` p ) e. ZZ -> 3 e. RR )
41		zre 11381	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` p ) e. ZZ -> ( # ` p ) e. RR )
42	38, 40, 41	3jca 1242	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` p ) e. ZZ -> ( 1 e. RR /\ 3 e. RR /\ ( # ` p ) e. RR ) )
43		1lt3 11196	. . . . . . . . . . . . . . . . . . . 20 |- 1 < 3
44		ltletr 10129	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( 1 e. RR /\ 3 e. RR /\ ( # ` p ) e. RR ) -> ( ( 1 < 3 /\ 3 <_ ( # ` p ) ) -> 1 < ( # ` p ) ) )
45	44	expd 452	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 1 e. RR /\ 3 e. RR /\ ( # ` p ) e. RR ) -> ( 1 < 3 -> ( 3 <_ ( # ` p ) -> 1 < ( # ` p ) ) ) )
46	42, 43, 45	mpisyl 21	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` p ) e. ZZ -> ( 3 <_ ( # ` p ) -> 1 < ( # ` p ) ) )
47	46	imp 445	. . . . . . . . . . . . . . . . . 18 |- ( ( ( # ` p ) e. ZZ /\ 3 <_ ( # ` p ) ) -> 1 < ( # ` p ) )
48	47	3adant1 1079	. . . . . . . . . . . . . . . . 17 |- ( ( 3 e. ZZ /\ ( # ` p ) e. ZZ /\ 3 <_ ( # ` p ) ) -> 1 < ( # ` p ) )
49	37, 48	sylbi 207	. . . . . . . . . . . . . . . 16 |- ( ( # ` p ) e. ( ZZ>= ` 3 ) -> 1 < ( # ` p ) )
50	36, 49	syl6bi 243	. . . . . . . . . . . . . . 15 |- ( ( # ` p ) = N -> ( N e. ( ZZ>= ` 3 ) -> 1 < ( # ` p ) ) )
51	50	com12 32	. . . . . . . . . . . . . 14 |- ( N e. ( ZZ>= ` 3 ) -> ( ( # ` p ) = N -> 1 < ( # ` p ) ) )
52	51	3ad2ant3 1084	. . . . . . . . . . . . 13 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( # ` p ) = N -> 1 < ( # ` p ) ) )
53	52	com12 32	. . . . . . . . . . . 12 |- ( ( # ` p ) = N -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> 1 < ( # ` p ) ) )
54	53	ad3antlr 767	. . . . . . . . . . 11 |- ( ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> 1 < ( # ` p ) ) )
55	54	impcom 446	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> 1 < ( # ` p ) )
56		2swrd2eqwrdeq 13696	. . . . . . . . . 10 |- ( ( p e. Word V /\ a e. Word V /\ 1 < ( # ` p ) ) -> ( p = a <-> ( ( # ` p ) = ( # ` a ) /\ ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) /\ ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( lastS ` p ) = ( lastS ` a ) ) ) ) )
57	32, 34, 55, 56	syl3anc 1326	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> ( p = a <-> ( ( # ` p ) = ( # ` a ) /\ ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) /\ ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( lastS ` p ) = ( lastS ` a ) ) ) ) )
58		eqtr3 2643	. . . . . . . . . . . . . . . 16 |- ( ( ( # ` p ) = N /\ ( # ` a ) = N ) -> ( # ` p ) = ( # ` a ) )
59	58	expcom 451	. . . . . . . . . . . . . . 15 |- ( ( # ` a ) = N -> ( ( # ` p ) = N -> ( # ` p ) = ( # ` a ) ) )
60	59	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) -> ( ( # ` p ) = N -> ( # ` p ) = ( # ` a ) ) )
61	60	com12 32	. . . . . . . . . . . . 13 |- ( ( # ` p ) = N -> ( ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) -> ( # ` p ) = ( # ` a ) ) )
62	61	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) -> ( ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) -> ( # ` p ) = ( # ` a ) ) )
63	62	imp 445	. . . . . . . . . . 11 |- ( ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) -> ( # ` p ) = ( # ` a ) )
64	63	adantl 482	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> ( # ` p ) = ( # ` a ) )
65	64	biantrurd 529	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> ( ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) /\ ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( lastS ` p ) = ( lastS ` a ) ) <-> ( ( # ` p ) = ( # ` a ) /\ ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) /\ ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( lastS ` p ) = ( lastS ` a ) ) ) ) )
