Theorem upgr3v3e3cycl 27040

Description: If there is a cycle of length 3 in a pseudograph, there are three distinct vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)

Hypotheses

Ref	Expression
upgr3v3e3cycl.e	|- E = (Edg ` G )
upgr3v3e3cycl.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
upgr3v3e3cycl	|- ( ( G e. UPGraph /\ F (Cycles ` G ) P /\ ( # ` F ) = 3 ) -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) )

Distinct variable groups:   E, a, b, c   P, a, b, c   V, a, b, c

Allowed substitution hints:   F( a, b, c)   G( a, b, c)

Proof of Theorem upgr3v3e3cycl

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cyclprop 26688	. . 3 |- ( F (Cycles ` G ) P -> ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
2		pthiswlk 26623	. . . . 5 |- ( F (Paths ` G ) P -> F (Walks ` G ) P )
3		upgr3v3e3cycl.e	. . . . . . . . . 10 |- E = (Edg ` G )
4	3	upgrwlkvtxedg 26541	. . . . . . . . 9 |- ( ( G e. UPGraph /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E )
5		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 3 -> ( P ` ( # ` F ) ) = ( P ` 3 ) )
6	5	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( ( # ` F ) = 3 -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) <-> ( P ` 0 ) = ( P ` 3 ) ) )
7	6	anbi2d 740	. . . . . . . . . . . . 13 |- ( ( # ` F ) = 3 -> ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) <-> ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) ) )
8		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 3 -> ( 0..^ ( # ` F ) ) = ( 0..^ 3 ) )
9		fzo0to3tp 12554	. . . . . . . . . . . . . . . 16 |- ( 0..^ 3 ) = { 0 , 1 , 2 }
10	8, 9	syl6eq 2672	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 3 -> ( 0..^ ( # ` F ) ) = { 0 , 1 , 2 } )
11	10	raleqdv 3144	. . . . . . . . . . . . . 14 |- ( ( # ` F ) = 3 -> ( A. k e. ( 0..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> A. k e. { 0 , 1 , 2 } { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) )
12		c0ex 10034	. . . . . . . . . . . . . . 15 |- 0 e. _V
13		1ex 10035	. . . . . . . . . . . . . . 15 |- 1 e. _V
14		2ex 11092	. . . . . . . . . . . . . . 15 |- 2 e. _V
15		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
16		oveq1 6657	. . . . . . . . . . . . . . . . . . 19 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
17		0p1e1 11132	. . . . . . . . . . . . . . . . . . 19 |- ( 0 + 1 ) = 1
18	16, 17	syl6eq 2672	. . . . . . . . . . . . . . . . . 18 |- ( k = 0 -> ( k + 1 ) = 1 )
19	18	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
20	15, 19	preq12d 4276	. . . . . . . . . . . . . . . 16 |- ( k = 0 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
21	20	eleq1d 2686	. . . . . . . . . . . . . . 15 |- ( k = 0 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> { ( P ` 0 ) , ( P ` 1 ) } e. E ) )
22		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
23		oveq1 6657	. . . . . . . . . . . . . . . . . . 19 |- ( k = 1 -> ( k + 1 ) = ( 1 + 1 ) )
24		1p1e2 11134	. . . . . . . . . . . . . . . . . . 19 |- ( 1 + 1 ) = 2
25	23, 24	syl6eq 2672	. . . . . . . . . . . . . . . . . 18 |- ( k = 1 -> ( k + 1 ) = 2 )
26	25	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( k = 1 -> ( P ` ( k + 1 ) ) = ( P ` 2 ) )
27	22, 26	preq12d 4276	. . . . . . . . . . . . . . . 16 |- ( k = 1 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 1 ) , ( P ` 2 ) } )
28	27	eleq1d 2686	. . . . . . . . . . . . . . 15 |- ( k = 1 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> { ( P ` 1 ) , ( P ` 2 ) } e. E ) )
29		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( k = 2 -> ( P ` k ) = ( P ` 2 ) )
