Theorem clwwlkn2 16867

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 9477	. . 3 |- 2 e. NN
2		eqid 2238	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2238	. . . 4 |- (Edg ` G ) = (Edg ` G )
4	2, 3	isclwwlknx 16862	. . 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 1013	. . . 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 6095	. . . . . . . . . . . . 13 |- ( (♯ ` W ) = 2 -> ( (♯ ` W ) - 1 ) = ( 2 - 1 ) )
8		2m1e1 9431	. . . . . . . . . . . . 13 |- ( 2 - 1 ) = 1
9	7, 8	eqtrdi 2287	. . . . . . . . . . . 12 |- ( (♯ ` W ) = 2 -> ( (♯ ` W ) - 1 ) = 1 )
10	9	oveq2d 6104	. . . . . . . . . . 11 |- ( (♯ ` W ) = 2 -> ( 0..^ ( (♯ ` W ) - 1 ) ) = ( 0..^ 1 ) )
11		fzo01 10651	. . . . . . . . . . 11 |- ( 0..^ 1 ) = { 0 }
12	10, 11	eqtrdi 2287	. . . . . . . . . 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 2755	. . . . . . . 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 8326	. . . . . . . . 9 |- 0 e. _V
16		fveq2 5698	. . . . . . . . . . 11 |- ( i = 0 -> ( W ` i ) = ( W ` 0 ) )
17		fv0p1e1 9428	. . . . . . . . . . 11 |- ( i = 0 -> ( W ` ( i + 1 ) ) = ( W ` 1 ) )
18	16, 17	preq12d 3796	. . . . . . . . . 10 |- ( i = 0 -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( W ` 0 ) , ( W ` 1 ) } )
19	18	eleq1d 2307	. . . . . . . . 9 |- ( i = 0 -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )
20	15, 19	ralsn 3752	. . . . . . . 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 3787	. . . . . . . . 9 |- { (lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) , (lastS ` W ) }
23		lswwrd 11373	. . . . . . . . . . 11 |- ( W e. Word (Vtx ` G ) -> (lastS ` W ) = ( W ` ( (♯ ` W ) - 1 ) ) )
24	9	fveq2d 5702	. . . . . . . . . . 11 |- ( (♯ ` W ) = 2 -> ( W ` ( (♯ ` W ) - 1 ) ) = ( W ` 1 ) )
25	23, 24	sylan9eqr 2293	. . . . . . . . . 10 |- ( ( (♯ ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> (lastS ` W ) = ( W ` 1 ) )
26	25	preq2d 3795	. . . . . . . . 9 |- ( ( (♯ ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> { ( W ` 0 ) , (lastS ` W ) } = { ( W ` 0 ) , ( W ` 1 ) } )
27	22, 26	eqtrid 2283	. . . . . . . 8 |- ( ( (♯ ` W ) = 2 /\ W e. Word (Vtx ` G ) ) -> { (lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) , ( W ` 1 ) } )
28	27	eleq1d 2307	. . . . . . 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 477	. . . . . 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 400	. . . . . 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 456	. . . 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 459	. 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 1013	. . 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 ) ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   {csn 3709   {cpr 3710   ` cfv 5380  (class class class)co 6088   0cc0 8185   1c1 8186   + caddc 8188   - cmin 8504   NNcn 9313   2c2 9364  ..^cfzo 10566  ♯chash 11236  Word cword 11326  lastSclsw 11371  Vtxcvtx 16458  Edgcedg 16503   ClWWalksN cclwwlkn 16849

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-map 6928  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-ihash 11237  df-word 11327  df-lsw 11372  df-ndx 13413  df-slot 13414  df-base 13416  df-vtx 16460  df-clwwlk 16838  df-clwwlkn 16850

This theorem is used by:  clwwlknon2x  16881