Theorem usgr2wlkneq 26652

Description: The vertices and edges are pairwise different in a walk of length 2 in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.)

Assertion

Ref	Expression
usgr2wlkneq	|- ( ( ( G e. USGraph /\ F (Walks ` G ) P ) /\ ( ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) )

Proof of Theorem usgr2wlkneq

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrupgr 26077	. . . 4 |- ( G e. USGraph -> G e. UPGraph )
2		eqid 2622	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2622	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
4	2, 3	upgriswlk 26537	. . . 4 |- ( G e. UPGraph -> ( F (Walks ` G ) P <-> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
5	1, 4	syl 17	. . 3 |- ( G e. USGraph -> ( F (Walks ` G ) P <-> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
6		2wlklem 26563	. . . . . . . . . . . 12 |- ( A. k e. { 0 , 1 } ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
7		simplll 798	. . . . . . . . . . . . . . 15 |- ( ( ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) /\ ( P ` 0 ) =/= ( P ` 2 ) ) /\ P : ( 0 ... 2 ) --> (Vtx ` G ) ) -> G e. USGraph )
8		fvex 6201	. . . . . . . . . . . . . . 15 |- ( P ` 0 ) e. _V
9	3	usgrnloopv 26092	. . . . . . . . . . . . . . 15 |- ( ( G e. USGraph /\ ( P ` 0 ) e. _V ) -> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } -> ( P ` 0 ) =/= ( P ` 1 ) ) )
10	7, 8, 9	sylancl 694	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) /\ ( P ` 0 ) =/= ( P ` 2 ) ) /\ P : ( 0 ... 2 ) --> (Vtx ` G ) ) -> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } -> ( P ` 0 ) =/= ( P ` 1 ) ) )
11		fvex 6201	. . . . . . . . . . . . . . 15 |- ( P ` 1 ) e. _V
12	3	usgrnloopv 26092	. . . . . . . . . . . . . . 15 |- ( ( G e. USGraph /\ ( P ` 1 ) e. _V ) -> ( ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } -> ( P ` 1 ) =/= ( P ` 2 ) ) )
13	7, 11, 12	sylancl 694	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) /\ ( P ` 0 ) =/= ( P ` 2 ) ) /\ P : ( 0 ... 2 ) --> (Vtx ` G ) ) -> ( ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } -> ( P ` 1 ) =/= ( P ` 2 ) ) )
14	10, 13	anim12d 586	. . . . . . . . . . . . 13 |- ( ( ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) /\ ( P ` 0 ) =/= ( P ` 2 ) ) /\ P : ( 0 ... 2 ) --> (Vtx ` G ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) )
15		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( (iEdg ` G ) ` ( F ` 0 ) ) = ( (iEdg ` G ) ` ( F ` 1 ) ) )
16	15	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } <-> ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) )
17		eqtr2 2642	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> { ( P ` 0 ) , ( P ` 1 ) } = { ( P ` 1 ) , ( P ` 2 ) } )
18		prcom 4267	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- { ( P ` 1 ) , ( P ` 2 ) } = { ( P ` 2 ) , ( P ` 1 ) }
19	18	eqeq2i 2634	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( { ( P ` 0 ) , ( P ` 1 ) } = { ( P ` 1 ) , ( P ` 2 ) } <-> { ( P ` 0 ) , ( P ` 1 ) } = { ( P ` 2 ) , ( P ` 1 ) } )
20		fvex 6201	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( P ` 2 ) e. _V
21	8, 20	preqr1 4379	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( { ( P ` 0 ) , ( P ` 1 ) } = { ( P ` 2 ) , ( P ` 1 ) } -> ( P ` 0 ) = ( P ` 2 ) )
22	19, 21	sylbi 207	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( { ( P ` 0 ) , ( P ` 1 ) } = { ( P ` 1 ) , ( P ` 2 ) } -> ( P ` 0 ) = ( P ` 2 ) )
23	17, 22	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 0 ) = ( P ` 2 ) )
24	23	ex 450	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 0 ) , ( P ` 1 ) } -> ( ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } -> ( P ` 0 ) = ( P ` 2 ) ) )
25	16, 24	syl6bi 243	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } -> ( ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } -> ( P ` 0 ) = ( P ` 2 ) ) ) )
26	25	impd 447	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 0 ) = ( P ` 2 ) ) )
27	26	com12 32	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( F ` 0 ) = ( F ` 1 ) -> ( P ` 0 ) = ( P ` 2 ) ) )
28	27	necon3d 2815	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( P ` 0 ) =/= ( P ` 2 ) -> ( F ` 0 ) =/= ( F ` 1 ) ) )
29	28	com12 32	. . . . . . . . . . . . . . . . . . 19 |- ( ( P ` 0 ) =/= ( P ` 2 ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( F ` 0 ) =/= ( F ` 1 ) ) )
30	29	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( P ` 0 ) =/= ( P ` 2 ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( F ` 0 ) =/= ( F ` 1 ) ) )
31		simpl 473	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) -> ( P ` 0 ) =/= ( P ` 1 ) )
32	31	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( P ` 0 ) =/= ( P ` 2 ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) -> ( P ` 0 ) =/= ( P ` 1 ) )
33		simpl 473	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( P ` 0 ) =/= ( P ` 2 ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) -> ( P ` 0 ) =/= ( P ` 2 ) )
34		simprr 796	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( P ` 0 ) =/= ( P ` 2 ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) -> ( P ` 1 ) =/= ( P ` 2 ) )
35	32, 33, 34	3jca 1242	. . . . . . . . . . . . . . . . . 18 |- ( ( ( P ` 0 ) =/= ( P ` 2 ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) )