30		oveq1 6657	. . . . . . . . . . . . . . . . . . 19 |- ( k = 2 -> ( k + 1 ) = ( 2 + 1 ) )
31		2p1e3 11151	. . . . . . . . . . . . . . . . . . 19 |- ( 2 + 1 ) = 3
32	30, 31	syl6eq 2672	. . . . . . . . . . . . . . . . . 18 |- ( k = 2 -> ( k + 1 ) = 3 )
33	32	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( k = 2 -> ( P ` ( k + 1 ) ) = ( P ` 3 ) )
34	29, 33	preq12d 4276	. . . . . . . . . . . . . . . 16 |- ( k = 2 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 2 ) , ( P ` 3 ) } )
35	34	eleq1d 2686	. . . . . . . . . . . . . . 15 |- ( k = 2 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> { ( P ` 2 ) , ( P ` 3 ) } e. E ) )
36	12, 13, 14, 21, 28, 35	raltp 4240	. . . . . . . . . . . . . 14 |- ( A. k e. { 0 , 1 , 2 } { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) )
37	11, 36	syl6bb 276	. . . . . . . . . . . . 13 |- ( ( # ` F ) = 3 -> ( A. k e. ( 0..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) )
38	7, 37	anbi12d 747	. . . . . . . . . . . 12 |- ( ( # ` F ) = 3 -> ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) /\ A. k e. ( 0..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) <-> ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) ) )
39		upgr3v3e3cycl.v	. . . . . . . . . . . . . . . . . . 19 |- V = (Vtx ` G )
40	39	wlkp 26512	. . . . . . . . . . . . . . . . . 18 |- ( F (Walks ` G ) P -> P : ( 0 ... ( # ` F ) ) --> V )
41		oveq2 6658	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` F ) = 3 -> ( 0 ... ( # ` F ) ) = ( 0 ... 3 ) )
42	41	feq2d 6031	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 3 -> ( P : ( 0 ... ( # ` F ) ) --> V <-> P : ( 0 ... 3 ) --> V ) )
43		id 22	. . . . . . . . . . . . . . . . . . . . . 22 |- ( P : ( 0 ... 3 ) --> V -> P : ( 0 ... 3 ) --> V )
44		3nn0 11310	. . . . . . . . . . . . . . . . . . . . . . 23 |- 3 e. NN0
45		0elfz 12436	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( 3 e. NN0 -> 0 e. ( 0 ... 3 ) )
46	44, 45	mp1i 13	. . . . . . . . . . . . . . . . . . . . . 22 |- ( P : ( 0 ... 3 ) --> V -> 0 e. ( 0 ... 3 ) )
47	43, 46	ffvelrnd 6360	. . . . . . . . . . . . . . . . . . . . 21 |- ( P : ( 0 ... 3 ) --> V -> ( P ` 0 ) e. V )
48		1nn0 11308	. . . . . . . . . . . . . . . . . . . . . 22 |- 1 e. NN0
49		1lt3 11196	. . . . . . . . . . . . . . . . . . . . . 22 |- 1 < 3
50		fvffz0 12457	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( 3 e. NN0 /\ 1 e. NN0 /\ 1 < 3 ) /\ P : ( 0 ... 3 ) --> V ) -> ( P ` 1 ) e. V )
51	50	ex 450	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( 3 e. NN0 /\ 1 e. NN0 /\ 1 < 3 ) -> ( P : ( 0 ... 3 ) --> V -> ( P ` 1 ) e. V ) )
52	44, 48, 49, 51	mp3an 1424	. . . . . . . . . . . . . . . . . . . . 21 |- ( P : ( 0 ... 3 ) --> V -> ( P ` 1 ) e. V )
53		2nn0 11309	. . . . . . . . . . . . . . . . . . . . . 22 |- 2 e. NN0
54		2lt3 11195	. . . . . . . . . . . . . . . . . . . . . 22 |- 2 < 3
55		fvffz0 12457	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( 3 e. NN0 /\ 2 e. NN0 /\ 2 < 3 ) /\ P : ( 0 ... 3 ) --> V ) -> ( P ` 2 ) e. V )
56	55	ex 450	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( 3 e. NN0 /\ 2 e. NN0 /\ 2 < 3 ) -> ( P : ( 0 ... 3 ) --> V -> ( P ` 2 ) e. V ) )
57	44, 53, 54, 56	mp3an 1424	. . . . . . . . . . . . . . . . . . . . 21 |- ( P : ( 0 ... 3 ) --> V -> ( P ` 2 ) e. V )
58	47, 52, 57	3jca 1242	. . . . . . . . . . . . . . . . . . . 20 |- ( P : ( 0 ... 3 ) --> V -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) )
59	42, 58	syl6bi 243	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` F ) = 3 -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) )
