Theorem clwwlkccatlem 16137

Description: Lemma for clwwlkccat 16138: index j is shifted up by (♯ ` A ), and the case i = ( (♯ ` A ) - 1 ) is covered by the "bridge" { (lastS ` A ) , ( B ` 0 ) } = { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ). (Contributed by AV, 23-Apr-2022.)

Assertion

Ref

Expression

clwwlkccatlem

|- ( ( ( ( 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 ) )

Distinct variable groups:   A, i, j   B, i, j   i, G, j

Proof of Theorem clwwlkccatlem

Step	Hyp	Ref	Expression
1		simplll 533	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) /\ i e. ( 0..^ ( (♯ ` A ) - 1 ) ) ) -> A e. Word (Vtx ` G ) )
2		simplr 528	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) /\ i e. ( 0..^ ( (♯ ` A ) - 1 ) ) ) -> B e. Word (Vtx ` G ) )
3		lencl 11088	. . . . . . . . . . . . . . . . . . . . . 22 |- ( A e. Word (Vtx ` G ) -> (♯ ` A ) e. NN0 )
4	3	nn0zd 9578	. . . . . . . . . . . . . . . . . . . . 21 |- ( A e. Word (Vtx ` G ) -> (♯ ` A ) e. ZZ )
5		fzossrbm1 10383	. . . . . . . . . . . . . . . . . . . . 21 |- ( (♯ ` A ) e. ZZ -> ( 0..^ ( (♯ ` A ) - 1 ) ) C_ ( 0..^ (♯ ` A ) ) )
6	4, 5	syl 14	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. Word (Vtx ` G ) -> ( 0..^ ( (♯ ` A ) - 1 ) ) C_ ( 0..^ (♯ ` A ) ) )
7	6	ad2antrr 488	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) -> ( 0..^ ( (♯ ` A ) - 1 ) ) C_ ( 0..^ (♯ ` A ) ) )
8	7	sselda 3224	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) /\ i e. ( 0..^ ( (♯ ` A ) - 1 ) ) ) -> i e. ( 0..^ (♯ ` A ) ) )
9		ccatval1 11145	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) /\ i e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ B ) ` i ) = ( A ` i ) )
10	1, 2, 8, 9	syl3anc 1271	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) /\ i e. ( 0..^ ( (♯ ` A ) - 1 ) ) ) -> ( ( A ++ B ) ` i ) = ( A ` i ) )
11	4	ad2antrr 488	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) -> (♯ ` A ) e. ZZ )
12		elfzom1elp1fzo 10420	. . . . . . . . . . . . . . . . . . 19 |- ( ( (♯ ` A ) e. ZZ /\ i e. ( 0..^ ( (♯ ` A ) - 1 ) ) ) -> ( i + 1 ) e. ( 0..^ (♯ ` A ) ) )
13	11, 12	sylan 283	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) /\ i e. ( 0..^ ( (♯ ` A ) - 1 ) ) ) -> ( i + 1 ) e. ( 0..^ (♯ ` A ) ) )
14		ccatval1 11145	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) /\ ( i + 1 ) e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ B ) ` ( i + 1 ) ) = ( A ` ( i + 1 ) ) )
15	1, 2, 13, 14	syl3anc 1271	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) /\ i e. ( 0..^ ( (♯ ` A ) - 1 ) ) ) -> ( ( A ++ B ) ` ( i + 1 ) ) = ( A ` ( i + 1 ) ) )
16	10, 15	preq12d 3751	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) /\ i e. ( 0..^ ( (♯ ` A ) - 1 ) ) ) -> { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } = { ( A ` i ) , ( A ` ( i + 1 ) ) } )
17	16	eleq1d 2298	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) /\ i e. ( 0..^ ( (♯ ` A ) - 1 ) ) ) -> ( { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) ) )
18	17	biimprd 158	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) /\ i e. ( 0..^ ( (♯ ` A ) - 1 ) ) ) -> ( { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) -> { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
19	18	ralimdva 2597	. . . . . . . . . . . . 13 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ B e. Word (Vtx ` G ) ) -> ( A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) -> A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
20	19	impancom 260	. . . . . . . . . . . 12 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) ) -> ( B e. Word (Vtx ` G ) -> A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
21	20	3adant3 1041	. . . . . . . . . . 11 |- ( ( ( 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 ) -> A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
22	21	com12 30	. . . . . . . . . 10 |- ( B e. Word (Vtx ` 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 ) ) -> A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
23	22	adantr 276	. . . . . . . . 9 |- ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) -> ( ( ( 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. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
24	23	3ad2ant1 1042	. . . . . . . 8 |- ( ( ( 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 ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) -> A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
