Theorem uspgrn2crct 26700

Description: In a simple pseudograph there are no circuits with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 3-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.)

Assertion

Ref	Expression
uspgrn2crct	|- ( ( G e. USPGraph /\ F (Circuits ` G ) P ) -> ( # ` F ) =/= 2 )

Proof of Theorem uspgrn2crct

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

Step	Hyp	Ref	Expression
1		crctprop 26687	. . 3 |- ( F (Circuits ` G ) P -> ( F (Trails ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
2		istrl 26593	. . . . . . 7 |- ( F (Trails ` G ) P <-> ( F (Walks ` G ) P /\ Fun `' F ) )
3		uspgrupgr 26071	. . . . . . . . 9 |- ( G e. USPGraph -> G e. UPGraph )
4		eqid 2622	. . . . . . . . . . . . 13 |- (Vtx ` G ) = (Vtx ` G )
5		eqid 2622	. . . . . . . . . . . . 13 |- (iEdg ` G ) = (iEdg ` G )
6	4, 5	upgriswlk 26537	. . . . . . . . . . . 12 |- ( 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 ) ) } ) ) )
7		preq2 4269	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( P ` 2 ) = ( P ` 0 ) -> { ( P ` 1 ) , ( P ` 2 ) } = { ( P ` 1 ) , ( P ` 0 ) } )
8		prcom 4267	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- { ( P ` 1 ) , ( P ` 0 ) } = { ( P ` 0 ) , ( P ` 1 ) }
9	7, 8	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( P ` 2 ) = ( P ` 0 ) -> { ( P ` 1 ) , ( P ` 2 ) } = { ( P ` 0 ) , ( P ` 1 ) } )
10	9	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( P ` 0 ) = ( P ` 2 ) -> { ( P ` 1 ) , ( P ` 2 ) } = { ( P ` 0 ) , ( P ` 1 ) } )
11	10	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( P ` 0 ) = ( P ` 2 ) -> ( ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } <-> ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) )
12	11	anbi2d 740	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( P ` 0 ) = ( P ` 2 ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) <-> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) ) )
13	12	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( P ` 0 ) = ( P ` 2 ) /\ ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) ) /\ F e. Word dom (iEdg ` G ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) <-> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) ) )
14		eqtr3 2643	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) -> ( (iEdg ` G ) ` ( F ` 0 ) ) = ( (iEdg ` G ) ` ( F ` 1 ) ) )
15	4, 5	uspgrf 26049	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
16	15	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( Fun `' F /\ G e. USPGraph ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
17	16	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
18	17	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) /\ F e. Word dom (iEdg ` G ) ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
19		df-f1 5893	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) <-> ( F : ( 0..^ ( # ` F ) ) --> dom (iEdg ` G ) /\ Fun `' F ) )
20	19	simplbi2 655	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( F : ( 0..^ ( # ` F ) ) --> dom (iEdg ` G ) -> ( Fun `' F -> F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) ) )
21		wrdf 13310	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( F e. Word dom (iEdg ` G ) -> F : ( 0..^ ( # ` F ) ) --> dom (iEdg ` G ) )
22	20, 21	syl11 33	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( Fun `' F -> ( F e. Word dom (iEdg ` G ) -> F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) ) )
23	22	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( Fun `' F /\ G e. USPGraph ) -> ( F e. Word dom (iEdg ` G ) -> F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) ) )
24	23	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) -> ( F e. Word dom (iEdg ` G ) -> F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) ) )
25	24	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) /\ F e. Word dom (iEdg ` G ) ) -> F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) )
26		2nn 11185	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 2 e. NN
27		lbfzo0 12507	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( 0 e. ( 0..^ 2 ) <-> 2 e. NN )
28	26, 27	mpbir 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- 0 e. ( 0..^ 2 )
29		1nn0 11308	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 1 e. NN0
30		1lt2 11194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 1 < 2
