Theorem wlkl1loop 26534

Description: A walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.)

Assertion

Ref	Expression
wlkl1loop	|- ( ( ( Fun (iEdg ` G ) /\ F (Walks ` G ) P ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) )

Proof of Theorem wlkl1loop

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkv 26508	. . . . 5 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
2		simp3l 1089	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> Fun (iEdg ` G ) )
3		simp2 1062	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> F (Walks ` G ) P )
4		c0ex 10034	. . . . . . . . . . . . 13 |- 0 e. _V
5	4	snid 4208	. . . . . . . . . . . 12 |- 0 e. { 0 }
6		oveq2 6658	. . . . . . . . . . . . 13 |- ( ( # ` F ) = 1 -> ( 0..^ ( # ` F ) ) = ( 0..^ 1 ) )
7		fzo01 12550	. . . . . . . . . . . . 13 |- ( 0..^ 1 ) = { 0 }
8	6, 7	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( # ` F ) = 1 -> ( 0..^ ( # ` F ) ) = { 0 } )
9	5, 8	syl5eleqr 2708	. . . . . . . . . . 11 |- ( ( # ` F ) = 1 -> 0 e. ( 0..^ ( # ` F ) ) )
10	9	ad2antrl 764	. . . . . . . . . 10 |- ( ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> 0 e. ( 0..^ ( # ` F ) ) )
11	10	3ad2ant3 1084	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> 0 e. ( 0..^ ( # ` F ) ) )
12		eqid 2622	. . . . . . . . . 10 |- (iEdg ` G ) = (iEdg ` G )
13	12	iedginwlk 26533	. . . . . . . . 9 |- ( ( Fun (iEdg ` G ) /\ F (Walks ` G ) P /\ 0 e. ( 0..^ ( # ` F ) ) ) -> ( (iEdg ` G ) ` ( F ` 0 ) ) e. ran (iEdg ` G ) )
14	2, 3, 11, 13	syl3anc 1326	. . . . . . . 8 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> ( (iEdg ` G ) ` ( F ` 0 ) ) e. ran (iEdg ` G ) )
15		eqid 2622	. . . . . . . . . . 11 |- (Vtx ` G ) = (Vtx ` G )
16	15, 12	iswlkg 26509	. . . . . . . . . 10 |- ( G e. _V -> ( F (Walks ` G ) P <-> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) ) )
17	8	raleqdv 3144	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 1 -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> A. k e. { 0 }if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) )
18		oveq1 6657	. . . . . . . . . . . . . . . . . 18 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
19		0p1e1 11132	. . . . . . . . . . . . . . . . . 18 |- ( 0 + 1 ) = 1
20	18, 19	syl6eq 2672	. . . . . . . . . . . . . . . . 17 |- ( k = 0 -> ( k + 1 ) = 1 )
21		wkslem2 26504	. . . . . . . . . . . . . . . . 17 |- ( ( k = 0 /\ ( k + 1 ) = 1 ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) )
22	20, 21	mpdan 702	. . . . . . . . . . . . . . . 16 |- ( k = 0 -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) )
23	4, 22	ralsn 4222	. . . . . . . . . . . . . . 15 |- ( A. k e. { 0 }if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) )
24	17, 23	syl6bb 276	. . . . . . . . . . . . . 14 |- ( ( # ` F ) = 1 -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) )
25	24	ad2antrl 764	. . . . . . . . . . . . 13 |- ( ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) )
26		ifptru 1023	. . . . . . . . . . . . . . . . 17 |- ( ( P ` 0 ) = ( P ` 1 ) -> (if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) <-> ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } ) )
27	26	biimpa 501	. . . . . . . . . . . . . . . 16 |- ( ( ( P ` 0 ) = ( P ` 1 ) /\ if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) -> ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } )
28	27	eqcomd 2628	. . . . . . . . . . . . . . 15 |- ( ( ( P ` 0 ) = ( P ` 1 ) /\ if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) )
29	28	ex 450	. . . . . . . . . . . . . 14 |- ( ( P ` 0 ) = ( P ` 1 ) -> (if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) ) )
30	29	ad2antll 765	. . . . . . . . . . . . 13 |- ( ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> (if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) ) )
31	25, 30	sylbid 230	. . . . . . . . . . . 12 |- ( ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) ) )
32	31	com12 32	. . . . . . . . . . 11 |- ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) -> ( ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) ) )
33	32	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> ( ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) ) )
34	16, 33	syl6bi 243	. . . . . . . . 9 |- ( G e. _V -> ( F (Walks ` G ) P -> ( ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) )
35	34	3imp 1256	. . . . . . . 8 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) )
36		edgval 25941	. . . . . . . . 9 |- (Edg ` G ) = ran (iEdg ` G )
37	36	a1i 11	. . . . . . . 8 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> (Edg ` G ) = ran (iEdg ` G ) )
38	14, 35, 37	3eltr4d 2716	. . . . . . 7 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) )
39	38	3exp 1264	. . . . . 6 |- ( G e. _V -> ( F (Walks ` G ) P -> ( ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) ) ) )
40	39	3ad2ant1 1082	. . . . 5 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F (Walks ` G ) P -> ( ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) ) ) )
41	1, 40	mpcom 38	. . . 4 |- ( F (Walks ` G ) P -> ( ( Fun (iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) ) )
42	41	expd 452	. . 3 |- ( F (Walks ` G ) P -> ( Fun (iEdg ` G ) -> ( ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) -> { ( P ` 0 ) } e. (Edg ` G ) ) ) )
43	42	impcom 446	. 2 |- ( ( Fun (iEdg ` G ) /\ F (Walks ` G ) P ) -> ( ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) -> { ( P ` 0 ) } e. (Edg ` G ) ) )
44	43	imp 445	1 |- ( ( ( Fun (iEdg ` G ) /\ F (Walks ` G ) P ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384  if-wif 1012   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   {csn 4177   {cpr 4179   class class class wbr 4653   dom cdm 5114   ran crn 5115   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939  Walkscwlks 26492

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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-wlks 26495

This theorem is referenced by:  clwlkl1loop  26678