Theorem erclwwlkstr 26936

Description: .~ is a transitive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)

Hypothesis

Ref	Expression
erclwwlks.r	|- .~ = { <. u , w >. | ( u e. (ClWWalks ` G ) /\ w e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` w ) ) u = ( w cyclShift n ) ) }

Assertion

Ref	Expression
erclwwlkstr	|- ( ( x .~ y /\ y .~ z ) -> x .~ z )

Distinct variable groups:   n, G, u, w   x, n, u, w, y   z, n, u, w, x

Allowed substitution hints:   .~ ( x, y, z, w, u, n)   G( x, y, z)

Proof of Theorem erclwwlkstr

Dummy variables m k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. 2 |- x e. _V
2		vex 3203	. 2 |- y e. _V
3		vex 3203	. 2 |- z e. _V
4		erclwwlks.r	. . . . . 6 |- .~ = { <. u , w >. | ( u e. (ClWWalks ` G ) /\ w e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` w ) ) u = ( w cyclShift n ) ) }
5	4	erclwwlkseqlen 26933	. . . . 5 |- ( ( x e. _V /\ y e. _V ) -> ( x .~ y -> ( # ` x ) = ( # ` y ) ) )
6	5	3adant3 1081	. . . 4 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( x .~ y -> ( # ` x ) = ( # ` y ) ) )
7	4	erclwwlkseqlen 26933	. . . . . . 7 |- ( ( y e. _V /\ z e. _V ) -> ( y .~ z -> ( # ` y ) = ( # ` z ) ) )
8	7	3adant1 1079	. . . . . 6 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( y .~ z -> ( # ` y ) = ( # ` z ) ) )
9	4	erclwwlkseq 26932	. . . . . . . 8 |- ( ( y e. _V /\ z e. _V ) -> ( y .~ z <-> ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) ) )
10	9	3adant1 1079	. . . . . . 7 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( y .~ z <-> ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) ) )
11	4	erclwwlkseq 26932	. . . . . . . . . 10 |- ( ( x e. _V /\ y e. _V ) -> ( x .~ y <-> ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) ) ) )
12	11	3adant3 1081	. . . . . . . . 9 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( x .~ y <-> ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) ) ) )
13		simpr1 1067	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) ) /\ ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) ) ) -> x e. (ClWWalks ` G ) )
14		simplr2 1104	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) ) /\ ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) ) ) -> z e. (ClWWalks ` G ) )
15		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( n = m -> ( y cyclShift n ) = ( y cyclShift m ) )
16	15	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( n = m -> ( x = ( y cyclShift n ) <-> x = ( y cyclShift m ) ) )
17	16	cbvrexv 3172	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) <-> E. m e. ( 0 ... ( # ` y ) ) x = ( y cyclShift m ) )
18		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( n = k -> ( z cyclShift n ) = ( z cyclShift k ) )
19	18	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( n = k -> ( y = ( z cyclShift n ) <-> y = ( z cyclShift k ) ) )
20	19	cbvrexv 3172	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) <-> E. k e. ( 0 ... ( # ` z ) ) y = ( z cyclShift k ) )
21		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (Vtx ` G ) = (Vtx ` G )
22	21	clwwlkbp 26883	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( z e. (ClWWalks ` G ) -> ( G e. _V /\ z e. Word (Vtx ` G ) /\ z =/= (/) ) )
23	22	simp2d 1074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( z e. (ClWWalks ` G ) -> z e. Word (Vtx ` G ) )
24	23	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) /\ z e. (ClWWalks ` G ) ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> z e. Word (Vtx ` G ) )
25		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) /\ z e. (ClWWalks ` G ) ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) )
26	24, 25	cshwcsh2id 13574	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) /\ z e. (ClWWalks ` G ) ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( ( ( m e. ( 0 ... ( # ` y ) ) /\ x = ( y cyclShift m ) ) /\ ( k e. ( 0 ... ( # ` z ) ) /\ y = ( z cyclShift k ) ) ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) )
27	26	exp5l 646	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) /\ z e. (ClWWalks ` G ) ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( m e. ( 0 ... ( # ` y ) ) -> ( x = ( y cyclShift m ) -> ( k e. ( 0 ... ( # ` z ) ) -> ( y = ( z cyclShift k ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) ) ) )
28	27	imp41 619	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( ( ( ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) /\ z e. (ClWWalks ` G ) ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) /\ m e. ( 0 ... ( # ` y ) ) ) /\ x = ( y cyclShift m ) ) /\ k e. ( 0 ... ( # ` z ) ) ) -> ( y = ( z cyclShift k ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) )
29	28	rexlimdva 3031	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( ( ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) /\ z e. (ClWWalks ` G ) ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) /\ m e. ( 0 ... ( # ` y ) ) ) /\ x = ( y cyclShift m ) ) -> ( E. k e. ( 0 ... ( # ` z ) ) y = ( z cyclShift k ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) )
30	29	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) /\ z e. (ClWWalks ` G ) ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) /\ m e. ( 0 ... ( # ` y ) ) ) -> ( x = ( y cyclShift m ) -> ( E. k e. ( 0 ... ( # ` z ) ) y = ( z cyclShift k ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) )
31	30	rexlimdva 3031	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) /\ z e. (ClWWalks ` G ) ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( E. m e. ( 0 ... ( # ` y ) ) x = ( y cyclShift m ) -> ( E. k e. ( 0 ... ( # ` z ) ) y = ( z cyclShift k ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) )
32	20, 31	syl7bi 245	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) /\ z e. (ClWWalks ` G ) ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( E. m e. ( 0 ... ( # ` y ) ) x = ( y cyclShift m ) -> ( E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) )