31		elfzo0 12508	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( 1 e. ( 0..^ 2 ) <-> ( 1 e. NN0 /\ 2 e. NN /\ 1 < 2 ) )
32	29, 26, 30, 31	mpbir3an 1244	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- 1 e. ( 0..^ 2 )
33	28, 32	pm3.2i 471	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( 0 e. ( 0..^ 2 ) /\ 1 e. ( 0..^ 2 ) )
34		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = ( 0..^ 2 ) )
35	34	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( # ` F ) = 2 -> ( 0 e. ( 0..^ ( # ` F ) ) <-> 0 e. ( 0..^ 2 ) ) )
36	34	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( # ` F ) = 2 -> ( 1 e. ( 0..^ ( # ` F ) ) <-> 1 e. ( 0..^ 2 ) ) )
37	35, 36	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( # ` F ) = 2 -> ( ( 0 e. ( 0..^ ( # ` F ) ) /\ 1 e. ( 0..^ ( # ` F ) ) ) <-> ( 0 e. ( 0..^ 2 ) /\ 1 e. ( 0..^ 2 ) ) ) )
38	33, 37	mpbiri 248	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( # ` F ) = 2 -> ( 0 e. ( 0..^ ( # ` F ) ) /\ 1 e. ( 0..^ ( # ` F ) ) ) )
39	38	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) /\ F e. Word dom (iEdg ` G ) ) -> ( 0 e. ( 0..^ ( # ` F ) ) /\ 1 e. ( 0..^ ( # ` F ) ) ) )
40		f1cofveqaeq 6515	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } /\ F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) ) /\ ( 0 e. ( 0..^ ( # ` F ) ) /\ 1 e. ( 0..^ ( # ` F ) ) ) ) -> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = ( (iEdg ` G ) ` ( F ` 1 ) ) -> 0 = 1 ) )
41	18, 25, 39, 40	syl21anc 1325	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) /\ F e. Word dom (iEdg ` G ) ) -> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = ( (iEdg ` G ) ` ( F ` 1 ) ) -> 0 = 1 ) )
42		0ne1 11088	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 0 =/= 1
43		eqneqall 2805	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 0 = 1 -> ( 0 =/= 1 -> ( P ` 0 ) =/= ( P ` 2 ) ) )
44	41, 42, 43	syl6mpi 67	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) /\ F e. Word dom (iEdg ` G ) ) -> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = ( (iEdg ` G ) ` ( F ` 1 ) ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
45	44	adantll 750	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( P ` 0 ) = ( P ` 2 ) /\ ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) ) /\ F e. Word dom (iEdg ` G ) ) -> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = ( (iEdg ` G ) ` ( F ` 1 ) ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
46	14, 45	syl5 34	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( P ` 0 ) = ( P ` 2 ) /\ ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) ) /\ F e. Word dom (iEdg ` G ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
47	13, 46	sylbid 230	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( P ` 0 ) = ( P ` 2 ) /\ ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) ) /\ F e. Word dom (iEdg ` G ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
48	47	expimpd 629	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( P ` 0 ) = ( P ` 2 ) /\ ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) ) -> ( ( F e. Word dom (iEdg ` G ) /\ ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
49	48	ex 450	. . . . . . . . . . . . . . . . . 18 |- ( ( P ` 0 ) = ( P ` 2 ) -> ( ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) -> ( ( F e. Word dom (iEdg ` G ) /\ ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
50		2a1 28	. . . . . . . . . . . . . . . . . 18 |- ( ( P ` 0 ) =/= ( P ` 2 ) -> ( ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) -> ( ( F e. Word dom (iEdg ` G ) /\ ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
51	49, 50	pm2.61ine 2877	. . . . . . . . . . . . . . . . 17 |- ( ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) -> ( ( F e. Word dom (iEdg ` G ) /\ ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
52		fzo0to2pr 12553	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( 0..^ 2 ) = { 0 , 1 }
53	34, 52	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = { 0 , 1 } )
54	53	raleqdv 3144	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` 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 ) ) } ) )
55		2wlklem 26563	. . . . . . . . . . . . . . . . . . . . 21 |- ( 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 ) } ) )
