Theorem wlkp1lem8 26577

Description: Lemma for wlkp1 26578. (Contributed by AV, 6-Mar-2021.)

Hypotheses

Ref	Expression
wlkp1.v	|- V = (Vtx ` G )
wlkp1.i	|- I = (iEdg ` G )
wlkp1.f	|- ( ph -> Fun I )
wlkp1.a	|- ( ph -> I e. Fin )
wlkp1.b	|- ( ph -> B e. _V )
wlkp1.c	|- ( ph -> C e. V )
wlkp1.d	|- ( ph -> -. B e. dom I )
wlkp1.w	|- ( ph -> F (Walks ` G ) P )
wlkp1.n	|- N = ( # ` F )
wlkp1.e	|- ( ph -> E e. (Edg ` G ) )
wlkp1.x	|- ( ph -> { ( P ` N ) , C } C_ E )
wlkp1.u	|- ( ph -> (iEdg ` S ) = ( I u. { <. B , E >. } ) )
wlkp1.h	|- H = ( F u. { <. N , B >. } )
wlkp1.q	|- Q = ( P u. { <. ( N + 1 ) , C >. } )
wlkp1.s	|- ( ph -> (Vtx ` S ) = V )
wlkp1.l	|- ( ( ph /\ C = ( P ` N ) ) -> E = { C } )

Assertion

Ref	Expression
wlkp1lem8	|- ( ph -> A. k e. ( 0..^ ( # ` H ) )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) )

Distinct variable groups:   ph, k   k, N   P, k   Q, k   k, F   k, G   k, H   S, k

Allowed substitution hints:   B( k)   C( k)   E( k)   I( k)   V( k)

