Theorem umgrclwwlkge2 16139

Description: A closed walk in a multigraph has a length of at least 2 (because it cannot have a loop). (Contributed by Alexander van der Vekens, 16-Sep-2018.) (Revised by AV, 24-Apr-2021.)

Assertion

Ref	Expression
umgrclwwlkge2	|- ( G e. UMGraph -> ( P e. (ClWWalks ` G ) -> 2 <_ (♯ ` P ) ) )

Proof of Theorem umgrclwwlkge2

Step	Hyp	Ref	Expression
1		eqid 2229	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
2	1	clwwlkbp 16133	. . . . 5 |- ( P e. (ClWWalks ` G ) -> ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) )
3	2	adantl 277	. . . 4 |- ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) -> ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) )
4		lencl 11088	. . . . . . 7 |- ( P e. Word (Vtx ` G ) -> (♯ ` P ) e. NN0 )
5	4	3ad2ant2 1043	. . . . . 6 |- ( ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) -> (♯ ` P ) e. NN0 )
6	5	adantl 277	. . . . 5 |- ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> (♯ ` P ) e. NN0 )
7		wrdfin 11103	. . . . . . . . . . . 12 |- ( P e. Word (Vtx ` G ) -> P e. Fin )
8		fihasheq0 11027	. . . . . . . . . . . 12 |- ( P e. Fin -> ( (♯ ` P ) = 0 <-> P = (/) ) )
9	7, 8	syl 14	. . . . . . . . . . 11 |- ( P e. Word (Vtx ` G ) -> ( (♯ ` P ) = 0 <-> P = (/) ) )
10	9	bicomd 141	. . . . . . . . . 10 |- ( P e. Word (Vtx ` G ) -> ( P = (/) <-> (♯ ` P ) = 0 ) )
11	10	necon3bid 2441	. . . . . . . . 9 |- ( P e. Word (Vtx ` G ) -> ( P =/= (/) <-> (♯ ` P ) =/= 0 ) )
12	11	biimpd 144	. . . . . . . 8 |- ( P e. Word (Vtx ` G ) -> ( P =/= (/) -> (♯ ` P ) =/= 0 ) )
13	12	a1i 9	. . . . . . 7 |- ( G e. _V -> ( P e. Word (Vtx ` G ) -> ( P =/= (/) -> (♯ ` P ) =/= 0 ) ) )
14	13	3imp 1217	. . . . . 6 |- ( ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) -> (♯ ` P ) =/= 0 )
15	14	adantl 277	. . . . 5 |- ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> (♯ ` P ) =/= 0 )
16	6	nn0zd 9578	. . . . . . . 8 |- ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> (♯ ` P ) e. ZZ )
17		1z 9483	. . . . . . . 8 |- 1 e. ZZ
18		zdceq 9533	. . . . . . . 8 |- ( ( (♯ ` P ) e. ZZ /\ 1 e. ZZ ) -> DECID (♯ ` P ) = 1 )
19	16, 17, 18	sylancl 413	. . . . . . 7 |- ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> DECID (♯ ` P ) = 1 )
20		exmiddc 841	. . . . . . 7 |- (DECID (♯ ` P ) = 1 -> ( (♯ ` P ) = 1 \/ -. (♯ ` P ) = 1 ) )
21	19, 20	syl 14	. . . . . 6 |- ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> ( (♯ ` P ) = 1 \/ -. (♯ ` P ) = 1 ) )
22		clwwlk1loop 16136	. . . . . . . . . . 11 |- ( ( P e. (ClWWalks ` G ) /\ (♯ ` P ) = 1 ) -> { ( P ` 0 ) , ( P ` 0 ) } e. (Edg ` G ) )
23	22	expcom 116	. . . . . . . . . 10 |- ( (♯ ` P ) = 1 -> ( P e. (ClWWalks ` G ) -> { ( P ` 0 ) , ( P ` 0 ) } e. (Edg ` G ) ) )
24		eqid 2229	. . . . . . . . . . . 12 |- ( P ` 0 ) = ( P ` 0 )
25		eqid 2229	. . . . . . . . . . . . 13 |- (Edg ` G ) = (Edg ` G )
