Theorem umgrclwwlkge2 16643

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 2238	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
2	1	clwwlkbp 16636	. . . . 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 11308	. . . . . . 7 |- ( P e. Word (Vtx ` G ) -> (♯ ` P ) e. NN0 )
5	4	3ad2ant2 1050	. . . . . 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 11323	. . . . . . . . . . . 12 |- ( P e. Word (Vtx ` G ) -> P e. Fin )
8		fihasheq0 11232	. . . . . . . . . . . 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 2461	. . . . . . . . 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 1224	. . . . . 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 9766	. . . . . . . 8 |- ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> (♯ ` P ) e. ZZ )
17		1z 9670	. . . . . . . 8 |- 1 e. ZZ
18		zdceq 9720	. . . . . . . 8 |- ( ( (♯ ` P ) e. ZZ /\ 1 e. ZZ ) -> DECID (♯ ` P ) = 1 )
19	16, 17, 18	sylancl 417	. . . . . . 7 |- ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> DECID (♯ ` P ) = 1 )
20		exmiddc 848	. . . . . . 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 16640	. . . . . . . . . . 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 2238	. . . . . . . . . . . 12 |- ( P ` 0 ) = ( P ` 0 )
25		eqid 2238	. . . . . . . . . . . . 13 |- (Edg ` G ) = (Edg ` G )
26	25	umgredgne 16391	. . . . . . . . . . . 12 |- ( ( G e. UMGraph /\ { ( P ` 0 ) , ( P ` 0 ) } e. (Edg ` G ) ) -> ( P ` 0 ) =/= ( P ` 0 ) )
27		eqneqall 2430	. . . . . . . . . . . 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 2428	. . . . . . . 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 728	. . . . . 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 1208	. . . 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 425	. . 3 |- ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) -> ( (♯ ` P ) e. NN0 /\ (♯ ` P ) =/= 0 /\ (♯ ` P ) =/= 1 ) )
39		nn0n0n1ge2 9715	. . 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 ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   (/)c0 3520   {cpr 3710   class class class wbr 4130   ` cfv 5377   Fincfn 7022   0cc0 8179   1c1 8180   <_ cle 8361   2c2 9355   NN0cn0 9563   ZZcz 9644  ♯chash 11214  Word cword 11304  Vtxcvtx 16253  Edgcedg 16298  UMGraphcumgr 16333  ClWWalkscclwwlk 16632

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-lsw 11350  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-umgren 16335  df-clwwlk 16633

This theorem is used by: (None)