66		3anan12 1051	. . . . . . . . . . 11 |- ( ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) /\ ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( lastS ` p ) = ( lastS ` a ) ) <-> ( ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) /\ ( lastS ` p ) = ( lastS ` a ) ) ) )
67	66	a1i 11	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> ( ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) /\ ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( lastS ` p ) = ( lastS ` a ) ) <-> ( ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) /\ ( lastS ` p ) = ( lastS ` a ) ) ) ) )
68		eqeq2 2633	. . . . . . . . . . . . . . . . . . 19 |- ( ( p ` 0 ) = X -> ( ( p ` ( N - 2 ) ) = ( p ` 0 ) <-> ( p ` ( N - 2 ) ) = X ) )
69		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N = ( # ` p ) -> ( N - 2 ) = ( ( # ` p ) - 2 ) )
70	69	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` p ) = N -> ( N - 2 ) = ( ( # ` p ) - 2 ) )
71	70	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` p ) = N -> ( p ` ( N - 2 ) ) = ( p ` ( ( # ` p ) - 2 ) ) )
72	71	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` p ) = N -> ( ( p ` ( N - 2 ) ) = X <-> ( p ` ( ( # ` p ) - 2 ) ) = X ) )
73	72	biimpcd 239	. . . . . . . . . . . . . . . . . . 19 |- ( ( p ` ( N - 2 ) ) = X -> ( ( # ` p ) = N -> ( p ` ( ( # ` p ) - 2 ) ) = X ) )
74	68, 73	syl6bi 243	. . . . . . . . . . . . . . . . . 18 |- ( ( p ` 0 ) = X -> ( ( p ` ( N - 2 ) ) = ( p ` 0 ) -> ( ( # ` p ) = N -> ( p ` ( ( # ` p ) - 2 ) ) = X ) ) )
75	74	imp 445	. . . . . . . . . . . . . . . . 17 |- ( ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) -> ( ( # ` p ) = N -> ( p ` ( ( # ` p ) - 2 ) ) = X ) )
76	75	com12 32	. . . . . . . . . . . . . . . 16 |- ( ( # ` p ) = N -> ( ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) -> ( p ` ( ( # ` p ) - 2 ) ) = X ) )
77	76	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( p e. Word V /\ ( # ` p ) = N ) -> ( ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) -> ( p ` ( ( # ` p ) - 2 ) ) = X ) )
78	77	imp 445	. . . . . . . . . . . . . 14 |- ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) -> ( p ` ( ( # ` p ) - 2 ) ) = X )
79	78	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) -> ( p ` ( ( # ` p ) - 2 ) ) = X )
80		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( ( # ` p ) = N -> ( ( # ` p ) - 2 ) = ( N - 2 ) )
81	80	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( # ` p ) = N -> ( a ` ( ( # ` p ) - 2 ) ) = ( a ` ( N - 2 ) ) )
82		eqeq1 2626	. . . . . . . . . . . . . . . . . 18 |- ( ( a ` 0 ) = ( a ` ( N - 2 ) ) -> ( ( a ` 0 ) = X <-> ( a ` ( N - 2 ) ) = X ) )
83	82	eqcoms 2630	. . . . . . . . . . . . . . . . 17 |- ( ( a ` ( N - 2 ) ) = ( a ` 0 ) -> ( ( a ` 0 ) = X <-> ( a ` ( N - 2 ) ) = X ) )
84	83	biimpac 503	. . . . . . . . . . . . . . . 16 |- ( ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) -> ( a ` ( N - 2 ) ) = X )
85	84	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) -> ( a ` ( N - 2 ) ) = X )
86	81, 85	sylan9eq 2676	. . . . . . . . . . . . . 14 |- ( ( ( # ` p ) = N /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) -> ( a ` ( ( # ` p ) - 2 ) ) = X )
87	86	ad4ant24 1298	. . . . . . . . . . . . 13 |- ( ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) -> ( a ` ( ( # ` p ) - 2 ) ) = X )
88	79, 87	eqtr4d 2659	. . . . . . . . . . . 12 |- ( ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) -> ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) )