25	24	impcom 125	. . . . . . 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. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) )
26	25	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. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) )
27		simprl 529	. . . . . . . . . . . . . . . . 17 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) -> A e. Word (Vtx ` G ) )
28		simpll 527	. . . . . . . . . . . . . . . . 17 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) -> B e. Word (Vtx ` G ) )
29		simprr 531	. . . . . . . . . . . . . . . . 17 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) -> A =/= (/) )
30		ccatval1lsw 11152	. . . . . . . . . . . . . . . . 17 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) /\ A =/= (/) ) -> ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) = (lastS ` A ) )
31	27, 28, 29, 30	syl3anc 1271	. . . . . . . . . . . . . . . 16 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) -> ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) = (lastS ` A ) )
32	31	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) = (lastS ` A ) )
33	3	nn0cnd 9435	. . . . . . . . . . . . . . . . . . . . 21 |- ( A e. Word (Vtx ` G ) -> (♯ ` A ) e. CC )
34		npcan1 8535	. . . . . . . . . . . . . . . . . . . . 21 |- ( (♯ ` A ) e. CC -> ( ( (♯ ` A ) - 1 ) + 1 ) = (♯ ` A ) )
35	33, 34	syl 14	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. Word (Vtx ` G ) -> ( ( (♯ ` A ) - 1 ) + 1 ) = (♯ ` A ) )
36	35	ad2antrl 490	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) -> ( ( (♯ ` A ) - 1 ) + 1 ) = (♯ ` A ) )
37	36	fveq2d 5633	. . . . . . . . . . . . . . . . . 18 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) -> ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) = ( ( A ++ B ) ` (♯ ` A ) ) )
38		simplr 528	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) -> B =/= (/) )
39		ccatval21sw 11153	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) /\ B =/= (/) ) -> ( ( A ++ B ) ` (♯ ` A ) ) = ( B ` 0 ) )
40	27, 28, 38, 39	syl3anc 1271	. . . . . . . . . . . . . . . . . 18 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) -> ( ( A ++ B ) ` (♯ ` A ) ) = ( B ` 0 ) )
41	37, 40	eqtrd 2262	. . . . . . . . . . . . . . . . 17 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) -> ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) = ( B ` 0 ) )
42	41	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) = ( B ` 0 ) )
43		simpr 110	. . . . . . . . . . . . . . . 16 |- ( ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ` 0 ) = ( B ` 0 ) )
44	42, 43	eqtr4d 2265	. . . . . . . . . . . . . . 15 |- ( ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) = ( A ` 0 ) )
45	32, 44	preq12d 3751	. . . . . . . . . . . . . 14 |- ( ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> { ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } = { (lastS ` A ) , ( A ` 0 ) } )
46	45	eleq1d 2298	. . . . . . . . . . . . 13 |- ( ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( { ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } e. (Edg ` G ) <-> { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) )
47	46	exbiri 382	. . . . . . . . . . . 12 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> ( { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) -> { ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } e. (Edg ` G ) ) ) )
48	47	com23 78	. . . . . . . . . . 11 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word (Vtx ` G ) /\ A =/= (/) ) ) -> ( { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) -> ( ( A ` 0 ) = ( B ` 0 ) -> { ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } e. (Edg ` G ) ) ) )
49	48	expimpd 363	. . . . . . . . . 10 |- ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) -> ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> { ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } e. (Edg ` G ) ) ) )
50	49	3ad2ant1 1042	. . . . . . . . 9 |- ( ( ( 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 ) /\ A =/= (/) ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> { ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } e. (Edg ` G ) ) ) )
51	50	com12 30	. . . . . . . 8 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ { (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 ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } e. (Edg ` G ) ) ) )