60	59	com12 32	. . . . . . . . . . . . . . . . . 18 |- ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( # ` F ) = 3 -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) )
61	2, 40, 60	3syl 18	. . . . . . . . . . . . . . . . 17 |- ( F (Paths ` G ) P -> ( ( # ` F ) = 3 -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) )
62	61	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) -> ( ( # ` F ) = 3 -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) )
63	62	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) -> ( ( # ` F ) = 3 -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) )
64	63	impcom 446	. . . . . . . . . . . . . 14 |- ( ( ( # ` F ) = 3 /\ ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) ) -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) )
65		preq2 4269	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` 3 ) = ( P ` 0 ) -> { ( P ` 2 ) , ( P ` 3 ) } = { ( P ` 2 ) , ( P ` 0 ) } )
66	65	eqcoms 2630	. . . . . . . . . . . . . . . . . . 19 |- ( ( P ` 0 ) = ( P ` 3 ) -> { ( P ` 2 ) , ( P ` 3 ) } = { ( P ` 2 ) , ( P ` 0 ) } )
67	66	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) -> { ( P ` 2 ) , ( P ` 3 ) } = { ( P ` 2 ) , ( P ` 0 ) } )
68	67	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) -> ( { ( P ` 2 ) , ( P ` 3 ) } e. E <-> { ( P ` 2 ) , ( P ` 0 ) } e. E ) )
69	68	3anbi3d 1405	. . . . . . . . . . . . . . . 16 |- ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) -> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 0 ) } e. E ) ) )
70	69	biimpa 501	. . . . . . . . . . . . . . 15 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) -> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 0 ) } e. E ) )
71	70	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( # ` F ) = 3 /\ ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) ) -> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 0 ) } e. E ) )
72		simpll 790	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> F (Paths ` G ) P )
73		breq2 4657	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` F ) = 3 -> ( 1 < ( # ` F ) <-> 1 < 3 ) )
74	49, 73	mpbiri 248	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 3 -> 1 < ( # ` F ) )
75	74	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> 1 < ( # ` F ) )
76		3nn 11186	. . . . . . . . . . . . . . . . . . . . . 22 |- 3 e. NN
77		lbfzo0 12507	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 0 e. ( 0..^ 3 ) <-> 3 e. NN )
78	76, 77	mpbir 221	. . . . . . . . . . . . . . . . . . . . 21 |- 0 e. ( 0..^ 3 )
79	78, 8	syl5eleqr 2708	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 3 -> 0 e. ( 0..^ ( # ` F ) ) )
80	79	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> 0 e. ( 0..^ ( # ` F ) ) )
81		pthdadjvtx 26626	. . . . . . . . . . . . . . . . . . . 20 |- ( ( F (Paths ` G ) P /\ 1 < ( # ` F ) /\ 0 e. ( 0..^ ( # ` F ) ) ) -> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) )
82		1e0p1 11552	. . . . . . . . . . . . . . . . . . . . . 22 |- 1 = ( 0 + 1 )
83	82	fveq2i 6194	. . . . . . . . . . . . . . . . . . . . 21 |- ( P ` 1 ) = ( P ` ( 0 + 1 ) )
84	83	neeq2i 2859	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` 0 ) =/= ( P ` 1 ) <-> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) )
85	81, 84	sylibr 224	. . . . . . . . . . . . . . . . . . 19 |- ( ( F (Paths ` G ) P /\ 1 < ( # ` F ) /\ 0 e. ( 0..^ ( # ` F ) ) ) -> ( P ` 0 ) =/= ( P ` 1 ) )
86	72, 75, 80, 85	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> ( P ` 0 ) =/= ( P ` 1 ) )
87		elfzo0 12508	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 1 e. ( 0..^ 3 ) <-> ( 1 e. NN0 /\ 3 e. NN /\ 1 < 3 ) )