Proof of Theorem wlkp1lem8

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkp1.v	. . . 4 |- V = (Vtx ` G )
2		wlkp1.i	. . . 4 |- I = (iEdg ` G )
3		wlkp1.f	. . . 4 |- ( ph -> Fun I )
4		wlkp1.a	. . . 4 |- ( ph -> I e. Fin )
5		wlkp1.b	. . . 4 |- ( ph -> B e. _V )
6		wlkp1.c	. . . 4 |- ( ph -> C e. V )
7		wlkp1.d	. . . 4 |- ( ph -> -. B e. dom I )
8		wlkp1.w	. . . 4 |- ( ph -> F (Walks ` G ) P )
9		wlkp1.n	. . . 4 |- N = ( # ` F )
10		wlkp1.e	. . . 4 |- ( ph -> E e. (Edg ` G ) )
11		wlkp1.x	. . . 4 |- ( ph -> { ( P ` N ) , C } C_ E )
12		wlkp1.u	. . . 4 |- ( ph -> (iEdg ` S ) = ( I u. { <. B , E >. } ) )
13		wlkp1.h	. . . 4 |- H = ( F u. { <. N , B >. } )
14		wlkp1.q	. . . 4 |- Q = ( P u. { <. ( N + 1 ) , C >. } )
15		wlkp1.s	. . . 4 |- ( ph -> (Vtx ` S ) = V )
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	wlkp1lem6 26575	. . 3 |- ( ph -> A. k e. ( 0..^ N ) ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) )
17	10	elfvexd 6222	. . . . . 6 |- ( ph -> G e. _V )
18	1, 2	iswlkg 26509	. . . . . 6 |- ( G e. _V -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )
19	17, 18	syl 17	. . . . 5 |- ( ph -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )
20	9	eqcomi 2631	. . . . . . . . 9 |- ( # ` F ) = N
21	20	oveq2i 6661	. . . . . . . 8 |- ( 0..^ ( # ` F ) ) = ( 0..^ N )
22	21	raleqi 3142	. . . . . . 7 |- ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) <-> A. k e. ( 0..^ N )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
23	22	biimpi 206	. . . . . 6 |- ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> A. k e. ( 0..^ N )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
24	23	3ad2ant3 1084	. . . . 5 |- ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> A. k e. ( 0..^ N )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
25	19, 24	syl6bi 243	. . . 4 |- ( ph -> ( F (Walks ` G ) P -> A. k e. ( 0..^ N )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
26	8, 25	mpd 15	. . 3 |- ( ph -> A. k e. ( 0..^ N )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
27		eqeq12 2635	. . . . . . 7 |- ( ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) ) -> ( ( Q ` k ) = ( Q ` ( k + 1 ) ) <-> ( P ` k ) = ( P ` ( k + 1 ) ) ) )
28	27	3adant3 1081	. . . . . 6 |- ( ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) -> ( ( Q ` k ) = ( Q ` ( k + 1 ) ) <-> ( P ` k ) = ( P ` ( k + 1 ) ) ) )
29		simp3 1063	. . . . . . 7 |- ( ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) -> ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) )
30		simp1 1061	. . . . . . . 8 |- ( ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) -> ( Q ` k ) = ( P ` k ) )
31	30	sneqd 4189	. . . . . . 7 |- ( ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) -> { ( Q ` k ) } = { ( P ` k ) } )
32	29, 31	eqeq12d 2637	. . . . . 6 |- ( ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) -> ( ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } <-> ( I ` ( F ` k ) ) = { ( P ` k ) } ) )
33		preq12 4270	. . . . . . . 8 |- ( ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) ) -> { ( Q ` k ) , ( Q ` ( k + 1 ) ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
34	33	3adant3 1081	. . . . . . 7 |- ( ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) -> { ( Q ` k ) , ( Q ` ( k + 1 ) ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
35	34, 29	sseq12d 3634	. . . . . 6 |- ( ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) -> ( { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) <-> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