52	51	3adant2 1040	. . . . . . 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 ` 0 ) = ( B ` 0 ) -> { ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } e. (Edg ` G ) ) ) )
53	52	3imp 1217	. . . . . 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 ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } e. (Edg ` G ) )
54	26, 53	jca 306	. . . . 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. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } e. (Edg ` G ) ) )
55		peano2zm 9495	. . . . . . . . . 10 |- ( (♯ ` A ) e. ZZ -> ( (♯ ` A ) - 1 ) e. ZZ )
56	4, 55	syl 14	. . . . . . . . 9 |- ( A e. Word (Vtx ` G ) -> ( (♯ ` A ) - 1 ) e. ZZ )
57	56	adantr 276	. . . . . . . 8 |- ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> ( (♯ ` A ) - 1 ) e. ZZ )
58	57	3ad2ant1 1042	. . . . . . 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 ) ) -> ( (♯ ` A ) - 1 ) e. ZZ )
59	58	3ad2ant1 1042	. . . . . 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 ) - 1 ) e. ZZ )
60		fveq2 5629	. . . . . . . . 9 |- ( i = ( (♯ ` A ) - 1 ) -> ( ( A ++ B ) ` i ) = ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) )
61		fvoveq1 6030	. . . . . . . . 9 |- ( i = ( (♯ ` A ) - 1 ) -> ( ( A ++ B ) ` ( i + 1 ) ) = ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) )
62	60, 61	preq12d 3751	. . . . . . . 8 |- ( i = ( (♯ ` A ) - 1 ) -> { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } = { ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } )
63	62	eleq1d 2298	. . . . . . 7 |- ( i = ( (♯ ` A ) - 1 ) -> ( { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } e. (Edg ` G ) ) )
64	63	ralunsn 3876	. . . . . 6 |- ( ( (♯ ` A ) - 1 ) e. ZZ -> ( A. i e. ( ( 0..^ ( (♯ ` A ) - 1 ) ) u. { ( (♯ ` A ) - 1 ) } ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) <-> ( A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } e. (Edg ` G ) ) ) )
65	59, 64	syl 14	. . . . 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. i e. ( ( 0..^ ( (♯ ` A ) - 1 ) ) u. { ( (♯ ` A ) - 1 ) } ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) <-> ( A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( (♯ ` A ) - 1 ) + 1 ) ) } e. (Edg ` G ) ) ) )
66	54, 65	mpbird 167	. . . 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 ) ) -> A. i e. ( ( 0..^ ( (♯ ` A ) - 1 ) ) u. { ( (♯ ` A ) - 1 ) } ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) )
67		0z 9468	. . . . . . . 8 |- 0 e. ZZ
68		lennncl 11104	. . . . . . . . 9 |- ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> (♯ ` A ) e. NN )
69		0p1e1 9235	. . . . . . . . . . . 12 |- ( 0 + 1 ) = 1
70	69	fveq2i 5632	. . . . . . . . . . 11 |- ( ZZ>= ` ( 0 + 1 ) ) = ( ZZ>= ` 1 )
71	70	eleq2i 2296	. . . . . . . . . 10 |- ( (♯ ` A ) e. ( ZZ>= ` ( 0 + 1 ) ) <-> (♯ ` A ) e. ( ZZ>= ` 1 ) )
72		elnnuz 9771	. . . . . . . . . 10 |- ( (♯ ` A ) e. NN <-> (♯ ` A ) e. ( ZZ>= ` 1 ) )
73	71, 72	bitr4i 187	. . . . . . . . 9 |- ( (♯ ` A ) e. ( ZZ>= ` ( 0 + 1 ) ) <-> (♯ ` A ) e. NN )
74	68, 73	sylibr 134	. . . . . . . 8 |- ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> (♯ ` A ) e. ( ZZ>= ` ( 0 + 1 ) ) )
75		fzosplitsnm1 10427	. . . . . . . 8 |- ( ( 0 e. ZZ /\ (♯ ` A ) e. ( ZZ>= ` ( 0 + 1 ) ) ) -> ( 0..^ (♯ ` A ) ) = ( ( 0..^ ( (♯ ` A ) - 1 ) ) u. { ( (♯ ` A ) - 1 ) } ) )
76	67, 74, 75	sylancr 414	. . . . . . 7 |- ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> ( 0..^ (♯ ` A ) ) = ( ( 0..^ ( (♯ ` A ) - 1 ) ) u. { ( (♯ ` A ) - 1 ) } ) )
77	76	raleqdv 2734	. . . . . 6 |- ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> ( A. i e. ( 0..^ (♯ ` A ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. ( ( 0..^ ( (♯ ` A ) - 1 ) ) u. { ( (♯ ` A ) - 1 ) } ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
78	77	3ad2ant1 1042	. . . . 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 ) ) -> ( A. i e. ( 0..^ (♯ ` A ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. ( ( 0..^ ( (♯ ` A ) - 1 ) ) u. { ( (♯ ` A ) - 1 ) } ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
79	78	3ad2ant1 1042	. . . 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 ) ) -> ( A. i e. ( 0..^ (♯ ` A ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. ( ( 0..^ ( (♯ ` A ) - 1 ) ) u. { ( (♯ ` A ) - 1 ) } ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