36	30, 35	jctild 566	. . . . . . . . . . . . . . . . 17 |- ( ( ( P ` 0 ) =/= ( P ` 2 ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) )
37	36	ex 450	. . . . . . . . . . . . . . . 16 |- ( ( P ` 0 ) =/= ( P ` 2 ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
38	37	com23 86	. . . . . . . . . . . . . . 15 |- ( ( P ` 0 ) =/= ( P ` 2 ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
39	38	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) /\ ( P ` 0 ) =/= ( P ` 2 ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
40	39	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) /\ ( P ` 0 ) =/= ( P ` 2 ) ) /\ P : ( 0 ... 2 ) --> (Vtx ` G ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
41	14, 40	mpdd 43	. . . . . . . . . . . 12 |- ( ( ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) /\ ( P ` 0 ) =/= ( P ` 2 ) ) /\ P : ( 0 ... 2 ) --> (Vtx ` G ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) )
42	6, 41	syl5bi 232	. . . . . . . . . . 11 |- ( ( ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) /\ ( P ` 0 ) =/= ( P ` 2 ) ) /\ P : ( 0 ... 2 ) --> (Vtx ` G ) ) -> ( A. k e. { 0 , 1 } ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) )
43	42	ex 450	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) /\ ( P ` 0 ) =/= ( P ` 2 ) ) -> ( P : ( 0 ... 2 ) --> (Vtx ` G ) -> ( A. k e. { 0 , 1 } ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
44	43	com23 86	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) /\ ( P ` 0 ) =/= ( P ` 2 ) ) -> ( A. k e. { 0 , 1 } ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( P : ( 0 ... 2 ) --> (Vtx ` G ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
45	44	ex 450	. . . . . . . 8 |- ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) -> ( ( P ` 0 ) =/= ( P ` 2 ) -> ( A. k e. { 0 , 1 } ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( P : ( 0 ... 2 ) --> (Vtx ` G ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) ) )
46		fveq2 6191	. . . . . . . . . 10 |- ( ( # ` F ) = 2 -> ( P ` ( # ` F ) ) = ( P ` 2 ) )
47	46	neeq2d 2854	. . . . . . . . 9 |- ( ( # ` F ) = 2 -> ( ( P ` 0 ) =/= ( P ` ( # ` F ) ) <-> ( P ` 0 ) =/= ( P ` 2 ) ) )
48		oveq2 6658	. . . . . . . . . . . 12 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = ( 0..^ 2 ) )
49		fzo0to2pr 12553	. . . . . . . . . . . 12 |- ( 0..^ 2 ) = { 0 , 1 }
50	48, 49	syl6eq 2672	. . . . . . . . . . 11 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = { 0 , 1 } )
51	50	raleqdv 3144	. . . . . . . . . 10 |- ( ( # ` F ) = 2 -> ( A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> A. k e. { 0 , 1 } ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
52		oveq2 6658	. . . . . . . . . . . 12 |- ( ( # ` F ) = 2 -> ( 0 ... ( # ` F ) ) = ( 0 ... 2 ) )
53	52	feq2d 6031	. . . . . . . . . . 11 |- ( ( # ` F ) = 2 -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) <-> P : ( 0 ... 2 ) --> (Vtx ` G ) ) )
54	53	imbi1d 331	. . . . . . . . . 10 |- ( ( # ` F ) = 2 -> ( ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) <-> ( P : ( 0 ... 2 ) --> (Vtx ` G ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
55	51, 54	imbi12d 334	. . . . . . . . 9 |- ( ( # ` F ) = 2 -> ( ( A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) <-> ( A. k e. { 0 , 1 } ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( P : ( 0 ... 2 ) --> (Vtx ` G ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) ) )
56	47, 55	imbi12d 334	. . . . . . . 8 |- ( ( # ` F ) = 2 -> ( ( ( P ` 0 ) =/= ( P ` ( # ` F ) ) -> ( A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) ) <-> ( ( P ` 0 ) =/= ( P ` 2 ) -> ( A. k e. { 0 , 1 } ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( P : ( 0 ... 2 ) --> (Vtx ` G ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) ) ) )
57	45, 56	syl5ibrcom 237	. . . . . . 7 |- ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) -> ( ( # ` F ) = 2 -> ( ( P ` 0 ) =/= ( P ` ( # ` F ) ) -> ( A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) ) ) )
58	57	impd 447	. . . . . 6 |- ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) -> ( ( ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> ( A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) ) )
59	58	com24 95	. . . . 5 |- ( ( G e. USGraph /\ F e. Word dom (iEdg ` G ) ) -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( ( ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) ) )
60	59	ex 450	. . . 4 |- ( G e. USGraph -> ( F e. Word dom (iEdg ` G ) -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( ( ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) ) ) )
61	60	3impd 1281	. . 3 |- ( G e. USGraph -> ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( ( ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
62	5, 61	sylbid 230	. 2 |- ( G e. USGraph -> ( F (Walks ` G ) P -> ( ( ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
63	62	imp31 448	1 |- ( ( ( G e. USGraph /\ F (Walks ` G ) P ) /\ ( ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   {cpr 4179   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975   USGraph cusgr 26044  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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-wlks 26495

This theorem is referenced by:  usgr2wlkspthlem1  26653  usgr2wlkspthlem2  26654