Theorem umgrclwwlkge2 16252

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 2231	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
2	1	clwwlkbp 16245	. . . . 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 11116	. . . . . . 7 |- ( P e. Word (Vtx ` G ) -> (♯ ` P ) e. NN0 )
5	4	3ad2ant2 1045	. . . . . 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 11131	. . . . . . . . . . . 12 |- ( P e. Word (Vtx ` G ) -> P e. Fin )
8		fihasheq0 11054	. . . . . . . . . . . 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 2443	. . . . . . . . 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 1219	. . . . . 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 9599	. . . . . . . 8 |- ( ( ( G e. UMGraph /\ P e. (ClWWalks ` G ) ) /\ ( G e. _V /\ P e. Word (Vtx ` G ) /\ P =/= (/) ) ) -> (♯ ` P ) e. ZZ )
17		1z 9504	. . . . . . . 8 |- 1 e. ZZ
18		zdceq 9554	. . . . . . . 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 843	. . . . . . 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 16249	. . . . . . . . . . 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 2231	. . . . . . . . . . . 12 |- ( P ` 0 ) = ( P ` 0 )
25		eqid 2231	. . . . . . . . . . . . 13 |- (Edg ` G ) = (Edg ` G )
26	25	umgredgne 16000	. . . . . . . . . . . 12 |- ( ( G e. UMGraph /\ { ( P ` 0 ) , ( P ` 0 ) } e. (Edg ` G ) ) -> ( P ` 0 ) =/= ( P ` 0 ) )
27		eqneqall 2412	. . . . . . . . . . . 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 2410	. . . . . . . 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 723	. . . . . 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 1203	. . . 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 9549	. . 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 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   _Vcvv 2802   (/)c0 3494   {cpr 3670   class class class wbr 4088   ` cfv 5326   Fincfn 6908   0cc0 8031   1c1 8032   <_ cle 8214   2c2 9193   NN0cn0 9401   ZZcz 9478  ♯chash 11036  Word cword 11112  Vtxcvtx 15862  Edgcedg 15907  UMGraphcumgr 15942  ClWWalkscclwwlk 16241

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-2o 6582  df-er 6701  df-map 6818  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-z 9479  df-dec 9611  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-lsw 11158  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-edg 15908  df-umgren 15944  df-clwwlk 16242

This theorem is referenced by: (None)