80	66, 79	mpbird 167	. . 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 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) )
81		lencl 11088	. . . . . . . . . . . . . . . . . . . . 21 |- ( B e. Word (Vtx ` G ) -> (♯ ` B ) e. NN0 )
82	81	nn0zd 9578	. . . . . . . . . . . . . . . . . . . 20 |- ( B e. Word (Vtx ` G ) -> (♯ ` B ) e. ZZ )
83		peano2zm 9495	. . . . . . . . . . . . . . . . . . . 20 |- ( (♯ ` B ) e. ZZ -> ( (♯ ` B ) - 1 ) e. ZZ )
84	82, 83	syl 14	. . . . . . . . . . . . . . . . . . 19 |- ( B e. Word (Vtx ` G ) -> ( (♯ ` B ) - 1 ) e. ZZ )
85	84	ad2antrl 490	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( (♯ ` B ) - 1 ) e. ZZ )
86	85	adantr 276	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( (♯ ` B ) - 1 ) e. ZZ )
87	86	anim1ci 341	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> ( i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) /\ ( (♯ ` B ) - 1 ) e. ZZ ) )
88		fzosubel3 10414	. . . . . . . . . . . . . . . 16 |- ( ( i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) /\ ( (♯ ` B ) - 1 ) e. ZZ ) -> ( i - (♯ ` A ) ) e. ( 0..^ ( (♯ ` B ) - 1 ) ) )
89		fveq2 5629	. . . . . . . . . . . . . . . . . . 19 |- ( j = ( i - (♯ ` A ) ) -> ( B ` j ) = ( B ` ( i - (♯ ` A ) ) ) )
90		fvoveq1 6030	. . . . . . . . . . . . . . . . . . 19 |- ( j = ( i - (♯ ` A ) ) -> ( B ` ( j + 1 ) ) = ( B ` ( ( i - (♯ ` A ) ) + 1 ) ) )
91	89, 90	preq12d 3751	. . . . . . . . . . . . . . . . . 18 |- ( j = ( i - (♯ ` A ) ) -> { ( B ` j ) , ( B ` ( j + 1 ) ) } = { ( B ` ( i - (♯ ` A ) ) ) , ( B ` ( ( i - (♯ ` A ) ) + 1 ) ) } )
92	91	eleq1d 2298	. . . . . . . . . . . . . . . . 17 |- ( j = ( i - (♯ ` A ) ) -> ( { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) <-> { ( B ` ( i - (♯ ` A ) ) ) , ( B ` ( ( i - (♯ ` A ) ) + 1 ) ) } e. (Edg ` G ) ) )
93	92	rspcv 2903	. . . . . . . . . . . . . . . 16 |- ( ( i - (♯ ` A ) ) e. ( 0..^ ( (♯ ` B ) - 1 ) ) -> ( A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) -> { ( B ` ( i - (♯ ` A ) ) ) , ( B ` ( ( i - (♯ ` A ) ) + 1 ) ) } e. (Edg ` G ) ) )
94	87, 88, 93	3syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> ( A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) -> { ( B ` ( i - (♯ ` A ) ) ) , ( B ` ( ( i - (♯ ` A ) ) + 1 ) ) } e. (Edg ` G ) ) )
95		simp-4l 541	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> A e. Word (Vtx ` G ) )
96		simprl 529	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> B e. Word (Vtx ` G ) )
97	96	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> B e. Word (Vtx ` G ) )
98	3	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> (♯ ` A ) e. NN0 )
99	81	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) -> (♯ ` B ) e. NN0 )
100		nn0addcl 9415	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( (♯ ` A ) e. NN0 /\ (♯ ` B ) e. NN0 ) -> ( (♯ ` A ) + (♯ ` B ) ) e. NN0 )
101	100	nn0zd 9578	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( (♯ ` A ) e. NN0 /\ (♯ ` B ) e. NN0 ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ZZ )
102	98, 99, 101	syl2an 289	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ZZ )
103		1nn0 9396	. . . . . . . . . . . . . . . . . . . . . . 23 |- 1 e. NN0
104		eluzmn 9740	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( (♯ ` A ) + (♯ ` B ) ) e. ZZ /\ 1 e. NN0 ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ( ZZ>= ` ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) )
105	102, 103, 104	sylancl 413	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ( ZZ>= ` ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) )
106	33	ad2antrr 488	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> (♯ ` A ) e. CC )
107	81	nn0cnd 9435	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( B e. Word (Vtx ` G ) -> (♯ ` B ) e. CC )
108	107	ad2antrl 490	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> (♯ ` B ) e. CC )
109		1cnd 8173	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> 1 e. CC )
110	106, 108, 109	addsubassd 8488	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) = ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) )
111	110	fveq2d 5633	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( ZZ>= ` ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) = ( ZZ>= ` ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) )
112	105, 111	eleqtrd 2308	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ( ZZ>= ` ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) )
