Theorem usgrcyclnl1 23349

Description: In an undirected simple graph (with no loops!) there are no cycles with length 1 (consisting of one edge ). (Contributed by Alexander van der Vekens, 7-Nov-2017.)

Assertion

Ref	Expression
usgrcyclnl1	|- ( ( V USGrph E /\ F ( V Cycles E ) P ) -> ( # ` F ) =/= 1 )

Proof of Theorem usgrcyclnl1

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycliswlk 23341	. . 3 |- ( F ( V Cycles E ) P -> F ( V Walks E ) P )
2		wlkbprop 23256	. . . 4 |- ( F ( V Walks E ) P -> ( ( # ` F ) e. NN0 /\ ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )
3		iscycl 23334	. . . . . 6 |- ( ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( V Cycles E ) P <-> ( F ( V Paths E ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) )
4	3	3adant1 999	. . . . 5 |- ( ( ( # ` F ) e. NN0 /\ ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( V Cycles E ) P <-> ( F ( V Paths E ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) )
5		pthistrl 23294	. . . . . . . 8 |- ( F ( V Paths E ) P -> F ( V Trails E ) P )
6		trliswlk 23261	. . . . . . . 8 |- ( F ( V Trails E ) P -> F ( V Walks E ) P )
7	5, 6	syl 16	. . . . . . 7 |- ( F ( V Paths E ) P -> F ( V Walks E ) P )
8		usgrnloop 23285	. . . . . . . . . . 11 |- ( ( V USGrph E /\ F ( V Walks E ) P ) -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )
9		nne 2602	. . . . . . . . . . . . . . . 16 |- ( -. ( # ` F ) =/= 1 <-> ( # ` F ) = 1 )
10		0z 10645	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 0 e. ZZ
11		1z 10664	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 1 e. ZZ
12		0lt1 9850	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 0 < 1
13		fzolb 11542	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( 0 e. ( 0..^ 1 ) <-> ( 0 e. ZZ /\ 1 e. ZZ /\ 0 < 1 ) )
14	10, 11, 12, 13	mpbir3an 1163	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 0 e. ( 0..^ 1 )
15		oveq2 6088	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` F ) = 1 -> ( 0..^ ( # ` F ) ) = ( 0..^ 1 ) )
16	14, 15	syl5eleqr 2520	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( # ` F ) = 1 -> 0 e. ( 0..^ ( # ` F ) ) )
17	16	anim2i 564	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) = 1 ) -> ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ 0 e. ( 0..^ ( # ` F ) ) ) )
18	17	ex 434	. . . . . . . . . . . . . . . . . . . . . 22 |- ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( ( # ` F ) = 1 -> ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ 0 e. ( 0..^ ( # ` F ) ) ) ) )
19	18	adantr 462	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) e. NN0 ) -> ( ( # ` F ) = 1 -> ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ 0 e. ( 0..^ ( # ` F ) ) ) ) )
20	19	impcom 430	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( # ` F ) = 1 /\ ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) e. NN0 ) ) -> ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ 0 e. ( 0..^ ( # ` F ) ) ) )
21		fveq2 5679	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
22		oveq1 6087	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
23	22	fveq2d 5683	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` ( 0 + 1 ) ) )
24	21, 23	neeq12d 2613	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = 0 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) ) )
25	24	rspccva 3061	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ 0 e. ( 0..^ ( # ` F ) ) ) -> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) )
26	20, 25	syl 16	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( # ` F ) = 1 /\ ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) e. NN0 ) ) -> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) )
27		0p1e1 10421	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( 0 + 1 ) = 1
28		id 22	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 1 = ( # ` F ) -> 1 = ( # ` F ) )
29	28	eqcoms 2436	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( # ` F ) = 1 -> 1 = ( # ` F ) )
30	27, 29	syl5eq 2477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` F ) = 1 -> ( 0 + 1 ) = ( # ` F ) )
31	30	fveq2d 5683	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` F ) = 1 -> ( P ` ( 0 + 1 ) ) = ( P ` ( # ` F ) ) )