89	88	adantl 482	. . . . . . . . . . 11 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) )
90	89	biantrurd 529	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> ( ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) /\ ( lastS ` p ) = ( lastS ` a ) ) <-> ( ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) /\ ( lastS ` p ) = ( lastS ` a ) ) ) ) )
91	80	opeq2d 4409	. . . . . . . . . . . . . . 15 |- ( ( # ` p ) = N -> <. 0 , ( ( # ` p ) - 2 ) >. = <. 0 , ( N - 2 ) >. )
92	91	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( # ` p ) = N -> ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( p substr <. 0 , ( N - 2 ) >. ) )
93	91	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( # ` p ) = N -> ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( N - 2 ) >. ) )
94	92, 93	eqeq12d 2637	. . . . . . . . . . . . 13 |- ( ( # ` p ) = N -> ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) <-> ( p substr <. 0 , ( N - 2 ) >. ) = ( a substr <. 0 , ( N - 2 ) >. ) ) )
95	94	ad3antlr 767	. . . . . . . . . . . 12 |- ( ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) -> ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) <-> ( p substr <. 0 , ( N - 2 ) >. ) = ( a substr <. 0 , ( N - 2 ) >. ) ) )
96	95	adantl 482	. . . . . . . . . . 11 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) <-> ( p substr <. 0 , ( N - 2 ) >. ) = ( a substr <. 0 , ( N - 2 ) >. ) ) )
97		lsw 13351	. . . . . . . . . . . . . . 15 |- ( p e. Word V -> ( lastS ` p ) = ( p ` ( ( # ` p ) - 1 ) ) )
98		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( ( # ` p ) = N -> ( ( # ` p ) - 1 ) = ( N - 1 ) )
99	98	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( # ` p ) = N -> ( p ` ( ( # ` p ) - 1 ) ) = ( p ` ( N - 1 ) ) )
100	97, 99	sylan9eq 2676	. . . . . . . . . . . . . 14 |- ( ( p e. Word V /\ ( # ` p ) = N ) -> ( lastS ` p ) = ( p ` ( N - 1 ) ) )
101	100	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) -> ( lastS ` p ) = ( p ` ( N - 1 ) ) )
102		lsw 13351	. . . . . . . . . . . . . . . 16 |- ( a e. Word V -> ( lastS ` a ) = ( a ` ( ( # ` a ) - 1 ) ) )
103	102	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( a e. Word V /\ ( # ` a ) = N ) -> ( lastS ` a ) = ( a ` ( ( # ` a ) - 1 ) ) )
104		oveq1 6657	. . . . . . . . . . . . . . . . . . 19 |- ( N = ( # ` a ) -> ( N - 1 ) = ( ( # ` a ) - 1 ) )
105	104	eqcoms 2630	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` a ) = N -> ( N - 1 ) = ( ( # ` a ) - 1 ) )
106	105	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( ( # ` a ) = N -> ( a ` ( N - 1 ) ) = ( a ` ( ( # ` a ) - 1 ) ) )
107	106	eqeq2d 2632	. . . . . . . . . . . . . . . 16 |- ( ( # ` a ) = N -> ( ( lastS ` a ) = ( a ` ( N - 1 ) ) <-> ( lastS ` a ) = ( a ` ( ( # ` a ) - 1 ) ) ) )
108	107	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( a e. Word V /\ ( # ` a ) = N ) -> ( ( lastS ` a ) = ( a ` ( N - 1 ) ) <-> ( lastS ` a ) = ( a ` ( ( # ` a ) - 1 ) ) ) )
109	103, 108	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( a e. Word V /\ ( # ` a ) = N ) -> ( lastS ` a ) = ( a ` ( N - 1 ) ) )
110	109	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) -> ( lastS ` a ) = ( a ` ( N - 1 ) ) )
111	101, 110	eqeqan12d 2638	. . . . . . . . . . . 12 |- ( ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) -> ( ( lastS ` p ) = ( lastS ` a ) <-> ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) )
