Theorem usgrcyclnl1 23677

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 23669	. . 3 |- ( F ( V Cycles E ) P -> F ( V Walks E ) P )
2		wlkbprop 23584	. . . 4 |- ( F ( V Walks E ) P -> ( ( # ` F ) e. NN0 /\ ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )
3		iscycl 23662	. . . . . 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 1006	. . . . 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 23622	. . . . . . . 8 |- ( F ( V Paths E ) P -> F ( V Trails E ) P )
6		trliswlk 23589	. . . . . . . 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 23613	. . . . . . . . . . 11 |- ( ( V USGrph E /\ F ( V Walks E ) P ) -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )
9		nne 2653	. . . . . . . . . . . . . . . 16 |- ( -. ( # ` F ) =/= 1 <-> ( # ` F ) = 1 )
10		0z 10767	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 0 e. ZZ
11		1z 10786	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 1 e. ZZ
12		0lt1 9972	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 0 < 1
13		fzolb 11674	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( 0 e. ( 0..^ 1 ) <-> ( 0 e. ZZ /\ 1 e. ZZ /\ 0 < 1 ) )
14	10, 11, 12, 13	mpbir3an 1170	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 0 e. ( 0..^ 1 )
15		oveq2 6207	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` F ) = 1 -> ( 0..^ ( # ` F ) ) = ( 0..^ 1 ) )
16	14, 15	syl5eleqr 2549	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( # ` F ) = 1 -> 0 e. ( 0..^ ( # ` F ) ) )
17	16	anim2i 569	. . . . . . . . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . . . . . . . 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 5798	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
22		oveq1 6206	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
23	22	fveq2d 5802	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` ( 0 + 1 ) ) )
24	21, 23	neeq12d 2730	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = 0 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) ) )
25	24	rspccva 3176	. . . . . . . . . . . . . . . . . . . 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 10543	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( 0 + 1 ) = 1
28		id 22	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 1 = ( # ` F ) -> 1 = ( # ` F ) )
29	28	eqcoms 2466	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( # ` F ) = 1 -> 1 = ( # ` F ) )
30	27, 29	syl5eq 2507	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` F ) = 1 -> ( 0 + 1 ) = ( # ` F ) )
31	30	fveq2d 5802	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` F ) = 1 -> ( P ` ( 0 + 1 ) ) = ( P ` ( # ` F ) ) )
32	31	neeq2d 2729	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 1 -> ( ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) <-> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) )
33		df-ne 2649	. . . . . . . . . . . . . . . . . . . 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 1009	. . . . . . 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	impd 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 965   = wceq 1370   e. wcel 1758   =/= wne 2647   A.wral 2798   _Vcvv 3076   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   0cc0 9392   1c1 9393   + caddc 9395   < clt 9528   NN0cn0 10689   ZZcz 10756  ..^cfzo 11664   #chash 12219   USGrph cusg 23415   Walks cwalk 23556   Trails ctrail 23557   Paths cpath 23558   Cycles ccycl 23565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-hash 12220  df-word 12346  df-usgra 23417  df-wlk 23566  df-trail 23567  df-pth 23568  df-cycl 23571

This theorem is referenced by: (None)