Theorem clwwlkn2 25582

Description: In an undirected simple graph, a closed walk of length 2 represented as word is a word consisting of 2 symbols representing vertices connected by an edge. (Contributed by Alexander van der Vekens, 19-Sep-2018.)

Assertion

Ref	Expression
clwwlkn2	|- ( V USGrph E -> ( W e. ( ( V ClWWalksN E ) ` 2 ) <-> ( ( # ` W ) = 2 /\ W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) ) )

Proof of Theorem clwwlkn2

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrav 25144	. . 3 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
2		2nn0 10910	. . . . 5 |- 2 e. NN0
3		isclwwlkn 25576	. . . . 5 |- ( ( V e. _V /\ E e. _V /\ 2 e. NN0 ) -> ( W e. ( ( V ClWWalksN E ) ` 2 ) <-> ( W e. ( V ClWWalks E ) /\ ( # ` W ) = 2 ) ) )
4	2, 3	mp3an3 1379	. . . 4 |- ( ( V e. _V /\ E e. _V ) -> ( W e. ( ( V ClWWalksN E ) ` 2 ) <-> ( W e. ( V ClWWalks E ) /\ ( # ` W ) = 2 ) ) )
5		isclwwlk 25575	. . . . 5 |- ( ( V e. _V /\ E e. _V ) -> ( W e. ( V ClWWalks E ) <-> ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) ) )
6	5	anbi1d 719	. . . 4 |- ( ( V e. _V /\ E e. _V ) -> ( ( W e. ( V ClWWalks E ) /\ ( # ` W ) = 2 ) <-> ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) /\ ( # ` W ) = 2 ) ) )
7	4, 6	bitrd 261	. . 3 |- ( ( V e. _V /\ E e. _V ) -> ( W e. ( ( V ClWWalksN E ) ` 2 ) <-> ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) /\ ( # ` W ) = 2 ) ) )
8	1, 7	syl 17	. 2 |- ( V USGrph E -> ( W e. ( ( V ClWWalksN E ) ` 2 ) <-> ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) /\ ( # ` W ) = 2 ) ) )
9		3anass 1011	. . . . 5 |- ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) <-> ( W e. Word V /\ ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) ) )
10		oveq1 6315	. . . . . . . . . . . 12 |- ( ( # ` W ) = 2 -> ( ( # ` W ) - 1 ) = ( 2 - 1 ) )
11	10	oveq2d 6324	. . . . . . . . . . 11 |- ( ( # ` W ) = 2 -> ( 0..^ ( ( # ` W ) - 1 ) ) = ( 0..^ ( 2 - 1 ) ) )
12	11	raleqdv 2979	. . . . . . . . . 10 |- ( ( # ` W ) = 2 -> ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E <-> A. i e. ( 0..^ ( 2 - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E ) )
13	12	ad2antlr 741	. . . . . . . . 9 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E <-> A. i e. ( 0..^ ( 2 - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E ) )
14		2m1e1 10746	. . . . . . . . . . . . 13 |- ( 2 - 1 ) = 1
15	14	oveq2i 6319	. . . . . . . . . . . 12 |- ( 0..^ ( 2 - 1 ) ) = ( 0..^ 1 )
16		fzo01 12024	. . . . . . . . . . . 12 |- ( 0..^ 1 ) = { 0 }
17	15, 16	eqtri 2493	. . . . . . . . . . 11 |- ( 0..^ ( 2 - 1 ) ) = { 0 }
18	17	a1i 11	. . . . . . . . . 10 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( 0..^ ( 2 - 1 ) ) = { 0 } )
19	18	raleqdv 2979	. . . . . . . . 9 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( A. i e. ( 0..^ ( 2 - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E <-> A. i e. { 0 } { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E ) )
20		c0ex 9655	. . . . . . . . . 10 |- 0 e. _V
21		fveq2 5879	. . . . . . . . . . . . 13 |- ( i = 0 -> ( W ` i ) = ( W ` 0 ) )
22		oveq1 6315	. . . . . . . . . . . . . . 15 |- ( i = 0 -> ( i + 1 ) = ( 0 + 1 ) )
23		0p1e1 10743	. . . . . . . . . . . . . . 15 |- ( 0 + 1 ) = 1
24	22, 23	syl6eq 2521	. . . . . . . . . . . . . 14 |- ( i = 0 -> ( i + 1 ) = 1 )
25	24	fveq2d 5883	. . . . . . . . . . . . 13 |- ( i = 0 -> ( W ` ( i + 1 ) ) = ( W ` 1 ) )
26	21, 25	preq12d 4050	. . . . . . . . . . . 12 |- ( i = 0 -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( W ` 0 ) , ( W ` 1 ) } )
27	26	eleq1d 2533	. . . . . . . . . . 11 |- ( i = 0 -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E <-> { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) )
28	27	ralsng 3997	. . . . . . . . . 10 |- ( 0 e. _V -> ( A. i e. { 0 } { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E <-> { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) )
29	20, 28	mp1i 13	. . . . . . . . 9 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( A. i e. { 0 } { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E <-> { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) )
30	13, 19, 29	3bitrd 287	. . . . . . . 8 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E <-> { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) )