36	28, 32, 35	ifpbi123d 1027	. . . . 5 |- ( ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) -> (if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) <-> if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
37	36	biimprd 238	. . . 4 |- ( ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) ) )
38	37	ral2imi 2947	. . 3 |- ( A. k e. ( 0..^ N ) ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) -> ( A. k e. ( 0..^ N )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> A. k e. ( 0..^ N )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) ) )
39	16, 26, 38	sylc 65	. 2 |- ( ph -> A. k e. ( 0..^ N )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) )
40	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	wlkp1lem3 26572	. . . . 5 |- ( ph -> ( (iEdg ` S ) ` ( H ` N ) ) = ( ( I u. { <. B , E >. } ) ` B ) )
41	40	adantr 481	. . . 4 |- ( ( ph /\ ( Q ` N ) = ( Q ` ( N + 1 ) ) ) -> ( (iEdg ` S ) ` ( H ` N ) ) = ( ( I u. { <. B , E >. } ) ` B ) )
42	5, 10, 7	3jca 1242	. . . . . 6 |- ( ph -> ( B e. _V /\ E e. (Edg ` G ) /\ -. B e. dom I ) )
43	42	adantr 481	. . . . 5 |- ( ( ph /\ ( Q ` N ) = ( Q ` ( N + 1 ) ) ) -> ( B e. _V /\ E e. (Edg ` G ) /\ -. B e. dom I ) )
44		fsnunfv 6453	. . . . 5 |- ( ( B e. _V /\ E e. (Edg ` G ) /\ -. B e. dom I ) -> ( ( I u. { <. B , E >. } ) ` B ) = E )
45	43, 44	syl 17	. . . 4 |- ( ( ph /\ ( Q ` N ) = ( Q ` ( N + 1 ) ) ) -> ( ( I u. { <. B , E >. } ) ` B ) = E )
46		fveq2 6191	. . . . . . . 8 |- ( x = N -> ( Q ` x ) = ( Q ` N ) )
47		fveq2 6191	. . . . . . . 8 |- ( x = N -> ( P ` x ) = ( P ` N ) )
48	46, 47	eqeq12d 2637	. . . . . . 7 |- ( x = N -> ( ( Q ` x ) = ( P ` x ) <-> ( Q ` N ) = ( P ` N ) ) )
49	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	wlkp1lem5 26574	. . . . . . 7 |- ( ph -> A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) )
50	2	wlkf 26510	. . . . . . . . . 10 |- ( F (Walks ` G ) P -> F e. Word dom I )
51		lencl 13324	. . . . . . . . . . 11 |- ( F e. Word dom I -> ( # ` F ) e. NN0 )
52	9	eleq1i 2692	. . . . . . . . . . . 12 |- ( N e. NN0 <-> ( # ` F ) e. NN0 )
53		elnn0uz 11725	. . . . . . . . . . . 12 |- ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) )
54	52, 53	sylbb1 227	. . . . . . . . . . 11 |- ( ( # ` F ) e. NN0 -> N e. ( ZZ>= ` 0 ) )
55	51, 54	syl 17	. . . . . . . . . 10 |- ( F e. Word dom I -> N e. ( ZZ>= ` 0 ) )
56	8, 50, 55	3syl 18	. . . . . . . . 9 |- ( ph -> N e. ( ZZ>= ` 0 ) )
57	56, 53	sylibr 224	. . . . . . . 8 |- ( ph -> N e. NN0 )
58		nn0fz0 12437	. . . . . . . 8 |- ( N e. NN0 <-> N e. ( 0 ... N ) )
59	57, 58	sylib 208	. . . . . . 7 |- ( ph -> N e. ( 0 ... N ) )
60	48, 49, 59	rspcdva 3316	. . . . . 6 |- ( ph -> ( Q ` N ) = ( P ` N ) )
61	14	fveq1i 6192	. . . . . . . . . . 11 |- ( Q ` ( N + 1 ) ) = ( ( P u. { <. ( N + 1 ) , C >. } ) ` ( N + 1 ) )
62		ovex 6678	. . . . . . . . . . . 12 |- ( N + 1 ) e. _V
63	1, 2, 3, 4, 5, 6, 7, 8, 9	wlkp1lem1 26570	. . . . . . . . . . . 12 |- ( ph -> -. ( N + 1 ) e. dom P )
64		fsnunfv 6453	. . . . . . . . . . . 12 |- ( ( ( N + 1 ) e. _V /\ C e. V /\ -. ( N + 1 ) e. dom P ) -> ( ( P u. { <. ( N + 1 ) , C >. } ) ` ( N + 1 ) ) = C )
65	62, 6, 63, 64	mp3an2i 1429	. . . . . . . . . . 11 |- ( ph -> ( ( P u. { <. ( N + 1 ) , C >. } ) ` ( N + 1 ) ) = C )
66	61, 65	syl5eq 2668	. . . . . . . . . 10 |- ( ph -> ( Q ` ( N + 1 ) ) = C )
67	66	eqeq2d 2632	. . . . . . . . 9 |- ( ph -> ( ( P ` N ) = ( Q ` ( N + 1 ) ) <-> ( P ` N ) = C ) )
68		eqcom 2629	. . . . . . . . 9 |- ( ( P ` N ) = C <-> C = ( P ` N ) )
