Theorem clwwlkn2 24902

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 24465	. . 3 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
2		2nn0 10833	. . . . 5 |- 2 e. NN0
3		isclwwlkn 24896	. . . . 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 1313	. . . 4 |- ( ( V e. _V /\ E e. _V ) -> ( W e. ( ( V ClWWalksN E ) ` 2 ) <-> ( W e. ( V ClWWalks E ) /\ ( # ` W ) = 2 ) ) )
5		isclwwlk 24895	. . . . 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 704	. . . 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 253	. . 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 16	. 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 977	. . . . 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 6303	. . . . . . . . . . . 12 |- ( ( # ` W ) = 2 -> ( ( # ` W ) - 1 ) = ( 2 - 1 ) )
11	10	oveq2d 6312	. . . . . . . . . . 11 |- ( ( # ` W ) = 2 -> ( 0..^ ( ( # ` W ) - 1 ) ) = ( 0..^ ( 2 - 1 ) ) )
12	11	raleqdv 3060	. . . . . . . . . 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 726	. . . . . . . . 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 10671	. . . . . . . . . . . . 13 |- ( 2 - 1 ) = 1
15	14	oveq2i 6307	. . . . . . . . . . . 12 |- ( 0..^ ( 2 - 1 ) ) = ( 0..^ 1 )
16		fzo01 11900	. . . . . . . . . . . 12 |- ( 0..^ 1 ) = { 0 }
17	15, 16	eqtri 2486	. . . . . . . . . . 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 3060	. . . . . . . . 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 9607	. . . . . . . . . 10 |- 0 e. _V
21		fveq2 5872	. . . . . . . . . . . . 13 |- ( i = 0 -> ( W ` i ) = ( W ` 0 ) )
22		oveq1 6303	. . . . . . . . . . . . . . 15 |- ( i = 0 -> ( i + 1 ) = ( 0 + 1 ) )
23		0p1e1 10668	. . . . . . . . . . . . . . 15 |- ( 0 + 1 ) = 1
24	22, 23	syl6eq 2514	. . . . . . . . . . . . . 14 |- ( i = 0 -> ( i + 1 ) = 1 )
25	24	fveq2d 5876	. . . . . . . . . . . . 13 |- ( i = 0 -> ( W ` ( i + 1 ) ) = ( W ` 1 ) )
26	21, 25	preq12d 4119	. . . . . . . . . . . 12 |- ( i = 0 -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( W ` 0 ) , ( W ` 1 ) } )
27	26	eleq1d 2526	. . . . . . . . . . 11 |- ( i = 0 -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E <-> { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) )
28	27	ralsng 4067	. . . . . . . . . 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 12	. . . . . . . . 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 279	. . . . . . . 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 704	. . . . . . 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 12593	. . . . . . . . . . . . . 14 |- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
33	32	adantl 466	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
34	10	ad2antlr 726	. . . . . . . . . . . . . . 15 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( ( # ` W ) - 1 ) = ( 2 - 1 ) )
35	34, 14	syl6eq 2514	. . . . . . . . . . . . . 14 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( ( # ` W ) - 1 ) = 1 )
36	35	fveq2d 5876	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` 1 ) )
37	33, 36	eqtr2d 2499	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( W ` 1 ) = ( lastS ` W ) )
38	37	preq2d 4118	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> { ( W ` 0 ) , ( W ` 1 ) } = { ( W ` 0 ) , ( lastS ` W ) } )
39		prcom 4110	. . . . . . . . . . 11 |- { ( W ` 0 ) , ( lastS ` W ) } = { ( lastS ` W ) , ( W ` 0 ) }
40	38, 39	syl6eq 2514	. . . . . . . . . 10 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> { ( W ` 0 ) , ( W ` 1 ) } = { ( lastS ` W ) , ( W ` 0 ) } )
41	40	eleq1d 2526	. . . . . . . . 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 207	. . . . . . . 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 634	. . . . . . 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 256	. . . . . 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 641	. . . . 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 257	. . . 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 434	. . 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 640	. 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 977	. . . 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 450	. . . 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 250	. . 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 279	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 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   A.wral 2807   _Vcvv 3109   {csn 4032   {cpr 4034   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510   + caddc 9512   - cmin 9824   2c2 10606   NN0cn0 10816  ..^cfzo 11821   #chash 12408  Word cword 12538   lastS clsw 12539   USGrph cusg 24457   ClWWalks cclwwlk 24875   ClWWalksN cclwwlkn 24876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-lsw 12547  df-usgra 24460  df-clwwlk 24878  df-clwwlkn 24879

This theorem is referenced by:  numclwwlkovf2  25211