Theorem clwwlkn2 16542

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
clwwlkn2	|- ( W e. ( 2 ClWWalksN G ) <-> ( (♯ ` W ) = 2 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )

Proof of Theorem clwwlkn2

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn 9416	. . 3 |- 2 e. NN
2		eqid 2234	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2234	. . . 4 |- (Edg ` G ) = (Edg ` G )
4	2, 3	isclwwlknx 16537	. . 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 1009	. . . 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 6065	. . . . . . . . . . . . 13 |- ( (♯ ` W ) = 2 -> ( (♯ ` W ) - 1 ) = ( 2 - 1 ) )
8		2m1e1 9372	. . . . . . . . . . . . 13 |- ( 2 - 1 ) = 1
9	7, 8	eqtrdi 2283	. . . . . . . . . . . 12 |- ( (♯ ` W ) = 2 -> ( (♯ ` W ) - 1 ) = 1 )
10	9	oveq2d 6074	. . . . . . . . . . 11 |- ( (♯ ` W ) = 2 -> ( 0..^ ( (♯ ` W ) - 1 ) ) = ( 0..^ 1 ) )
11		fzo01 10583	. . . . . . . . . . 11 |- ( 0..^ 1 ) = { 0 }
12	10, 11	eqtrdi 2283	. . . . . . . . . 10 |- ( (♯ ` W ) = 2 -> ( 0..^ ( (♯ ` W ) - 1 ) ) = { 0 } )
13	12	adantr 276	. . . . . . . . 9 |- ( ( (♯ ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> ( 0..^ ( (♯ ` W ) - 1 ) ) = { 0 } )
14	13	raleqdv 2749	. . . . . . . 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 8284	. . . . . . . . 9 |- 0 e. _V
16		fveq2 5675	. . . . . . . . . . 11 |- ( i = 0 -> ( W ` i ) = ( W ` 0 ) )
17		fv0p1e1 9369	. . . . . . . . . . 11 |- ( i = 0 -> ( W ` ( i + 1 ) ) = ( W ` 1 ) )
18	16, 17	preq12d 3781	. . . . . . . . . 10 |- ( i = 0 -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( W ` 0 ) , ( W ` 1 ) } )
19	18	eleq1d 2303	. . . . . . . . 9 |- ( i = 0 -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )
20	15, 19	ralsn 3737	. . . . . . . 8 |- ( A. i e. { 0 } { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) )
21	14, 20	bitrdi 196	. . . . . . 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 ) ) )
22		prcom 3772	. . . . . . . . 9 |- { (lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) , (lastS ` W ) }
23		lswwrd 11296	. . . . . . . . . . 11 |- ( W e. Word (Vtx ` G ) -> (lastS ` W ) = ( W ` ( (♯ ` W ) - 1 ) ) )
24	9	fveq2d 5679	. . . . . . . . . . 11 |- ( (♯ ` W ) = 2 -> ( W ` ( (♯ ` W ) - 1 ) ) = ( W ` 1 ) )
25	23, 24	sylan9eqr 2289	. . . . . . . . . 10 |- ( ( (♯ ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> (lastS ` W ) = ( W ` 1 ) )
26	25	preq2d 3780	. . . . . . . . 9 |- ( ( (♯ ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> { ( W ` 0 ) , (lastS ` W ) } = { ( W ` 0 ) , ( W ` 1 ) } )
27	22, 26	eqtrid 2279	. . . . . . . 8 |- ( ( (♯ ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> { (lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) , ( W ` 1 ) } )
28	27	eleq1d 2303	. . . . . . 7 |- ( ( (♯ ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> ( { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )
29	21, 28	anbi12d 473	. . . . . 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 ) ) ) )
30		anidm 396	. . . . . 6 |- ( ( { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) )
31	29, 30	bitrdi 196	. . . . 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 ) ) )
32	31	pm5.32da 452	. . . 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 ) ) ) )
33	6, 32	bitrid 192	. . 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 ) ) ) )
34	33	pm5.32ri 455	. 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 ) )
35		3anass 1009	. . 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 ) ) ) )
36		ancom 266	. . 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 ) )
37	35, 36	bitr2i 185	. 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 ) ) )
38	5, 34, 37	3bitri 206	1 |- ( W e. ( 2 ClWWalksN G ) <-> ( (♯ ` W ) = 2 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   {csn 3694   {cpr 3695   ` cfv 5357  (class class class)co 6058   0cc0 8143   1c1 8144   + caddc 8146   - cmin 8460   NNcn 9254   2c2 9305  ..^cfzo 10498  ♯chash 11163  Word cword 11249  lastSclsw 11294  Vtxcvtx 16133  Edgcedg 16178   ClWWalksN cclwwlkn 16524

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-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-n0 9514  df-z 9595  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-vtx 16135  df-clwwlk 16513  df-clwwlkn 16525

This theorem is referenced by:  clwwlknon2x  16556