Theorem clwwlkn2 30606

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 23442	. . 3 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
2		2nn0 10710	. . . . 5 |- 2 e. NN0
3		isclwwlkn 30600	. . . . 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 1304	. . . 4 |- ( ( V e. _V /\ E e. _V ) -> ( W e. ( ( V ClWWalksN E ) ` 2 ) <-> ( W e. ( V ClWWalks E ) /\ ( # ` W ) = 2 ) ) )
5		isclwwlk 30599	. . . . 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 969	. . . . 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 6210	. . . . . . . . . . . 12 |- ( ( # ` W ) = 2 -> ( ( # ` W ) - 1 ) = ( 2 - 1 ) )
11	10	oveq2d 6219	. . . . . . . . . . 11 |- ( ( # ` W ) = 2 -> ( 0..^ ( ( # ` W ) - 1 ) ) = ( 0..^ ( 2 - 1 ) ) )
12	11	raleqdv 3029	. . . . . . . . . 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 10550	. . . . . . . . . . . . 13 |- ( 2 - 1 ) = 1
15	14	oveq2i 6214	. . . . . . . . . . . 12 |- ( 0..^ ( 2 - 1 ) ) = ( 0..^ 1 )
16		fzo01 11732	. . . . . . . . . . . 12 |- ( 0..^ 1 ) = { 0 }
17	15, 16	eqtri 2483	. . . . . . . . . . 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 3029	. . . . . . . . 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 9494	. . . . . . . . . 10 |- 0 e. _V
21		fveq2 5802	. . . . . . . . . . . . 13 |- ( i = 0 -> ( W ` i ) = ( W ` 0 ) )
22		oveq1 6210	. . . . . . . . . . . . . . 15 |- ( i = 0 -> ( i + 1 ) = ( 0 + 1 ) )
23		0p1e1 10547	. . . . . . . . . . . . . . 15 |- ( 0 + 1 ) = 1
24	22, 23	syl6eq 2511	. . . . . . . . . . . . . 14 |- ( i = 0 -> ( i + 1 ) = 1 )
25	24	fveq2d 5806	. . . . . . . . . . . . 13 |- ( i = 0 -> ( W ` ( i + 1 ) ) = ( W ` 1 ) )
26	21, 25	preq12d 4073	. . . . . . . . . . . 12 |- ( i = 0 -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( W ` 0 ) , ( W ` 1 ) } )
27	26	eleq1d 2523	. . . . . . . . . . 11 |- ( i = 0 -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E <-> { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) )
28	27	ralsng 4023	. . . . . . . . . 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 12387	. . . . . . . . . . . . . 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 2511	. . . . . . . . . . . . . 14 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( ( # ` W ) - 1 ) = 1 )
36	35	fveq2d 5806	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` 1 ) )
37	33, 36	eqtr2d 2496	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( W ` 1 ) = ( lastS ` W ) )
38	37	preq2d 4072	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> { ( W ` 0 ) , ( W ` 1 ) } = { ( W ` 0 ) , ( lastS ` W ) } )
39		prcom 4064	. . . . . . . . . . 11 |- { ( W ` 0 ) , ( lastS ` W ) } = { ( lastS ` W ) , ( W ` 0 ) }
40	38, 39	syl6eq 2511	. . . . . . . . . 10 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> { ( W ` 0 ) , ( W ` 1 ) } = { ( lastS ` W ) , ( W ` 0 ) } )
41	40	eleq1d 2523	. . . . . . . . 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 969	. . . 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 965   = wceq 1370   e. wcel 1758   A.wral 2799   _Vcvv 3078   {csn 3988   {cpr 3990   class class class wbr 4403   ran crn 4952   ` cfv 5529  (class class class)co 6203   0cc0 9396   1c1 9397   + caddc 9399   - cmin 9709   2c2 10485   NN0cn0 10693  ..^cfzo 11668   #chash 12223  Word cword 12342   lastS clsw 12343   USGrph cusg 23436   ClWWalks cclwwlk 30581   ClWWalksN cclwwlkn 30582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-word 12350  df-lsw 12351  df-usgra 23438  df-clwwlk 30584  df-clwwlkn 30585

This theorem is referenced by:  numclwwlkovf2  30845