Theorem usgrcyclnl1 24842

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 24834	. . 3 |- ( F ( V Cycles E ) P -> F ( V Walks E ) P )
2		wlkbprop 24725	. . . 4 |- ( F ( V Walks E ) P -> ( ( # ` F ) e. NN0 /\ ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )
3		iscycl 24827	. . . . . 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 1012	. . . . 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 24776	. . . . . . . 8 |- ( F ( V Paths E ) P -> F ( V Trails E ) P )
6		trliswlk 24743	. . . . . . . 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 24767	. . . . . . . . . . 11 |- ( ( V USGrph E /\ F ( V Walks E ) P ) -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )
9		nne 2655	. . . . . . . . . . . . . . . 16 |- ( -. ( # ` F ) =/= 1 <-> ( # ` F ) = 1 )
10		0z 10871	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 0 e. ZZ
11		1z 10890	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 1 e. ZZ
12		0lt1 10071	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 0 < 1
13		fzolb 11810	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( 0 e. ( 0..^ 1 ) <-> ( 0 e. ZZ /\ 1 e. ZZ /\ 0 < 1 ) )
14	10, 11, 12, 13	mpbir3an 1176	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 0 e. ( 0..^ 1 )
15		oveq2 6278	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` F ) = 1 -> ( 0..^ ( # ` F ) ) = ( 0..^ 1 ) )
16	14, 15	syl5eleqr 2549	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( # ` F ) = 1 -> 0 e. ( 0..^ ( # ` F ) ) )
17	16	anim2i 567	. . . . . . . . . . . . . . . . . . . . . . 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 432	. . . . . . . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . . . . . . 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 428	. . . . . . . . . . . . . . . . . . . 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 5848	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
22		oveq1 6277	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
23	22	fveq2d 5852	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` ( 0 + 1 ) ) )
24	21, 23	neeq12d 2733	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = 0 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) ) )
25	24	rspccva 3206	. . . . . . . . . . . . . . . . . . . 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 10643	. . . . . . . . . . . . . . . . . . . . . . 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 5852	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` F ) = 1 -> ( P ` ( 0 + 1 ) ) = ( P ` ( # ` F ) ) )
32	31	neeq2d 2732	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 1 -> ( ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) <-> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) )
33		df-ne 2651	. . . . . . . . . . . . . . . . . . . 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 432	. . . . . . . . . . . . . . . . 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 432	. . . . . . . . . . . 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 432	. . . . . . . . 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 1015	. . . . . . 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 429	. . . . 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 428	1 |- ( ( V USGrph E /\ F ( V Cycles E ) P ) -> ( # ` F ) =/= 1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   _Vcvv 3106   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482   + caddc 9484   < clt 9617   NN0cn0 10791   ZZcz 10860  ..^cfzo 11799   #chash 12387   USGrph cusg 24532   Walks cwalk 24700   Trails ctrail 24701   Paths cpath 24702   Cycles ccycl 24709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-usgra 24535  df-wlk 24710  df-trail 24711  df-pth 24712  df-cycl 24715

This theorem is referenced by: (None)