Theorem usgrcyclnl1 23494

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 23486	. . 3 |- ( F ( V Cycles E ) P -> F ( V Walks E ) P )
2		wlkbprop 23401	. . . 4 |- ( F ( V Walks E ) P -> ( ( # ` F ) e. NN0 /\ ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )
3		iscycl 23479	. . . . . 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 23439	. . . . . . . 8 |- ( F ( V Paths E ) P -> F ( V Trails E ) P )
6		trliswlk 23406	. . . . . . . 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 23430	. . . . . . . . . . 11 |- ( ( V USGrph E /\ F ( V Walks E ) P ) -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )
9		nne 2607	. . . . . . . . . . . . . . . 16 |- ( -. ( # ` F ) =/= 1 <-> ( # ` F ) = 1 )
10		0z 10649	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 0 e. ZZ
11		1z 10668	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 1 e. ZZ
12		0lt1 9854	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 0 < 1
13		fzolb 11550	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6094	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` F ) = 1 -> ( 0..^ ( # ` F ) ) = ( 0..^ 1 ) )
16	14, 15	syl5eleqr 2525	. . . . . . . . . . . . . . . . . . . . . . . 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 5686	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
22		oveq1 6093	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
23	22	fveq2d 5690	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` ( 0 + 1 ) ) )
24	21, 23	neeq12d 2618	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = 0 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) ) )
25	24	rspccva 3067	. . . . . . . . . . . . . . . . . . . 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 10425	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( 0 + 1 ) = 1
28		id 22	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 1 = ( # ` F ) -> 1 = ( # ` F ) )
29	28	eqcoms 2441	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( # ` F ) = 1 -> 1 = ( # ` F ) )
30	27, 29	syl5eq 2482	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` F ) = 1 -> ( 0 + 1 ) = ( # ` F ) )
31	30	fveq2d 5690	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` F ) = 1 -> ( P ` ( 0 + 1 ) ) = ( P ` ( # ` F ) ) )
32	31	neeq2d 2617	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 1 -> ( ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) <-> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) )
33		df-ne 2603	. . . . . . . . . . . . . . . . . . . 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 1369   e. wcel 1756   =/= wne 2601   A.wral 2710   _Vcvv 2967   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   0cc0 9274   1c1 9275   + caddc 9277   < clt 9410   NN0cn0 10571   ZZcz 10638  ..^cfzo 11540   #chash 12095   USGrph cusg 23232   Walks cwalk 23373   Trails ctrail 23374   Paths cpath 23375   Cycles ccycl 23382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-hash 12096  df-word 12221  df-usgra 23234  df-wlk 23383  df-trail 23384  df-pth 23385  df-cycl 23388

This theorem is referenced by: (None)