Theorem umgrclwwlkge2 16523

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 2234	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
2	1	clwwlkbp 16516	. . . . 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 11253	. . . . . . 7 |- ( P e. Word (Vtx ` G ) -> (♯ ` P ) e. NN0 )
5	4	3ad2ant2 1046	. . . . . 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 11268	. . . . . . . . . . . 12 |- ( P e. Word (Vtx ` G ) -> P e. Fin )
8		fihasheq0 11181	. . . . . . . . . . . 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 2455	. . . . . . . . 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 1220	. . . . . 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 9716	. . . . . . . 8 |- ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> (♯ ` P ) e. ZZ )
17		1z 9620	. . . . . . . 8 |- 1 e. ZZ
18		zdceq 9670	. . . . . . . 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 844	. . . . . . 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 16520	. . . . . . . . . . 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 2234	. . . . . . . . . . . 12 |- ( P ` 0 ) = ( P ` 0 )
25		eqid 2234	. . . . . . . . . . . . 13 |- (Edg ` G ) = (Edg ` G )
26	25	umgredgne 16271	. . . . . . . . . . . 12 |- ( ( G e. UMGraph /\ { ( P ` 0 ) , ( P ` 0 ) } e. (Edg ` G ) ) -> ( P ` 0 ) =/= ( P ` 0 ) )
27		eqneqall 2424	. . . . . . . . . . . 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 2422	. . . . . . . 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 724	. . . . . 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 1204	. . . 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 9665	. . 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 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   _Vcvv 2815   (/)c0 3512   {cpr 3695   class class class wbr 4114   ` cfv 5357   Fincfn 6988   0cc0 8143   1c1 8144   <_ cle 8325   2c2 9305   NN0cn0 9513   ZZcz 9594  ♯chash 11163  Word cword 11249  Vtxcvtx 16133  Edgcedg 16178  UMGraphcumgr 16213  ClWWalkscclwwlk 16512

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-lsw 11295  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-umgren 16215  df-clwwlk 16513

This theorem is referenced by: (None)