Theorem lfgrn1cycl 26697

Description: In a loop-free graph there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 2-Feb-2021.)

Hypotheses

Ref	Expression
lfgrn1cycl.v	|- V = (Vtx ` G )
lfgrn1cycl.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
lfgrn1cycl	|- ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } -> ( F (Cycles ` G ) P -> ( # ` F ) =/= 1 ) )

Distinct variable groups:   x, F   x, I   x, V

Allowed substitution hints:   P( x)   G( x)

Proof of Theorem lfgrn1cycl

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		cycliswlk 26693	. . 3 |- ( F (Cycles ` G ) P -> F (Walks ` G ) P )
3		lfgrn1cycl.i	. . . . . . . 8 |- I = (iEdg ` G )
4		lfgrn1cycl.v	. . . . . . . 8 |- V = (Vtx ` G )
5	3, 4	lfgrwlknloop 26586	. . . . . . 7 |- ( ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )
6		1nn 11031	. . . . . . . . . . . . . 14 |- 1 e. NN
7		eleq1 2689	. . . . . . . . . . . . . 14 |- ( ( # ` F ) = 1 -> ( ( # ` F ) e. NN <-> 1 e. NN ) )
8	6, 7	mpbiri 248	. . . . . . . . . . . . 13 |- ( ( # ` F ) = 1 -> ( # ` F ) e. NN )
9		lbfzo0 12507	. . . . . . . . . . . . 13 |- ( 0 e. ( 0..^ ( # ` F ) ) <-> ( # ` F ) e. NN )
10	8, 9	sylibr 224	. . . . . . . . . . . 12 |- ( ( # ` F ) = 1 -> 0 e. ( 0..^ ( # ` F ) ) )
11		fveq2 6191	. . . . . . . . . . . . . 14 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
12		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
13		0p1e1 11132	. . . . . . . . . . . . . . . 16 |- ( 0 + 1 ) = 1
14	12, 13	syl6eq 2672	. . . . . . . . . . . . . . 15 |- ( k = 0 -> ( k + 1 ) = 1 )
15	14	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
16	11, 15	neeq12d 2855	. . . . . . . . . . . . 13 |- ( k = 0 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 0 ) =/= ( P ` 1 ) ) )
17	16	rspcv 3305	. . . . . . . . . . . 12 |- ( 0 e. ( 0..^ ( # ` F ) ) -> ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( P ` 0 ) =/= ( P ` 1 ) ) )
18	10, 17	syl 17	. . . . . . . . . . 11 |- ( ( # ` F ) = 1 -> ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( P ` 0 ) =/= ( P ` 1 ) ) )
19	18	impcom 446	. . . . . . . . . 10 |- ( ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) = 1 ) -> ( P ` 0 ) =/= ( P ` 1 ) )
20		fveq2 6191	. . . . . . . . . . . 12 |- ( ( # ` F ) = 1 -> ( P ` ( # ` F ) ) = ( P ` 1 ) )
21	20	neeq2d 2854	. . . . . . . . . . 11 |- ( ( # ` F ) = 1 -> ( ( P ` 0 ) =/= ( P ` ( # ` F ) ) <-> ( P ` 0 ) =/= ( P ` 1 ) ) )
22	21	adantl 482	. . . . . . . . . 10 |- ( ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) = 1 ) -> ( ( P ` 0 ) =/= ( P ` ( # ` F ) ) <-> ( P ` 0 ) =/= ( P ` 1 ) ) )
23	19, 22	mpbird 247	. . . . . . . . 9 |- ( ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) /\ ( # ` F ) = 1 ) -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) )
24	23	ex 450	. . . . . . . 8 |- ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( ( # ` F ) = 1 -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) )
25	24	necon2d 2817	. . . . . . 7 |- ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( # ` F ) =/= 1 ) )
26	5, 25	syl 17	. . . . . 6 |- ( ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } /\ F (Walks ` G ) P ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( # ` F ) =/= 1 ) )
27	26	ex 450	. . . . 5 |- ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } -> ( F (Walks ` G ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( # ` F ) =/= 1 ) ) )
28	27	com13 88	. . . 4 |- ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( F (Walks ` G ) P -> ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } -> ( # ` F ) =/= 1 ) ) )
29	28	adantl 482	. . 3 |- ( ( F (Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( F (Walks ` G ) P -> ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } -> ( # ` F ) =/= 1 ) ) )
30	1, 2, 29	sylc 65	. 2 |- ( F (Cycles ` G ) P -> ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } -> ( # ` F ) =/= 1 ) )
31	30	com12 32	1 |- ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } -> ( F (Cycles ` G ) P -> ( # ` F ) =/= 1 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   NNcn 11020   2c2 11070  ..^cfzo 12465   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  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-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-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495  df-trls 26589  df-pths 26612  df-cycls 26682

This theorem is referenced by:  umgrn1cycl  26699