113		fzoss2 10382	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( (♯ ` A ) + (♯ ` B ) ) e. ( ZZ>= ` ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) -> ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) C_ ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
114	112, 113	syl 14	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) C_ ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
115	114	adantr 276	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) C_ ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
116	115	sselda 3224	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> i e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
117		ccatval2 11146	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( ( A ++ B ) ` i ) = ( B ` ( i - (♯ ` A ) ) ) )
118	95, 97, 116, 117	syl3anc 1271	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> ( ( A ++ B ) ` i ) = ( B ` ( i - (♯ ` A ) ) ) )
119	110	oveq2d 6023	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( (♯ ` A )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) = ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) )
120	119	eleq2d 2299	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( i e. ( (♯ ` A )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) <-> i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) )
121	120	adantr 276	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( i e. ( (♯ ` A )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) <-> i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) )
122		eluzmn 9740	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( (♯ ` A ) e. ZZ /\ 1 e. NN0 ) -> (♯ ` A ) e. ( ZZ>= ` ( (♯ ` A ) - 1 ) ) )
123	4, 103, 122	sylancl 413	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( A e. Word (Vtx ` G ) -> (♯ ` A ) e. ( ZZ>= ` ( (♯ ` A ) - 1 ) ) )
124	123	ad3antrrr 492	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> (♯ ` A ) e. ( ZZ>= ` ( (♯ ` A ) - 1 ) ) )
125		fzoss1 10381	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( (♯ ` A ) e. ( ZZ>= ` ( (♯ ` A ) - 1 ) ) -> ( (♯ ` A )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) C_ ( ( (♯ ` A ) - 1 )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) )
126	124, 125	syl 14	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( (♯ ` A )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) C_ ( ( (♯ ` A ) - 1 )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) )
127	126	sseld 3223	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( i e. ( (♯ ` A )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) -> i e. ( ( (♯ ` A ) - 1 )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) ) )
128	121, 127	sylbird 170	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) -> i e. ( ( (♯ ` A ) - 1 )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) ) )
129	128	imp 124	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> i e. ( ( (♯ ` A ) - 1 )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) )
130	4	adantr 276	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> (♯ ` A ) e. ZZ )
131	82	adantr 276	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) -> (♯ ` B ) e. ZZ )
132		simpl 109	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( (♯ ` A ) e. ZZ /\ (♯ ` B ) e. ZZ ) -> (♯ ` A ) e. ZZ )
133		zaddcl 9497	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( (♯ ` A ) e. ZZ /\ (♯ ` B ) e. ZZ ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ZZ )
134	132, 133	jca 306	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( (♯ ` A ) e. ZZ /\ (♯ ` B ) e. ZZ ) -> ( (♯ ` A ) e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ ) )
135	130, 131, 134	syl2an 289	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( (♯ ` A ) e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ ) )
136	135	adantr 276	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( (♯ ` A ) e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ ) )
137		elfzoelz 10355	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) -> i e. ZZ )
138		1zzd 9484	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) -> 1 e. ZZ )
139	137, 138	jca 306	. . . . . . . . . . . . . . . . . . . . 21 |- ( i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) -> ( i e. ZZ /\ 1 e. ZZ ) )