112	111	adantl 482	. . . . . . . . . . 11 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> ( ( lastS ` p ) = ( lastS ` a ) <-> ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) )
113	96, 112	anbi12d 747	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> ( ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) /\ ( lastS ` p ) = ( lastS ` a ) ) <-> ( ( p substr <. 0 , ( N - 2 ) >. ) = ( a substr <. 0 , ( N - 2 ) >. ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) )
114	67, 90, 113	3bitr2d 296	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> ( ( ( p substr <. 0 , ( ( # ` p ) - 2 ) >. ) = ( a substr <. 0 , ( ( # ` p ) - 2 ) >. ) /\ ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( lastS ` p ) = ( lastS ` a ) ) <-> ( ( p substr <. 0 , ( N - 2 ) >. ) = ( a substr <. 0 , ( N - 2 ) >. ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) )
115	57, 65, 114	3bitr2d 296	. . . . . . . 8 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) ) -> ( p = a <-> ( ( p substr <. 0 , ( N - 2 ) >. ) = ( a substr <. 0 , ( N - 2 ) >. ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) )
116	115	exbiri 652	. . . . . . 7 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) -> ( ( ( p substr <. 0 , ( N - 2 ) >. ) = ( a substr <. 0 , ( N - 2 ) >. ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) -> p = a ) ) )
117	22, 30, 116	syl2and 500	. . . . . 6 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( p e. ( X C N ) /\ a e. ( X C N ) ) -> ( ( ( p substr <. 0 , ( N - 2 ) >. ) = ( a substr <. 0 , ( N - 2 ) >. ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) -> p = a ) ) )
118	117	imp 445	. . . . 5 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( p e. ( X C N ) /\ a e. ( X C N ) ) ) -> ( ( ( p substr <. 0 , ( N - 2 ) >. ) = ( a substr <. 0 , ( N - 2 ) >. ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) -> p = a ) )
119	13, 118	syl5bi 232	. . . 4 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( p e. ( X C N ) /\ a e. ( X C N ) ) ) -> ( <. ( p substr <. 0 , ( N - 2 ) >. ) , ( p ` ( N - 1 ) ) >. = <. ( a substr <. 0 , ( N - 2 ) >. ) , ( a ` ( N - 1 ) ) >. -> p = a ) )
120	10, 119	sylbid 230	. . 3 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( p e. ( X C N ) /\ a e. ( X C N ) ) ) -> ( ( T ` p ) = ( T ` a ) -> p = a ) )
121	120	ralrimivva 2971	. 2 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> A. p e. ( X C N ) A. a e. ( X C N ) ( ( T ` p ) = ( T ` a ) -> p = a ) )
122		dff13 6512	. 2 |- ( T : ( X C N ) -1-1-> ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) <-> ( T : ( X C N ) --> ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) /\ A. p e. ( X C N ) A. a e. ( X C N ) ( ( T ` p ) = ( T ` a ) -> p = a ) ) )
123	5, 121, 122	sylanbrc 698	1 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> T : ( X C N ) -1-1-> ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   3c3 11071   ZZcz 11377   ZZ>=cuz 11687   #chash 13117  Word cword 13291   lastS clsw 13292   substr csubstr 13295  Vtxcvtx 25874   USGraph cusgr 26044   NeighbVtx cnbgr 26224   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-rmo 2920  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-2o 7561  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-cda 8990  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-2 11079  df-3 11080  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  df-s2 13593  df-edg 25940  df-upgr 25977  df-umgr 25978  df-usgr 26046  df-nbgr 26228  df-wwlks 26722  df-wwlksn 26723  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  numclwlk1lem2f1o  27229