Theorem 2pthnloop 26627

Description: A path of length at least 2 does not contain a loop. In contrast, a path of length 1 can contain/be a loop, see lppthon 27011. (Contributed by AV, 6-Feb-2021.)

Hypothesis

Ref	Expression
2pthnloop.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
2pthnloop	|- ( ( F (Paths ` G ) P /\ 1 < ( # ` F ) ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) )

Distinct variable groups:   i, F   i, G   i, I   P, i

Proof of Theorem 2pthnloop

Step	Hyp	Ref	Expression
1		pthiswlk 26623	. . . . 5 |- ( F (Paths ` G ) P -> F (Walks ` G ) P )
2		wlkv 26508	. . . . 5 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
3	1, 2	syl 17	. . . 4 |- ( F (Paths ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
4		ispth 26619	. . . . . . 7 |- ( F (Paths ` G ) P <-> ( F (Trails ` G ) P /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) )
5	4	a1i 11	. . . . . 6 |- ( G e. _V -> ( F (Paths ` G ) P <-> ( F (Trails ` G ) P /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) ) )
6		istrl 26593	. . . . . . . . . . . 12 |- ( F (Trails ` G ) P <-> ( F (Walks ` G ) P /\ Fun `' F ) )
7		eqid 2622	. . . . . . . . . . . . . 14 |- (Vtx ` G ) = (Vtx ` G )
8		2pthnloop.i	. . . . . . . . . . . . . 14 |- I = (iEdg ` G )
9	7, 8	iswlkg 26509	. . . . . . . . . . . . 13 |- ( G e. _V -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) )if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) ) ) )
10	9	anbi1d 741	. . . . . . . . . . . 12 |- ( G e. _V -> ( ( F (Walks ` G ) P /\ Fun `' F ) <-> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) )if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) ) /\ Fun `' F ) ) )
11	6, 10	syl5bb 272	. . . . . . . . . . 11 |- ( G e. _V -> ( F (Trails ` G ) P <-> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) )if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) ) /\ Fun `' F ) ) )
12		pthdadjvtx 26626	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( F (Paths ` G ) P /\ 1 < ( # ` F ) /\ i e. ( 0..^ ( # ` F ) ) ) -> ( P ` i ) =/= ( P ` ( i + 1 ) ) )
13	12	ad5ant245 1307	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ F (Paths ` G ) P ) /\ ( ( Fun `' F /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) ) ) /\ 1 < ( # ` F ) ) /\ i e. ( 0..^ ( # ` F ) ) ) -> ( P ` i ) =/= ( P ` ( i + 1 ) ) )
14	13	neneqd 2799	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ F (Paths ` G ) P ) /\ ( ( Fun `' F /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) ) ) /\ 1 < ( # ` F ) ) /\ i e. ( 0..^ ( # ` F ) ) ) -> -. ( P ` i ) = ( P ` ( i + 1 ) ) )
15		ifpfal 1024	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( -. ( P ` i ) = ( P ` ( i + 1 ) ) -> (if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) )
16	15	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ F (Paths ` G ) P ) /\ ( ( Fun `' F /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) ) ) /\ 1 < ( # ` F ) ) /\ i e. ( 0..^ ( # ` F ) ) ) /\ -. ( P ` i ) = ( P ` ( i + 1 ) ) ) -> (if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) )
17		fvexd 6203	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( -. ( P ` i ) = ( P ` ( i + 1 ) ) -> ( P ` i ) e. _V )
18		fvexd 6203	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( -. ( P ` i ) = ( P ` ( i + 1 ) ) -> ( P ` ( i + 1 ) ) e. _V )
19		neqne 2802	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( -. ( P ` i ) = ( P ` ( i + 1 ) ) -> ( P ` i ) =/= ( P ` ( i + 1 ) ) )
20		fvexd 6203	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( -. ( P ` i ) = ( P ` ( i + 1 ) ) -> ( I ` ( F ` i ) ) e. _V )
21		prsshashgt1 13198	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( P ` i ) e. _V /\ ( P ` ( i + 1 ) ) e. _V /\ ( P ` i ) =/= ( P ` ( i + 1 ) ) ) /\ ( I ` ( F ` i ) ) e. _V ) -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) -> 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) )
22	17, 18, 19, 20, 21	syl31anc 1329	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( -. ( P ` i ) = ( P ` ( i + 1 ) ) -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) -> 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) )
23	22	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ F (Paths ` G ) P ) /\ ( ( Fun `' F /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) ) ) /\ 1 < ( # ` F ) ) /\ i e. ( 0..^ ( # ` F ) ) ) /\ -. ( P ` i ) = ( P ` ( i + 1 ) ) ) -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) -> 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) )
24	16, 23	sylbid 230	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ F (Paths ` G ) P ) /\ ( ( Fun `' F /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) ) ) /\ 1 < ( # ` F ) ) /\ i e. ( 0..^ ( # ` F ) ) ) /\ -. ( P ` i ) = ( P ` ( i + 1 ) ) ) -> (if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) -> 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) )
25	14, 24	mpdan 702	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ F (Paths ` G ) P ) /\ ( ( Fun `' F /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) ) ) /\ 1 < ( # ` F ) ) /\ i e. ( 0..^ ( # ` F ) ) ) -> (if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) -> 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) )
