Theorem clwwlkn2 25496

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 25058	. . 3 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
2		2nn0 10883	. . . . 5 |- 2 e. NN0
3		isclwwlkn 25490	. . . . 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 1352	. . . 4 |- ( ( V e. _V /\ E e. _V ) -> ( W e. ( ( V ClWWalksN E ) ` 2 ) <-> ( W e. ( V ClWWalks E ) /\ ( # ` W ) = 2 ) ) )
5		isclwwlk 25489	. . . . 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 710	. . . 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 257	. . 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 988	. . . . 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 6295	. . . . . . . . . . . 12 |- ( ( # ` W ) = 2 -> ( ( # ` W ) - 1 ) = ( 2 - 1 ) )
11	10	oveq2d 6304	. . . . . . . . . . 11 |- ( ( # ` W ) = 2 -> ( 0..^ ( ( # ` W ) - 1 ) ) = ( 0..^ ( 2 - 1 ) ) )
12	11	raleqdv 2992	. . . . . . . . . 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 732	. . . . . . . . 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 10721	. . . . . . . . . . . . 13 |- ( 2 - 1 ) = 1
15	14	oveq2i 6299	. . . . . . . . . . . 12 |- ( 0..^ ( 2 - 1 ) ) = ( 0..^ 1 )
16		fzo01 11992	. . . . . . . . . . . 12 |- ( 0..^ 1 ) = { 0 }
17	15, 16	eqtri 2472	. . . . . . . . . . 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 2992	. . . . . . . . 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 9634	. . . . . . . . . 10 |- 0 e. _V
21		fveq2 5863	. . . . . . . . . . . . 13 |- ( i = 0 -> ( W ` i ) = ( W ` 0 ) )
22		oveq1 6295	. . . . . . . . . . . . . . 15 |- ( i = 0 -> ( i + 1 ) = ( 0 + 1 ) )
23		0p1e1 10718	. . . . . . . . . . . . . . 15 |- ( 0 + 1 ) = 1
24	22, 23	syl6eq 2500	. . . . . . . . . . . . . 14 |- ( i = 0 -> ( i + 1 ) = 1 )
25	24	fveq2d 5867	. . . . . . . . . . . . 13 |- ( i = 0 -> ( W ` ( i + 1 ) ) = ( W ` 1 ) )
26	21, 25	preq12d 4058	. . . . . . . . . . . 12 |- ( i = 0 -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( W ` 0 ) , ( W ` 1 ) } )
27	26	eleq1d 2512	. . . . . . . . . . 11 |- ( i = 0 -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ran E <-> { ( W ` 0 ) , ( W ` 1 ) } e. ran E ) )
28	27	ralsng 4005	. . . . . . . . . 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 283	. . . . . . . 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 710	. . . . . . 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 12708	. . . . . . . . . . . . . 14 |- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
33	32	adantl 468	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
34	10	ad2antlr 732	. . . . . . . . . . . . . . 15 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( ( # ` W ) - 1 ) = ( 2 - 1 ) )
35	34, 14	syl6eq 2500	. . . . . . . . . . . . . 14 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( ( # ` W ) - 1 ) = 1 )
36	35	fveq2d 5867	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` 1 ) )
37	33, 36	eqtr2d 2485	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> ( W ` 1 ) = ( lastS ` W ) )
38	37	preq2d 4057	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> { ( W ` 0 ) , ( W ` 1 ) } = { ( W ` 0 ) , ( lastS ` W ) } )
39		prcom 4049	. . . . . . . . . . 11 |- { ( W ` 0 ) , ( lastS ` W ) } = { ( lastS ` W ) , ( W ` 0 ) }
40	38, 39	syl6eq 2500	. . . . . . . . . 10 |- ( ( ( V USGrph E /\ ( # ` W ) = 2 ) /\ W e. Word V ) -> { ( W ` 0 ) , ( W ` 1 ) } = { ( lastS ` W ) , ( W ` 0 ) } )
41	40	eleq1d 2512	. . . . . . . . 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 211	. . . . . . . 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 639	. . . . . . 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 260	. . . . . 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 646	. . . . 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 261	. . . 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 436	. . 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 645	. 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 988	. . . 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 452	. . . 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 254	. . 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 283	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 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   A.wral 2736   _Vcvv 3044   {csn 3967   {cpr 3969   class class class wbr 4401   ran crn 4834   ` cfv 5581  (class class class)co 6288   0cc0 9536   1c1 9537   + caddc 9539   - cmin 9857   2c2 10656   NN0cn0 10866  ..^cfzo 11912   #chash 12512  Word cword 12653   lastS clsw 12654   USGrph cusg 25050   ClWWalks cclwwlk 25469   ClWWalksN cclwwlkn 25470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-fzo 11913  df-hash 12513  df-word 12661  df-lsw 12662  df-usgra 25053  df-clwwlk 25472  df-clwwlkn 25473

This theorem is referenced by:  numclwwlkovf2  25805