26	25	umgredgne 15963	. . . . . . . . . . . 12 |- ( ( G e. UMGraph /\ { ( P ` 0 ) , ( P ` 0 ) } e. (Edg ` G ) ) -> ( P ` 0 ) =/= ( P ` 0 ) )
27		eqneqall 2410	. . . . . . . . . . . 12 |- ( ( P ` 0 ) = ( P ` 0 ) -> ( ( P ` 0 ) =/= ( P ` 0 ) -> ( ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) -> (♯ ` P ) =/= 1 ) ) )
28	24, 26, 27	mpsyl 65	. . . . . . . . . . 11 |- ( ( G e. UMGraph /\ { ( P ` 0 ) , ( P ` 0 ) } e. (Edg ` G ) ) -> ( ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) -> (♯ ` P ) =/= 1 ) )
29	28	expcom 116	. . . . . . . . . 10 |- ( { ( P ` 0 ) , ( P ` 0 ) } e. (Edg ` G ) -> ( G e. UMGraph -> ( ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) -> (♯ ` P ) =/= 1 ) ) )
30	23, 29	syl6 33	. . . . . . . . 9 |- ( (♯ ` P ) = 1 -> ( P e. (ClWWalks ` G ) -> ( G e. UMGraph -> ( ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) -> (♯ ` P ) =/= 1 ) ) ) )
31	30	com23 78	. . . . . . . 8 |- ( (♯ ` P ) = 1 -> ( G e. UMGraph -> ( P e. (ClWWalks ` G ) -> ( ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) -> (♯ ` P ) =/= 1 ) ) ) )
32	31	imp4c 351	. . . . . . 7 |- ( (♯ ` P ) = 1 -> ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> (♯ ` P ) =/= 1 ) )
33		neqne 2408	. . . . . . . 8 |- ( -. (♯ ` P ) = 1 -> (♯ ` P ) =/= 1 )
34	33	a1d 22	. . . . . . 7 |- ( -. (♯ ` P ) = 1 -> ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> (♯ ` P ) =/= 1 ) )
35	32, 34	jaoi 721	. . . . . 6 |- ( ( (♯ ` P ) = 1 \/ -. (♯ ` P ) = 1 ) -> ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> (♯ ` P ) =/= 1 ) )
36	21, 35	mpcom 36	. . . . 5 |- ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> (♯ ` P ) =/= 1 )
37	6, 15, 36	3jca 1201	. . . 4 |- ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> ( (♯ ` P ) e. NN0 /\ (♯ ` P ) =/= 0 /\ (♯ ` P ) =/= 1 ) )
38	3, 37	mpdan 421	. . 3 |- ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) -> ( (♯ ` P ) e. NN0 /\ (♯ ` P ) =/= 0 /\ (♯ ` P ) =/= 1 ) )
39		nn0n0n1ge2 9528	. . 3 |- ( ( (♯ ` P ) e. NN0 /\ (♯ ` P ) =/= 0 /\ (♯ ` P ) =/= 1 ) -> 2 <_ (♯ ` P ) )
40	38, 39	syl 14	. 2 |- ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) -> 2 <_ (♯ ` P ) )
41	40	ex 115	1 |- ( G e. UMGraph -> ( P e. (ClWWalks ` G ) -> 2 <_ (♯ ` P ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   _Vcvv 2799   (/)c0 3491   {cpr 3667   class class class wbr 4083   ` cfv 5318   Fincfn 6895   0cc0 8010   1c1 8011   <_ cle 8193   2c2 9172   NN0cn0 9380   ZZcz 9457  ♯chash 11009  Word cword 11084  Vtxcvtx 15828  Edgcedg 15873  UMGraphcumgr 15907  ClWWalkscclwwlk 16129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-2o 6569  df-er 6688  df-map 6805  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-z 9458  df-dec 9590  df-uz 9734  df-fz 10217  df-fzo 10351  df-ihash 11010  df-word 11085  df-lsw 11130  df-ndx 13050  df-slot 13051  df-base 13053  df-edgf 15821  df-vtx 15830  df-iedg 15831  df-edg 15874  df-umgren 15909  df-clwwlk 16130

This theorem is referenced by: (None)