33	17, 32	syl5bi 232	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) /\ z e. (ClWWalks ` G ) ) /\ ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) -> ( E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) )
34	33	exp31 630	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) -> ( z e. (ClWWalks ` G ) -> ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) -> ( E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) ) ) )
35	34	com15 101	. . . . . . . . . . . . . . . . . . . . 21 |- ( E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) -> ( z e. (ClWWalks ` G ) -> ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) -> ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) ) ) )
36	35	impcom 446	. . . . . . . . . . . . . . . . . . . 20 |- ( ( z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) -> ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) -> ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) ) )
37	36	3adant1 1079	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) -> ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) -> ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) ) )
38	37	impcom 446	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) ) -> ( E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) -> ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) )
39	38	com13 88	. . . . . . . . . . . . . . . . 17 |- ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) ) -> ( E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) -> ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) )
40	39	3impia 1261	. . . . . . . . . . . . . . . 16 |- ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) ) -> ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) )
41	40	impcom 446	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) ) /\ ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) ) ) -> E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) )
42	13, 14, 41	3jca 1242	. . . . . . . . . . . . . 14 |- ( ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) ) /\ ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) ) ) -> ( x e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) )
43	4	erclwwlkseq 26932	. . . . . . . . . . . . . . 15 |- ( ( x e. _V /\ z e. _V ) -> ( x .~ z <-> ( x e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) )
44	43	3adant2 1080	. . . . . . . . . . . . . 14 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( x .~ z <-> ( x e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) x = ( z cyclShift n ) ) ) )
45	42, 44	syl5ibrcom 237	. . . . . . . . . . . . 13 |- ( ( ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) /\ ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) ) /\ ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) ) ) -> ( ( x e. _V /\ y e. _V /\ z e. _V ) -> x .~ z ) )
46	45	exp31 630	. . . . . . . . . . . 12 |- ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) -> ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) ) -> ( ( x e. _V /\ y e. _V /\ z e. _V ) -> x .~ z ) ) ) )
47	46	com24 95	. . . . . . . . . . 11 |- ( ( ( # ` y ) = ( # ` z ) /\ ( # ` x ) = ( # ` y ) ) -> ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) ) -> ( ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) -> x .~ z ) ) ) )
48	47	ex 450	. . . . . . . . . 10 |- ( ( # ` y ) = ( # ` z ) -> ( ( # ` x ) = ( # ` y ) -> ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) ) -> ( ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) -> x .~ z ) ) ) ) )
49	48	com4t 93	. . . . . . . . 9 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( ( x e. (ClWWalks ` G ) /\ y e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` y ) ) x = ( y cyclShift n ) ) -> ( ( # ` y ) = ( # ` z ) -> ( ( # ` x ) = ( # ` y ) -> ( ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) -> x .~ z ) ) ) ) )
50	12, 49	sylbid 230	. . . . . . . 8 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( x .~ y -> ( ( # ` y ) = ( # ` z ) -> ( ( # ` x ) = ( # ` y ) -> ( ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) -> x .~ z ) ) ) ) )
51	50	com25 99	. . . . . . 7 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( ( y e. (ClWWalks ` G ) /\ z e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` z ) ) y = ( z cyclShift n ) ) -> ( ( # ` y ) = ( # ` z ) -> ( ( # ` x ) = ( # ` y ) -> ( x .~ y -> x .~ z ) ) ) ) )
52	10, 51	sylbid 230	. . . . . 6 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( y .~ z -> ( ( # ` y ) = ( # ` z ) -> ( ( # ` x ) = ( # ` y ) -> ( x .~ y -> x .~ z ) ) ) ) )
53	8, 52	mpdd 43	. . . . 5 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( y .~ z -> ( ( # ` x ) = ( # ` y ) -> ( x .~ y -> x .~ z ) ) ) )
54	53	com24 95	. . . 4 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( x .~ y -> ( ( # ` x ) = ( # ` y ) -> ( y .~ z -> x .~ z ) ) ) )
55	6, 54	mpdd 43	. . 3 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( x .~ y -> ( y .~ z -> x .~ z ) ) )
56	55	impd 447	. 2 |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ( ( x .~ y /\ y .~ z ) -> x .~ z ) )
57	1, 2, 3, 56	mp3an 1424	1 |- ( ( x .~ y /\ y .~ z ) -> x .~ z )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   {copab 4712   ` cfv 5888  (class class class)co 6650   0cc0 9936   ...cfz 12326   #chash 13117  Word cword 13291   cyclShift ccsh 13534  Vtxcvtx 25874  ClWWalkscclwwlks 26875

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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-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-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-csh 13535  df-clwwlks 26877

This theorem is referenced by:  erclwwlks  26937