140		elfzomelpfzo 10449	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( (♯ ` A ) e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ ) /\ ( i e. ZZ /\ 1 e. ZZ ) ) -> ( i e. ( ( (♯ ` A ) - 1 )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) <-> ( i + 1 ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) )
141	136, 139, 140	syl2an 289	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> ( i e. ( ( (♯ ` A ) - 1 )..^ ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) ) <-> ( i + 1 ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) )
142	129, 141	mpbid 147	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> ( i + 1 ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
143		ccatval2 11146	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) /\ ( i + 1 ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( ( A ++ B ) ` ( i + 1 ) ) = ( B ` ( ( i + 1 ) - (♯ ` A ) ) ) )
144	95, 97, 142, 143	syl3anc 1271	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> ( ( A ++ B ) ` ( i + 1 ) ) = ( B ` ( ( i + 1 ) - (♯ ` A ) ) ) )
145	137	zcnd 9581	. . . . . . . . . . . . . . . . . . . . 21 |- ( i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) -> i e. CC )
146	145	adantl 277	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> i e. CC )
147		1cnd 8173	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> 1 e. CC )
148	106	ad2antrr 488	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> (♯ ` A ) e. CC )
149	146, 147, 148	addsubd 8489	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> ( ( i + 1 ) - (♯ ` A ) ) = ( ( i - (♯ ` A ) ) + 1 ) )
150	149	fveq2d 5633	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> ( B ` ( ( i + 1 ) - (♯ ` A ) ) ) = ( B ` ( ( i - (♯ ` A ) ) + 1 ) ) )
151	144, 150	eqtrd 2262	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> ( ( A ++ B ) ` ( i + 1 ) ) = ( B ` ( ( i - (♯ ` A ) ) + 1 ) ) )
152	118, 151	preq12d 3751	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } = { ( B ` ( i - (♯ ` A ) ) ) , ( B ` ( ( i - (♯ ` A ) ) + 1 ) ) } )
153	152	eleq1d 2298	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> ( { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( B ` ( i - (♯ ` A ) ) ) , ( B ` ( ( i - (♯ ` A ) ) + 1 ) ) } e. (Edg ` G ) ) )
154	94, 153	sylibrd 169	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) -> ( A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) -> { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
155	154	impancom 260	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) ) -> ( i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) -> { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
156	155	ralrimiv 2602	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) ) -> A. i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) )
157	156	exp31 364	. . . . . . . . . . 11 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> ( A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) -> A. i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) ) )
158	157	expcom 116	. . . . . . . . . 10 |- ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) -> ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> ( A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) -> A. i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) ) ) )
159	158	com23 78	. . . . . . . . 9 |- ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> ( A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) -> A. i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) ) ) )
160	159	com24 87	. . . . . . . 8 |- ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) -> ( A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) -> ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> A. i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) ) ) )
161	160	imp 124	. . . . . . 7 |- ( ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) ) -> ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> A. i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) ) )
162	161	3adant3 1041	. . . . . 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 ) ) -> ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> A. i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) ) )
163	162	com12 30	. . . . 5 |- ( ( 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 ` 0 ) = ( B ` 0 ) -> A. i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) ) )