31	30	anbi1d 719	. . . . . . 7 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) <-> ( { ( W ` 0 ) , ( W ` 1 ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) ) )
32		lsw 12762	. . . . . . . . . . . . . 14 |- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
33	32	adantl 473	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
34	10	ad2antlr 741	. . . . . . . . . . . . . . 15 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( ( # ` W ) - 1 ) = ( 2 - 1 ) )
35	34, 14	syl6eq 2521	. . . . . . . . . . . . . 14 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( ( # ` W ) - 1 ) = 1 )
36	35	fveq2d 5883	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` 1 ) )
37	33, 36	eqtr2d 2506	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( W ` 1 ) = ( lastS ` W ) )
38	37	preq2d 4049	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> { ( W ` 0 ) , ( W ` 1 ) } = { ( W ` 0 ) , ( lastS ` W ) } )
39		prcom 4041	. . . . . . . . . . 11 |- { ( W ` 0 ) , ( lastS ` W ) } = { ( lastS ` W ) , ( W ` 0 ) }
40	38, 39	syl6eq 2521	. . . . . . . . . 10 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> { ( W ` 0 ) , ( W ` 1 ) } = { ( lastS ` W ) , ( W ` 0 ) } )
41	40	eleq1d 2533	. . . . . . . . 9 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( { ( W ` 0 ) , ( W ` 1 ) } e. ran E <-> { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) )
42	41	biimpd 212	. . . . . . . 8 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( { ( W ` 0 ) , ( W ` 1 ) } e. ran E -> { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) )
43	42	pm4.71d 646	. . . . . . 7 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( { ( W ` 0 ) , ( W ` 1 ) } e. ran E <-> ( { ( W ` 0 ) , ( W ` 1 ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) ) )
44	31, 43	bitr4d 264	. . . . . 6 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) )
45	44	pm5.32da 653	. . . . 5 |- ( ( V USGrph E /\ ( # ` W ) = 2 ) -> ( ( W e. Word V /\ ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) ) <-> ( W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) ) )
46	9, 45	syl5bb 265	. . . 4 |- ( ( V USGrph E /\ ( # ` W ) = 2 ) -> ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) <-> ( W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) ) )
47	46	ex 441	. . 3 |- ( V USGrph E -> ( ( # ` W ) = 2 -> ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) <-> ( W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) ) ) )
48	47	pm5.32rd 652	. 2 |- ( V USGrph E -> ( ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` W ) , ( W ` 0 ) } e. ran E ) /\ ( # ` W ) = 2 ) <-> ( ( W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) /\ ( # ` W ) = 2 ) ) )
49		3anass 1011	. . . 4 |- ( ( ( # ` W ) = 2 /\ W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) <-> ( ( # ` W ) = 2 /\ ( W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) ) )
50		ancom 457	. . . 4 |- ( ( ( # ` W ) = 2 /\ ( W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) ) <-> ( ( W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) /\ ( # ` W ) = 2 ) )
51	49, 50	bitr2i 258	. . 3 |- ( ( ( W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) /\ ( # ` W ) = 2 ) <-> ( ( # ` W ) = 2 /\ W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) )
52	51	a1i 11	. 2 |- ( V USGrph E -> ( ( ( W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) /\ ( # ` W ) = 2 ) <-> ( ( # ` W ) = 2 /\ W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) ) )
53	8, 48, 52	3bitrd 287	1 |- ( V USGrph E -> ( W e. ( ( V ClWWalksN E ) ` 2 ) <-> ( ( # ` W ) = 2 /\ W e. Word V /\ { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   _Vcvv 3031   {csn 3959   {cpr 3961   class class class wbr 4395   ran crn 4840   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558   + caddc 9560   - cmin 9880   2c2 10681   NN0cn0 10893  ..^cfzo 11942   #chash 12553  Word cword 12703   lastS clsw 12704   USGrph cusg 25136   ClWWalks cclwwlk 25555   ClWWalksN cclwwlkn 25556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-lsw 12712  df-usgra 25139  df-clwwlk 25558  df-clwwlkn 25559

This theorem is referenced by:  numclwwlkovf2  25891