69	67, 68	syl6bb 276	. . . . . . . 8 |- ( ph -> ( ( P ` N ) = ( Q ` ( N + 1 ) ) <-> C = ( P ` N ) ) )
70		wlkp1.l	. . . . . . . . . 10 |- ( ( ph /\ C = ( P ` N ) ) -> E = { C } )
71		sneq 4187	. . . . . . . . . . 11 |- ( C = ( P ` N ) -> { C } = { ( P ` N ) } )
72	71	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ C = ( P ` N ) ) -> { C } = { ( P ` N ) } )
73	70, 72	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ C = ( P ` N ) ) -> E = { ( P ` N ) } )
74	73	ex 450	. . . . . . . 8 |- ( ph -> ( C = ( P ` N ) -> E = { ( P ` N ) } ) )
75	69, 74	sylbid 230	. . . . . . 7 |- ( ph -> ( ( P ` N ) = ( Q ` ( N + 1 ) ) -> E = { ( P ` N ) } ) )
76		eqeq1 2626	. . . . . . . 8 |- ( ( Q ` N ) = ( P ` N ) -> ( ( Q ` N ) = ( Q ` ( N + 1 ) ) <-> ( P ` N ) = ( Q ` ( N + 1 ) ) ) )
77		sneq 4187	. . . . . . . . 9 |- ( ( Q ` N ) = ( P ` N ) -> { ( Q ` N ) } = { ( P ` N ) } )
78	77	eqeq2d 2632	. . . . . . . 8 |- ( ( Q ` N ) = ( P ` N ) -> ( E = { ( Q ` N ) } <-> E = { ( P ` N ) } ) )
79	76, 78	imbi12d 334	. . . . . . 7 |- ( ( Q ` N ) = ( P ` N ) -> ( ( ( Q ` N ) = ( Q ` ( N + 1 ) ) -> E = { ( Q ` N ) } ) <-> ( ( P ` N ) = ( Q ` ( N + 1 ) ) -> E = { ( P ` N ) } ) ) )
80	75, 79	syl5ibrcom 237	. . . . . 6 |- ( ph -> ( ( Q ` N ) = ( P ` N ) -> ( ( Q ` N ) = ( Q ` ( N + 1 ) ) -> E = { ( Q ` N ) } ) ) )
81	60, 80	mpd 15	. . . . 5 |- ( ph -> ( ( Q ` N ) = ( Q ` ( N + 1 ) ) -> E = { ( Q ` N ) } ) )
82	81	imp 445	. . . 4 |- ( ( ph /\ ( Q ` N ) = ( Q ` ( N + 1 ) ) ) -> E = { ( Q ` N ) } )
83	41, 45, 82	3eqtrd 2660	. . 3 |- ( ( ph /\ ( Q ` N ) = ( Q ` ( N + 1 ) ) ) -> ( (iEdg ` S ) ` ( H ` N ) ) = { ( Q ` N ) } )
84	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	wlkp1lem7 26576	. . . 4 |- ( ph -> { ( Q ` N ) , ( Q ` ( N + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` N ) ) )
85	84	adantr 481	. . 3 |- ( ( ph /\ -. ( Q ` N ) = ( Q ` ( N + 1 ) ) ) -> { ( Q ` N ) , ( Q ` ( N + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` N ) ) )
86	83, 85	ifpimpda 1028	. 2 |- ( ph -> if- ( ( Q ` N ) = ( Q ` ( N + 1 ) ) , ( (iEdg ` S ) ` ( H ` N ) ) = { ( Q ` N ) } , { ( Q ` N ) , ( Q ` ( N + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` N ) ) ) )
87	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	wlkp1lem2 26571	. . . . . 6 |- ( ph -> ( # ` H ) = ( N + 1 ) )
88	87	oveq2d 6666	. . . . 5 |- ( ph -> ( 0..^ ( # ` H ) ) = ( 0..^ ( N + 1 ) ) )
89		fzosplitsn 12576	. . . . . 6 |- ( N e. ( ZZ>= ` 0 ) -> ( 0..^ ( N + 1 ) ) = ( ( 0..^ N ) u. { N } ) )
90	56, 89	syl 17	. . . . 5 |- ( ph -> ( 0..^ ( N + 1 ) ) = ( ( 0..^ N ) u. { N } ) )
91	88, 90	eqtrd 2656	. . . 4 |- ( ph -> ( 0..^ ( # ` H ) ) = ( ( 0..^ N ) u. { N } ) )
92	91	raleqdv 3144	. . 3 |- ( ph -> ( A. k e. ( 0..^ ( # ` H ) )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) <-> A. k e. ( ( 0..^ N ) u. { N } )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) ) )
93		ralunb 3794	. . . 4 |- ( A. k e. ( ( 0..^ N ) u. { N } )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) <-> ( A. k e. ( 0..^ N )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) /\ A. k e. { N }if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) ) )
94	93	a1i 11	. . 3 |- ( ph -> ( A. k e. ( ( 0..^ N ) u. { N } )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) <-> ( A. k e. ( 0..^ N )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) /\ A. k e. { N }if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) ) ) )
95		fvex 6201	. . . . . 6 |- ( # ` F ) e. _V