88	48, 76, 49, 87	mpbir3an 1244	. . . . . . . . . . . . . . . . . . . . 21 |- 1 e. ( 0..^ 3 )
89	88, 8	syl5eleqr 2708	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 3 -> 1 e. ( 0..^ ( # ` F ) ) )
90	89	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> 1 e. ( 0..^ ( # ` F ) ) )
91		pthdadjvtx 26626	. . . . . . . . . . . . . . . . . . . 20 |- ( ( F (Paths ` G ) P /\ 1 < ( # ` F ) /\ 1 e. ( 0..^ ( # ` F ) ) ) -> ( P ` 1 ) =/= ( P ` ( 1 + 1 ) ) )
92		df-2 11079	. . . . . . . . . . . . . . . . . . . . . 22 |- 2 = ( 1 + 1 )
93	92	fveq2i 6194	. . . . . . . . . . . . . . . . . . . . 21 |- ( P ` 2 ) = ( P ` ( 1 + 1 ) )
94	93	neeq2i 2859	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` 1 ) =/= ( P ` 2 ) <-> ( P ` 1 ) =/= ( P ` ( 1 + 1 ) ) )
95	91, 94	sylibr 224	. . . . . . . . . . . . . . . . . . 19 |- ( ( F (Paths ` G ) P /\ 1 < ( # ` F ) /\ 1 e. ( 0..^ ( # ` F ) ) ) -> ( P ` 1 ) =/= ( P ` 2 ) )
96	72, 75, 90, 95	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> ( P ` 1 ) =/= ( P ` 2 ) )
97		elfzo0 12508	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( 2 e. ( 0..^ 3 ) <-> ( 2 e. NN0 /\ 3 e. NN /\ 2 < 3 ) )
98	53, 76, 54, 97	mpbir3an 1244	. . . . . . . . . . . . . . . . . . . . . 22 |- 2 e. ( 0..^ 3 )
99	98, 8	syl5eleqr 2708	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` F ) = 3 -> 2 e. ( 0..^ ( # ` F ) ) )
100	99	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> 2 e. ( 0..^ ( # ` F ) ) )
101		pthdadjvtx 26626	. . . . . . . . . . . . . . . . . . . 20 |- ( ( F (Paths ` G ) P /\ 1 < ( # ` F ) /\ 2 e. ( 0..^ ( # ` F ) ) ) -> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) )
102	72, 75, 100, 101	syl3anc 1326	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) )
103		neeq2 2857	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( P ` 0 ) = ( P ` 3 ) -> ( ( P ` 2 ) =/= ( P ` 0 ) <-> ( P ` 2 ) =/= ( P ` 3 ) ) )
104		df-3 11080	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 3 = ( 2 + 1 )
105	104	fveq2i 6194	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( P ` 3 ) = ( P ` ( 2 + 1 ) )
106	105	neeq2i 2859	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( P ` 2 ) =/= ( P ` 3 ) <-> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) )
107	103, 106	syl6bb 276	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( P ` 0 ) = ( P ` 3 ) -> ( ( P ` 2 ) =/= ( P ` 0 ) <-> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) ) )
108	107	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) -> ( ( P ` 2 ) =/= ( P ` 0 ) <-> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) ) )
109	108	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> ( ( P ` 2 ) =/= ( P ` 0 ) <-> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) ) )
110	102, 109	mpbird 247	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> ( P ` 2 ) =/= ( P ` 0 ) )
111	86, 96, 110	3jca 1242	. . . . . . . . . . . . . . . . 17 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) )
112	111	ex 450	. . . . . . . . . . . . . . . 16 |- ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) -> ( ( # ` F ) = 3 -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) ) )
113	112	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) -> ( ( # ` F ) = 3 -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) ) )
114	113	impcom 446	. . . . . . . . . . . . . 14 |- ( ( ( # ` F ) = 3 /\ ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) )
115		preq1 4268	. . . . . . . . . . . . . . . . . 18 |- ( a = ( P ` 0 ) -> { a , b } = { ( P ` 0 ) , b } )
116	115	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( a = ( P ` 0 ) -> ( { a , b } e. E <-> { ( P ` 0 ) , b } e. E ) )
