Theorem clwwlksn2 26910

Description: A closed walk of length 2 represented as word is a word consisting of 2 symbols representing (not necessarily different) vertices connected by (at least) one edge. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Apr-2021.)

Assertion

Ref	Expression
clwwlksn2	|- ( W e. ( 2 ClWWalksN G ) <-> ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )

Proof of Theorem clwwlksn2

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn 11185	. . 3 |- 2 e. NN
2		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2622	. . . 4 |- (Edg ` G ) = (Edg ` G )
4	2, 3	isclwwlksnx 26889	. . 3 |- ( 2 e. NN -> ( W e. ( 2 ClWWalksN G ) <-> ( ( W e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ ( # ` W ) = 2 ) ) )
5	1, 4	ax-mp 5	. 2 |- ( W e. ( 2 ClWWalksN G ) <-> ( ( W e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ ( # ` W ) = 2 ) )
6		3anass 1042	. . . 4 |- ( ( W e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) <-> ( W e. Word (Vtx ` G ) /\ ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) ) )
7		oveq1 6657	. . . . . . . . . . . . 13 |- ( ( # ` W ) = 2 -> ( ( # ` W ) - 1 ) = ( 2 - 1 ) )
8		2m1e1 11135	. . . . . . . . . . . . 13 |- ( 2 - 1 ) = 1
9	7, 8	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( # ` W ) = 2 -> ( ( # ` W ) - 1 ) = 1 )
10	9	oveq2d 6666	. . . . . . . . . . 11 |- ( ( # ` W ) = 2 -> ( 0..^ ( ( # ` W ) - 1 ) ) = ( 0..^ 1 ) )
11		fzo01 12550	. . . . . . . . . . 11 |- ( 0..^ 1 ) = { 0 }
12	10, 11	syl6eq 2672	. . . . . . . . . 10 |- ( ( # ` W ) = 2 -> ( 0..^ ( ( # ` W ) - 1 ) ) = { 0 } )
13	12	adantr 481	. . . . . . . . 9 |- ( ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> ( 0..^ ( ( # ` W ) - 1 ) ) = { 0 } )
14	13	raleqdv 3144	. . . . . . . 8 |- ( ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. { 0 } { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) ) )
15		c0ex 10034	. . . . . . . . 9 |- 0 e. _V
16		fveq2 6191	. . . . . . . . . . 11 |- ( i = 0 -> ( W ` i ) = ( W ` 0 ) )
17		oveq1 6657	. . . . . . . . . . . . 13 |- ( i = 0 -> ( i + 1 ) = ( 0 + 1 ) )
18		0p1e1 11132	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
19	17, 18	syl6eq 2672	. . . . . . . . . . . 12 |- ( i = 0 -> ( i + 1 ) = 1 )
20	19	fveq2d 6195	. . . . . . . . . . 11 |- ( i = 0 -> ( W ` ( i + 1 ) ) = ( W ` 1 ) )
21	16, 20	preq12d 4276	. . . . . . . . . 10 |- ( i = 0 -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( W ` 0 ) , ( W ` 1 ) } )
22	21	eleq1d 2686	. . . . . . . . 9 |- ( i = 0 -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )
23	15, 22	ralsn 4222	. . . . . . . 8 |- ( A. i e. { 0 } { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) )
24	14, 23	syl6bb 276	. . . . . . 7 |- ( ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )
25		prcom 4267	. . . . . . . . 9 |- { ( lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) , ( lastS ` W ) }
26		lsw 13351	. . . . . . . . . . 11 |- ( W e. Word (Vtx ` G ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
27	9	fveq2d 6195	. . . . . . . . . . 11 |- ( ( # ` W ) = 2 -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` 1 ) )
28	26, 27	sylan9eqr 2678	. . . . . . . . . 10 |- ( ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> ( lastS ` W ) = ( W ` 1 ) )
29	28	preq2d 4275	. . . . . . . . 9 |- ( ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> { ( W ` 0 ) , ( lastS ` W ) } = { ( W ` 0 ) , ( W ` 1 ) } )
30	25, 29	syl5eq 2668	. . . . . . . 8 |- ( ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> { ( lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) , ( W ` 1 ) } )
31	30	eleq1d 2686	. . . . . . 7 |- ( ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> ( { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )
32	24, 31	anbi12d 747	. . . . . 6 |- ( ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> ( ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) <-> ( { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) ) )
33		anidm 676	. . . . . 6 |- ( ( { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) )
34	32, 33	syl6bb 276	. . . . 5 |- ( ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> ( ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )
35	34	pm5.32da 673	. . . 4 |- ( ( # ` W ) = 2 -> ( ( W e. Word (Vtx ` G ) /\ ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) ) <-> ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) ) )
36	6, 35	syl5bb 272	. . 3 |- ( ( # ` W ) = 2 -> ( ( W e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) <-> ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) ) )
37	36	pm5.32ri 670	. 2 |- ( ( ( W e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ ( # ` W ) = 2 ) <-> ( ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) /\ ( # ` W ) = 2 ) )
38		3anass 1042	. . 3 |- ( ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) <-> ( ( # ` W ) = 2 /\ ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) ) )
39		ancom 466	. . 3 |- ( ( ( # ` W ) = 2 /\ ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) ) <-> ( ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) /\ ( # ` W ) = 2 ) )
40	38, 39	bitr2i 265	. 2 |- ( ( ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) /\ ( # ` W ) = 2 ) <-> ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )
41	5, 37, 40	3bitri 286	1 |- ( W e. ( 2 ClWWalksN G ) <-> ( ( # ` W ) = 2 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   2c2 11070  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292  Vtxcvtx 25874  Edgcedg 25939   ClWWalksN cclwwlksn 26876

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-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-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-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-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  numclwwlkovf2  27217