96	9, 95	eqeltri 2697	. . . . 5 |- N e. _V
97		fveq2 6191	. . . . . . . 8 |- ( k = N -> ( Q ` k ) = ( Q ` N ) )
98		oveq1 6657	. . . . . . . . 9 |- ( k = N -> ( k + 1 ) = ( N + 1 ) )
99	98	fveq2d 6195	. . . . . . . 8 |- ( k = N -> ( Q ` ( k + 1 ) ) = ( Q ` ( N + 1 ) ) )
100	97, 99	eqeq12d 2637	. . . . . . 7 |- ( k = N -> ( ( Q ` k ) = ( Q ` ( k + 1 ) ) <-> ( Q ` N ) = ( Q ` ( N + 1 ) ) ) )
101		fveq2 6191	. . . . . . . . 9 |- ( k = N -> ( H ` k ) = ( H ` N ) )
102	101	fveq2d 6195	. . . . . . . 8 |- ( k = N -> ( (iEdg ` S ) ` ( H ` k ) ) = ( (iEdg ` S ) ` ( H ` N ) ) )
103	97	sneqd 4189	. . . . . . . 8 |- ( k = N -> { ( Q ` k ) } = { ( Q ` N ) } )
104	102, 103	eqeq12d 2637	. . . . . . 7 |- ( k = N -> ( ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } <-> ( (iEdg ` S ) ` ( H ` N ) ) = { ( Q ` N ) } ) )
105	97, 99	preq12d 4276	. . . . . . . 8 |- ( k = N -> { ( Q ` k ) , ( Q ` ( k + 1 ) ) } = { ( Q ` N ) , ( Q ` ( N + 1 ) ) } )
106	105, 102	sseq12d 3634	. . . . . . 7 |- ( k = N -> ( { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) <-> { ( Q ` N ) , ( Q ` ( N + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` N ) ) ) )
107	100, 104, 106	ifpbi123d 1027	. . . . . 6 |- ( k = N -> (if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) <-> if- ( ( Q ` N ) = ( Q ` ( N + 1 ) ) , ( (iEdg ` S ) ` ( H ` N ) ) = { ( Q ` N ) } , { ( Q ` N ) , ( Q ` ( N + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` N ) ) ) ) )
108	107	ralsng 4218	. . . . 5 |- ( N e. _V -> ( A. k e. { N }if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) <-> if- ( ( Q ` N ) = ( Q ` ( N + 1 ) ) , ( (iEdg ` S ) ` ( H ` N ) ) = { ( Q ` N ) } , { ( Q ` N ) , ( Q ` ( N + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` N ) ) ) ) )
109	96, 108	mp1i 13	. . . 4 |- ( ph -> ( A. k e. { N }if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) <-> if- ( ( Q ` N ) = ( Q ` ( N + 1 ) ) , ( (iEdg ` S ) ` ( H ` N ) ) = { ( Q ` N ) } , { ( Q ` N ) , ( Q ` ( N + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` N ) ) ) ) )
110	109	anbi2d 740	. . 3 |- ( ph -> ( ( A. k e. ( 0..^ N )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) /\ A. k e. { N }if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) ) <-> ( A. k e. ( 0..^ N )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) /\ if- ( ( Q ` N ) = ( Q ` ( N + 1 ) ) , ( (iEdg ` S ) ` ( H ` N ) ) = { ( Q ` N ) } , { ( Q ` N ) , ( Q ` ( N + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` N ) ) ) ) ) )
111	92, 94, 110	3bitrd 294	. 2 |- ( ph -> ( A. k e. ( 0..^ ( # ` H ) )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) <-> ( A. k e. ( 0..^ N )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) /\ if- ( ( Q ` N ) = ( Q ` ( N + 1 ) ) , ( (iEdg ` S ) ` ( H ` N ) ) = { ( Q ` N ) } , { ( Q ` N ) , ( Q ` ( N + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` N ) ) ) ) ) )
112	39, 86, 111	mpbir2and 957	1 |- ( ph -> A. k e. ( 0..^ ( # ` H ) )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384  if-wif 1012   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   C_ wss 3574   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   dom cdm 5114   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939  Walkscwlks 26492

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-ifp 1013  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-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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495

This theorem is referenced by:  wlkp1  26578