164	163	3ad2ant1 1042	. . . 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 ) -> A. i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) ) )
165	164	3imp 1217	. . 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. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) )
166		ralunb 3385	. . 3 |- ( A. i e. ( ( 0..^ (♯ ` A ) ) u. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) <-> ( A. i e. ( 0..^ (♯ ` A ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) /\ A. i e. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
167	80, 165, 166	sylanbrc 417	. 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. i e. ( ( 0..^ (♯ ` A ) ) u. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) )
168		ccatlen 11143	. . . . . . . . . 10 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
169	168	oveq1d 6022	. . . . . . . . 9 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) ) -> ( (♯ ` ( A ++ B ) ) - 1 ) = ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) )
170	169	ad2ant2r 509	. . . . . . . 8 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( (♯ ` ( A ++ B ) ) - 1 ) = ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) )
171	170, 110	eqtrd 2262	. . . . . . 7 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( (♯ ` ( A ++ B ) ) - 1 ) = ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) )
172	171	oveq2d 6023	. . . . . 6 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( 0..^ ( (♯ ` ( A ++ B ) ) - 1 ) ) = ( 0..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) )
173		elnn0uz 9772	. . . . . . . . 9 |- ( (♯ ` A ) e. NN0 <-> (♯ ` A ) e. ( ZZ>= ` 0 ) )
174	3, 173	sylib 122	. . . . . . . 8 |- ( A e. Word (Vtx ` G ) -> (♯ ` A ) e. ( ZZ>= ` 0 ) )
175	174	adantr 276	. . . . . . 7 |- ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) -> (♯ ` A ) e. ( ZZ>= ` 0 ) )
176		lennncl 11104	. . . . . . . 8 |- ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) -> (♯ ` B ) e. NN )
177		nnm1nn0 9421	. . . . . . . 8 |- ( (♯ ` B ) e. NN -> ( (♯ ` B ) - 1 ) e. NN0 )
178	176, 177	syl 14	. . . . . . 7 |- ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) -> ( (♯ ` B ) - 1 ) e. NN0 )
179		fzoun 10391	. . . . . . 7 |- ( ( (♯ ` A ) e. ( ZZ>= ` 0 ) /\ ( (♯ ` B ) - 1 ) e. NN0 ) -> ( 0..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) = ( ( 0..^ (♯ ` A ) ) u. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) )
180	175, 178, 179	syl2an 289	. . . . . 6 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( 0..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) = ( ( 0..^ (♯ ` A ) ) u. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) )
181	172, 180	eqtrd 2262	. . . . 5 |- ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word (Vtx ` G ) /\ B =/= (/) ) ) -> ( 0..^ ( (♯ ` ( A ++ B ) ) - 1 ) ) = ( ( 0..^ (♯ ` A ) ) u. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) )
182	181	3ad2antr1 1186	. . . 4 |- ( ( ( 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 ) ) ) -> ( 0..^ ( (♯ ` ( A ++ B ) ) - 1 ) ) = ( ( 0..^ (♯ ` A ) ) u. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) )
183	182	3ad2antl1 1183	. . 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 ) ) ) -> ( 0..^ ( (♯ ` ( A ++ B ) ) - 1 ) ) = ( ( 0..^ (♯ ` A ) ) u. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) )
184	183	3adant3 1041	. 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 ) ) -> ( 0..^ ( (♯ ` ( A ++ B ) ) - 1 ) ) = ( ( 0..^ (♯ ` A ) ) u. ( (♯ ` A )..^ ( (♯ ` A ) + ( (♯ ` B ) - 1 ) ) ) ) )
185	167, 184	raleqtrrdv 2738	1 |- ( ( ( ( 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   u. cun 3195   C_ wss 3197   (/)c0 3491   {csn 3666   {cpr 3667   ` cfv 5318  (class class class)co 6007   CCcc 8008   0cc0 8010   1c1 8011   + caddc 8013   - cmin 8328   NNcn 9121   NN0cn0 9380   ZZcz 9457   ZZ>=cuz 9733  ..^cfzo 10350  ♯chash 11009  Word cword 11084  lastSclsw 11129   ++ cconcat 11138  Vtxcvtx 15828  Edgcedg 15873

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-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126

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-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-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-fz 10217  df-fzo 10351  df-ihash 11010  df-word 11085  df-lsw 11130  df-concat 11139

This theorem is referenced by:  clwwlkccat  16138