117		preq2 4269	. . . . . . . . . . . . . . . . . 18 |- ( a = ( P ` 0 ) -> { c , a } = { c , ( P ` 0 ) } )
118	117	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( a = ( P ` 0 ) -> ( { c , a } e. E <-> { c , ( P ` 0 ) } e. E ) )
119	116, 118	3anbi13d 1401	. . . . . . . . . . . . . . . 16 |- ( a = ( P ` 0 ) -> ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) <-> ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E /\ { c , ( P ` 0 ) } e. E ) ) )
120		neeq1 2856	. . . . . . . . . . . . . . . . 17 |- ( a = ( P ` 0 ) -> ( a =/= b <-> ( P ` 0 ) =/= b ) )
121		neeq2 2857	. . . . . . . . . . . . . . . . 17 |- ( a = ( P ` 0 ) -> ( c =/= a <-> c =/= ( P ` 0 ) ) )
122	120, 121	3anbi13d 1401	. . . . . . . . . . . . . . . 16 |- ( a = ( P ` 0 ) -> ( ( a =/= b /\ b =/= c /\ c =/= a ) <-> ( ( P ` 0 ) =/= b /\ b =/= c /\ c =/= ( P ` 0 ) ) ) )
123	119, 122	anbi12d 747	. . . . . . . . . . . . . . 15 |- ( a = ( P ` 0 ) -> ( ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) <-> ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E /\ { c , ( P ` 0 ) } e. E ) /\ ( ( P ` 0 ) =/= b /\ b =/= c /\ c =/= ( P ` 0 ) ) ) ) )
124		preq2 4269	. . . . . . . . . . . . . . . . . 18 |- ( b = ( P ` 1 ) -> { ( P ` 0 ) , b } = { ( P ` 0 ) , ( P ` 1 ) } )
125	124	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( b = ( P ` 1 ) -> ( { ( P ` 0 ) , b } e. E <-> { ( P ` 0 ) , ( P ` 1 ) } e. E ) )
126		preq1 4268	. . . . . . . . . . . . . . . . . 18 |- ( b = ( P ` 1 ) -> { b , c } = { ( P ` 1 ) , c } )
127	126	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( b = ( P ` 1 ) -> ( { b , c } e. E <-> { ( P ` 1 ) , c } e. E ) )
128	125, 127	3anbi12d 1400	. . . . . . . . . . . . . . . 16 |- ( b = ( P ` 1 ) -> ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E /\ { c , ( P ` 0 ) } e. E ) <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E /\ { c , ( P ` 0 ) } e. E ) ) )
129		neeq2 2857	. . . . . . . . . . . . . . . . 17 |- ( b = ( P ` 1 ) -> ( ( P ` 0 ) =/= b <-> ( P ` 0 ) =/= ( P ` 1 ) ) )
130		neeq1 2856	. . . . . . . . . . . . . . . . 17 |- ( b = ( P ` 1 ) -> ( b =/= c <-> ( P ` 1 ) =/= c ) )
131	129, 130	3anbi12d 1400	. . . . . . . . . . . . . . . 16 |- ( b = ( P ` 1 ) -> ( ( ( P ` 0 ) =/= b /\ b =/= c /\ c =/= ( P ` 0 ) ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= c /\ c =/= ( P ` 0 ) ) ) )
132	128, 131	anbi12d 747	. . . . . . . . . . . . . . 15 |- ( b = ( P ` 1 ) -> ( ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E /\ { c , ( P ` 0 ) } e. E ) /\ ( ( P ` 0 ) =/= b /\ b =/= c /\ c =/= ( P ` 0 ) ) ) <-> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E /\ { c , ( P ` 0 ) } e. E ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= c /\ c =/= ( P ` 0 ) ) ) ) )
133		preq2 4269	. . . . . . . . . . . . . . . . . 18 |- ( c = ( P ` 2 ) -> { ( P ` 1 ) , c } = { ( P ` 1 ) , ( P ` 2 ) } )
134	133	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( c = ( P ` 2 ) -> ( { ( P ` 1 ) , c } e. E <-> { ( P ` 1 ) , ( P ` 2 ) } e. E ) )
135		preq1 4268	. . . . . . . . . . . . . . . . . 18 |- ( c = ( P ` 2 ) -> { c , ( P ` 0 ) } = { ( P ` 2 ) , ( P ` 0 ) } )
136	135	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( c = ( P ` 2 ) -> ( { c , ( P ` 0 ) } e. E <-> { ( P ` 2 ) , ( P ` 0 ) } e. E ) )
137	134, 136	3anbi23d 1402	. . . . . . . . . . . . . . . 16 |- ( c = ( P ` 2 ) -> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E /\ { c , ( P ` 0 ) } e. E ) <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 0 ) } e. E ) ) )
138		neeq2 2857	. . . . . . . . . . . . . . . . 17 |- ( c = ( P ` 2 ) -> ( ( P ` 1 ) =/= c <-> ( P ` 1 ) =/= ( P ` 2 ) ) )