26	25	ralimdva 2962	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ F (Paths ` G ) P ) /\ ( ( Fun `' F /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) ) ) /\ 1 < ( # ` F ) ) -> ( A. i e. ( 0..^ ( # ` F ) )if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) )
27	26	ex 450	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ F (Paths ` G ) P ) /\ ( ( Fun `' F /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) ) ) -> ( 1 < ( # ` F ) -> ( A. i e. ( 0..^ ( # ` F ) )if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) )
28	27	com23 86	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ F (Paths ` G ) P ) /\ ( ( Fun `' F /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) ) ) -> ( A. i e. ( 0..^ ( # ` F ) )if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) )
29	28	exp31 630	. . . . . . . . . . . . . . . 16 |- ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> ( F (Paths ` G ) P -> ( ( ( Fun `' F /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) ) -> ( A. i e. ( 0..^ ( # ` F ) )if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) ) )
30	29	com24 95	. . . . . . . . . . . . . . 15 |- ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> ( A. i e. ( 0..^ ( # ` F ) )if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) -> ( ( ( Fun `' F /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) ) -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) ) )
31	30	3impia 1261	. . . . . . . . . . . . . 14 |- ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) )if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) ) -> ( ( ( Fun `' F /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) ) -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) )
32	31	exp4c 636	. . . . . . . . . . . . 13 |- ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) )if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) ) -> ( Fun `' F -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) -> ( Fun `' ( P |` ( 1..^ ( # ` F ) ) ) -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) ) ) )
33	32	imp 445	. . . . . . . . . . . 12 |- ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) )if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) ) /\ Fun `' F ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) -> ( Fun `' ( P |` ( 1..^ ( # ` F ) ) ) -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) ) )
34	33	a1i 11	. . . . . . . . . . 11 |- ( G e. _V -> ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) )if- ( ( P ` i ) = ( P ` ( i + 1 ) ) , ( I ` ( F ` i ) ) = { ( P ` i ) } , { ( P ` i ) , ( P ` ( i + 1 ) ) } C_ ( I ` ( F ` i ) ) ) ) /\ Fun `' F ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) -> ( Fun `' ( P |` ( 1..^ ( # ` F ) ) ) -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) ) ) )
35	11, 34	sylbid 230	. . . . . . . . . 10 |- ( G e. _V -> ( F (Trails ` G ) P -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) -> ( Fun `' ( P |` ( 1..^ ( # ` F ) ) ) -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) ) ) )
36	35	com24 95	. . . . . . . . 9 |- ( G e. _V -> ( Fun `' ( P |` ( 1..^ ( # ` F ) ) ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) -> ( F (Trails ` G ) P -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) ) ) )
37	36	com14 96	. . . . . . . 8 |- ( F (Trails ` G ) P -> ( Fun `' ( P |` ( 1..^ ( # ` F ) ) ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) -> ( G e. _V -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) ) ) )
38	37	3imp 1256	. . . . . . 7 |- ( ( F (Trails ` G ) P /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) -> ( G e. _V -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) )
39	38	com12 32	. . . . . 6 |- ( G e. _V -> ( ( F (Trails ` G ) P /\ Fun `' ( P |` ( 1..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) )
40	5, 39	sylbid 230	. . . . 5 |- ( G e. _V -> ( F (Paths ` G ) P -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) )
41	40	3ad2ant1 1082	. . . 4 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F (Paths ` G ) P -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) ) )
42	3, 41	mpcom 38	. . 3 |- ( F (Paths ` G ) P -> ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) ) )
43	42	pm2.43i 52	. 2 |- ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) ) )
44	43	imp 445	1 |- ( ( F (Paths ` G ) P /\ 1 < ( # ` F ) ) -> A. i e. ( 0..^ ( # ` F ) ) 2 <_ ( # ` ( I ` ( F ` i ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384  if-wif 1012   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   2c2 11070   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Walkscwlks 26492  Trailsctrls 26587  Pathscpths 26608

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-rmo 2920  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-oadd 7564  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-cda 8990  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-xnn0 11364  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

This theorem is referenced by:  upgr2pthnlp  26628