32	31	neeq2d 2612	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 1 -> ( ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) <-> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) )
33		df-ne 2598	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` 0 ) =/= ( P ` ( # ` F ) ) <-> -. ( P ` 0 ) = ( P ` ( # ` F ) ) )
34	32, 33	syl6bb 261	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` F ) = 1 -> ( ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) <-> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
35	26, 34	syl5ibcom 220	. . . . . . . . . . . . . . . . . 18 |- ( ( ( # ` F ) = 1 /\ ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) e. NN0 ) ) -> ( ( # ` F ) = 1 -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
36	35	ex 434	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) = 1 -> ( ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) e. NN0 ) -> ( ( # ` F ) = 1 -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) )
37	36	pm2.43a 49	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 1 -> ( ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) e. NN0 ) -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
38	9, 37	sylbi 195	. . . . . . . . . . . . . . 15 |- ( -. ( # ` F ) =/= 1 -> ( ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) e. NN0 ) -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
39	38	com12 31	. . . . . . . . . . . . . 14 |- ( ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) e. NN0 ) -> ( -. ( # ` F ) =/= 1 -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
40	39	con4d 105	. . . . . . . . . . . . 13 |- ( ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) e. NN0 ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( # ` F ) =/= 1 ) )
41	40	ex 434	. . . . . . . . . . . 12 |- ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( ( # ` F ) e. NN0 -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( # ` F ) =/= 1 ) ) )
42	41	com23 78	. . . . . . . . . . 11 |- ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( ( # ` F ) e. NN0 -> ( # ` F ) =/= 1 ) ) )
43	8, 42	syl 16	. . . . . . . . . 10 |- ( ( V USGrph E /\ F ( V Walks E ) P ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( ( # ` F ) e. NN0 -> ( # ` F ) =/= 1 ) ) )
44	43	ex 434	. . . . . . . . 9 |- ( V USGrph E -> ( F ( V Walks E ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( ( # ` F ) e. NN0 -> ( # ` F ) =/= 1 ) ) ) )
45	44	com14 88	. . . . . . . 8 |- ( ( # ` F ) e. NN0 -> ( F ( V Walks E ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( V USGrph E -> ( # ` F ) =/= 1 ) ) ) )
46	45	3ad2ant1 1002	. . . . . . 7 |- ( ( ( # ` F ) e. NN0 /\ ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( V Walks E ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( V USGrph E -> ( # ` F ) =/= 1 ) ) ) )
47	7, 46	syl5 32	. . . . . 6 |- ( ( ( # ` F ) e. NN0 /\ ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( V Paths E ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( V USGrph E -> ( # ` F ) =/= 1 ) ) ) )
48	47	imp3a 431	. . . . 5 |- ( ( ( # ` F ) e. NN0 /\ ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) -> ( ( F ( V Paths E ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( V USGrph E -> ( # ` F ) =/= 1 ) ) )
49	4, 48	sylbid 215	. . . 4 |- ( ( ( # ` F ) e. NN0 /\ ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( V Cycles E ) P -> ( V USGrph E -> ( # ` F ) =/= 1 ) ) )
50	2, 49	syl 16	. . 3 |- ( F ( V Walks E ) P -> ( F ( V Cycles E ) P -> ( V USGrph E -> ( # ` F ) =/= 1 ) ) )
51	1, 50	mpcom 36	. 2 |- ( F ( V Cycles E ) P -> ( V USGrph E -> ( # ` F ) =/= 1 ) )
52	51	impcom 430	1 |- ( ( V USGrph E /\ F ( V Cycles E ) P ) -> ( # ` F ) =/= 1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 958   = wceq 1362   e. wcel 1755   =/= wne 2596   A.wral 2705   _Vcvv 2962   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   0cc0 9270   1c1 9271   + caddc 9273   < clt 9406   NN0cn0 10567   ZZcz 10634  ..^cfzo 11532   #chash 12087   USGrph cusg 23087   Walks cwalk 23228   Trails ctrail 23229   Paths cpath 23230   Cycles ccycl 23237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-fzo 11533  df-hash 12088  df-word 12213  df-usgra 23089  df-wlk 23238  df-trail 23239  df-pth 23240  df-cycl 23243

This theorem is referenced by: (None)