139		neeq1 2856	. . . . . . . . . . . . . . . . 17 |- ( c = ( P ` 2 ) -> ( c =/= ( P ` 0 ) <-> ( P ` 2 ) =/= ( P ` 0 ) ) )
140	138, 139	3anbi23d 1402	. . . . . . . . . . . . . . . 16 |- ( c = ( P ` 2 ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= c /\ c =/= ( P ` 0 ) ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) ) )
141	137, 140	anbi12d 747	. . . . . . . . . . . . . . 15 |- ( c = ( P ` 2 ) -> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E /\ { c , ( P ` 0 ) } e. E ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= c /\ c =/= ( P ` 0 ) ) ) <-> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 0 ) } e. E ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) ) ) )
142	123, 132, 141	rspc3ev 3326	. . . . . . . . . . . . . 14 |- ( ( ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 0 ) } e. E ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) ) ) -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) )
143	64, 71, 114, 142	syl12anc 1324	. . . . . . . . . . . . 13 |- ( ( ( # ` F ) = 3 /\ ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) ) -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) )
144	143	ex 450	. . . . . . . . . . . 12 |- ( ( # ` F ) = 3 -> ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) )
145	38, 144	sylbid 230	. . . . . . . . . . 11 |- ( ( # ` F ) = 3 -> ( ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) /\ A. k e. ( 0..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) )
146	145	expd 452	. . . . . . . . . 10 |- ( ( # ` F ) = 3 -> ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( A. k e. ( 0..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) )
147	146	com13 88	. . . . . . . . 9 |- ( A. k e. ( 0..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E -> ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) )
148	4, 147	syl 17	. . . . . . . 8 |- ( ( G e. UPGraph /\ F (Walks ` G ) P ) -> ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) )
149	148	expcom 451	. . . . . . 7 |- ( F (Walks ` G ) P -> ( G e. UPGraph -> ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) )
150	149	com23 86	. . . . . 6 |- ( F (Walks ` G ) P -> ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( G e. UPGraph -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) )
151	150	expd 452	. . . . 5 |- ( F (Walks ` G ) P -> ( F (Paths ` G ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( G e. UPGraph -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) ) )
152	2, 151	mpcom 38	. . . 4 |- ( F (Paths ` G ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( G e. UPGraph -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) )
153	152	imp 445	. . 3 |- ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( G e. UPGraph -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) )
154	1, 153	syl 17	. 2 |- ( F (Cycles ` G ) P -> ( G e. UPGraph -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) )
155	154	3imp21 1277	1 |- ( ( G e. UPGraph /\ F (Cycles ` G ) P /\ ( # ` F ) = 3 ) -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {cpr 4179   {ctp 4181   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   NNcn 11020   2c2 11070   3c3 11071   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Vtxcvtx 25874  Edgcedg 25939   UPGraph cupgr 25975  Walkscwlks 26492  Pathscpths 26608  Cyclesccycls 26680

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-3 11080  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-wlks 26495  df-trls 26589  df-pths 26612  df-cycls 26682

This theorem is referenced by:  umgr3v3e3cycl  27044