Theorem clwwlkccat 16138

Description: The concatenation of two words representing closed walks anchored at the same vertex represents a closed walk. The resulting walk is a "double loop", starting at the common vertex, coming back to the common vertex by the first walk, following the second walk and finally coming back to the common vertex again. (Contributed by AV, 23-Apr-2022.)

Assertion

Ref	Expression
clwwlkccat	|- ( ( A e. (ClWWalks ` G ) /\ B e. (ClWWalks ` G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ++ B ) e. (ClWWalks ` G ) )

Proof of Theorem clwwlkccat

Dummy variables i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1l 1045	. . . . . 6 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) -> A e. Word (Vtx ` G ) )
2		simp1l 1045	. . . . . 6 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) -> B e. Word (Vtx ` G ) )
3		ccatcl 11141	. . . . . 6 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) ) -> ( A ++ B ) e. Word (Vtx ` G ) )
4	1, 2, 3	syl2an 289	. . . . 5 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) ) -> ( A ++ B ) e. Word (Vtx ` G ) )
5		ccat0 11144	. . . . . . . . . . 11 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) ) -> ( ( A ++ B ) = (/) <-> ( A = (/) /\ B = (/) ) ) )
6	5	adantlr 477	. . . . . . . . . 10 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) -> ( ( A ++ B ) = (/) <-> ( A = (/) /\ B = (/) ) ) )
7		simpr 110	. . . . . . . . . 10 |- ( ( A = (/) /\ B = (/) ) -> B = (/) )
8	6, 7	biimtrdi 163	. . . . . . . . 9 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) -> ( ( A ++ B ) = (/) -> B = (/) ) )
9	8	necon3d 2444	. . . . . . . 8 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) -> ( B =/= (/) -> ( A ++ B ) =/= (/) ) )
10	9	impr 379	. . . . . . 7 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( A ++ B ) =/= (/) )
11	10	3ad2antr1 1186	. . . . . 6 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) ) -> ( A ++ B ) =/= (/) )
12	11	3ad2antl1 1183	. . . . 5 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) ) -> ( A ++ B ) =/= (/) )
13	4, 12	jca 306	. . . 4 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) ) -> ( ( A ++ B ) e. Word (Vtx ` G ) /\ ( A ++ B ) =/= (/) ) )
14	13	3adant3 1041	. . 3 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( A ++ B ) e. Word (Vtx ` G ) /\ ( A ++ B ) =/= (/) ) )
15		clwwlkccatlem 16137	. . 3 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> A. i e. ( 0..^ ( (♯ ` ( A ++ B ) ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) )
16		simpl1l 1072	. . . . . . 7 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) ) -> A e. Word (Vtx ` G ) )
17		simpr1l 1078	. . . . . . 7 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) ) -> B e. Word (Vtx ` G ) )
18		simpr1r 1079	. . . . . . 7 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) ) -> B =/= (/) )
19		lswccatn0lsw 11159	. . . . . . 7 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) /\ B =/= (/) ) -> (lastS ` ( A ++ B ) ) = (lastS ` B ) )
20	16, 17, 18, 19	syl3anc 1271	. . . . . 6 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) ) -> (lastS ` ( A ++ B ) ) = (lastS ` B ) )
21	20	3adant3 1041	. . . . 5 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> (lastS ` ( A ++ B ) ) = (lastS ` B ) )
22		wrdfin 11103	. . . . . . . . . . 11 |- ( A e. Word (Vtx ` G ) -> A e. Fin )
23		fihashgt0 11040	. . . . . . . . . . 11 |- ( ( A e. Fin /\ A =/= (/) ) -> 0 < (♯ ` A ) )
24	22, 23	sylan 283	. . . . . . . . . 10 |- ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> 0 < (♯ ` A ) )
25	24	3ad2ant1 1042	. . . . . . . . 9 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) -> 0 < (♯ ` A ) )
26	25	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) ) -> 0 < (♯ ` A ) )
27		ccatfv0 11151	. . . . . . . 8 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) /\ 0 < (♯ ` A ) ) -> ( ( A ++ B ) ` 0 ) = ( A ` 0 ) )
28	16, 17, 26, 27	syl3anc 1271	. . . . . . 7 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) ) -> ( ( A ++ B ) ` 0 ) = ( A ` 0 ) )
29	28	3adant3 1041	. . . . . 6 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( A ++ B ) ` 0 ) = ( A ` 0 ) )
30		simp3 1023	. . . . . 6 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ` 0 ) = ( B ` 0 ) )
31	29, 30	eqtrd 2262	. . . . 5 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( A ++ B ) ` 0 ) = ( B ` 0 ) )
32	21, 31	preq12d 3751	. . . 4 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> { (lastS ` ( A ++ B ) ) , ( ( A ++ B ) ` 0 ) } = { (lastS ` B ) , ( B ` 0 ) } )
33		simp23 1056	. . . 4 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) )
34	32, 33	eqeltrd 2306	. . 3 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> { (lastS ` ( A ++ B ) ) , ( ( A ++ B ) ` 0 ) } e. (Edg ` G ) )
35	14, 15, 34	3jca 1201	. 2 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( ( A ++ B ) e. Word (Vtx ` G ) /\ ( A ++ B ) =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` ( A ++ B ) ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` ( A ++ B ) ) , ( ( A ++ B ) ` 0 ) } e. (Edg ` G ) ) )
36		eqid 2229	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
37		eqid 2229	. . . 4 |- (Edg ` G ) = (Edg ` G )
38	36, 37	isclwwlk 16132	. . 3 |- ( A e. (ClWWalks ` G ) <-> ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) )
39	36, 37	isclwwlk 16132	. . 3 |- ( B e. (ClWWalks ` G ) <-> ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) )
40		biid 171	. . 3 |- ( ( A ` 0 ) = ( B ` 0 ) <-> ( A ` 0 ) = ( B ` 0 ) )
41	38, 39, 40	3anbi123i 1212	. 2 |- ( ( A e. (ClWWalks ` G ) /\ B e. (ClWWalks ` G ) /\ ( A ` 0 ) = ( B ` 0 ) ) <-> ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) )
42	36, 37	isclwwlk 16132	. 2 |- ( ( A ++ B ) e. (ClWWalks ` G ) <-> ( ( ( A ++ B ) e. Word (Vtx ` G ) /\ ( A ++ B ) =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` ( A ++ B ) ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` ( A ++ B ) ) , ( ( A ++ B ) ` 0 ) } e. (Edg ` G ) ) )
43	35, 41, 42	3imtr4i 201	1 |- ( ( A e. (ClWWalks ` G ) /\ B e. (ClWWalks ` G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ++ B ) e. (ClWWalks ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   (/)c0 3491   {cpr 3667   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   Fincfn 6895   0cc0 8010   1c1 8011   + caddc 8013   < clt 8192   - cmin 8328  ..^cfzo 10350  ♯chash 11009  Word cword 11084  lastSclsw 11129   ++ cconcat 11138  Vtxcvtx 15828  Edgcedg 15873  ClWWalkscclwwlk 16129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-map 6805  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-rp 9862  df-fz 10217  df-fzo 10351  df-ihash 11010  df-word 11085  df-lsw 11130  df-concat 11139  df-ndx 13050  df-slot 13051  df-base 13053  df-vtx 15830  df-clwwlk 16130

This theorem is referenced by: (None)