56	54, 55	syl6bb 276	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 2 -> ( A. k e. ( 0..^ ( # ` F ) ) ( (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 ) } ) ) )
57	56	anbi2d 740	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` F ) = 2 -> ( ( F e. Word dom (iEdg ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) <-> ( F e. Word dom (iEdg ` G ) /\ ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
58		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 2 -> ( P ` ( # ` F ) ) = ( P ` 2 ) )
59	58	neeq2d 2854	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` F ) = 2 -> ( ( P ` 0 ) =/= ( P ` ( # ` F ) ) <-> ( P ` 0 ) =/= ( P ` 2 ) ) )
60	57, 59	imbi12d 334	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` F ) = 2 -> ( ( ( F e. Word dom (iEdg ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) <-> ( ( F e. Word dom (iEdg ` G ) /\ ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
61	60	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) -> ( ( ( F e. Word dom (iEdg ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) <-> ( ( F e. Word dom (iEdg ` G ) /\ ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
62	51, 61	mpbird 247	. . . . . . . . . . . . . . . 16 |- ( ( ( # ` F ) = 2 /\ ( Fun `' F /\ G e. USPGraph ) ) -> ( ( F e. Word dom (iEdg ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) )
63	62	ex 450	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 2 -> ( ( Fun `' F /\ G e. USPGraph ) -> ( ( F e. Word dom (iEdg ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) )
64	63	com13 88	. . . . . . . . . . . . . 14 |- ( ( F e. Word dom (iEdg ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( ( Fun `' F /\ G e. USPGraph ) -> ( ( # ` F ) = 2 -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) )
65	64	expd 452	. . . . . . . . . . . . 13 |- ( ( F e. Word dom (iEdg ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( Fun `' F -> ( G e. USPGraph -> ( ( # ` F ) = 2 -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) ) )
66	65	3adant2 1080	. . . . . . . . . . . 12 |- ( ( 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 ) ) } ) -> ( Fun `' F -> ( G e. USPGraph -> ( ( # ` F ) = 2 -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) ) )
67	6, 66	syl6bi 243	. . . . . . . . . . 11 |- ( G e. UPGraph -> ( F (Walks ` G ) P -> ( Fun `' F -> ( G e. USPGraph -> ( ( # ` F ) = 2 -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) ) ) )
68	67	impd 447	. . . . . . . . . 10 |- ( G e. UPGraph -> ( ( F (Walks ` G ) P /\ Fun `' F ) -> ( G e. USPGraph -> ( ( # ` F ) = 2 -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) ) )
69	68	com23 86	. . . . . . . . 9 |- ( G e. UPGraph -> ( G e. USPGraph -> ( ( F (Walks ` G ) P /\ Fun `' F ) -> ( ( # ` F ) = 2 -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) ) )
70	3, 69	mpcom 38	. . . . . . . 8 |- ( G e. USPGraph -> ( ( F (Walks ` G ) P /\ Fun `' F ) -> ( ( # ` F ) = 2 -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) )
71	70	com12 32	. . . . . . 7 |- ( ( F (Walks ` G ) P /\ Fun `' F ) -> ( G e. USPGraph -> ( ( # ` F ) = 2 -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) )
72	2, 71	sylbi 207	. . . . . 6 |- ( F (Trails ` G ) P -> ( G e. USPGraph -> ( ( # ` F ) = 2 -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) )
73	72	imp 445	. . . . 5 |- ( ( F (Trails ` G ) P /\ G e. USPGraph ) -> ( ( # ` F ) = 2 -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) )
74	73	necon2d 2817	. . . 4 |- ( ( F (Trails ` G ) P /\ G e. USPGraph ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( # ` F ) =/= 2 ) )
75	74	impancom 456	. . 3 |- ( ( F (Trails ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( G e. USPGraph -> ( # ` F ) =/= 2 ) )
76	1, 75	syl 17	. 2 |- ( F (Circuits ` G ) P -> ( G e. USPGraph -> ( # ` F ) =/= 2 ) )
77	76	impcom 446	1 |- ( ( G e. USPGraph /\ F (Circuits ` G ) P ) -> ( # ` F ) =/= 2 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   NNcn 11020   2c2 11070   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975   USPGraph cuspgr 26043  Walkscwlks 26492  Trailsctrls 26587  Circuitsccrcts 26679

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-uspgr 26045  df-wlks 26495  df-trls 26589  df-crcts 26681

This theorem is referenced by:  usgrn2cycl  26701