Theorem isclwwlknx 16266

Description: Characterization of a word representing a closed walk of a fixed length, definition of ClWWalks expanded. (Contributed by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)

Hypotheses

Ref	Expression
isclwwlknx.v	|- V = (Vtx ` G )
isclwwlknx.e	|- E = (Edg ` G )

Assertion

Ref	Expression
isclwwlknx	|- ( N e. NN -> ( W e. ( N ClWWalksN G ) <-> ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ (♯ ` W ) = N ) ) )

Distinct variable groups:   i, G   i, W

Allowed substitution hints:   E( i)   N( i)   V( i)

Proof of Theorem isclwwlknx

Step	Hyp	Ref	Expression
1		eleq1 2294	. . . . . . . . . 10 |- ( (♯ ` W ) = N -> ( (♯ ` W ) e. NN <-> N e. NN ) )
2		len0nnbi 11147	. . . . . . . . . . 11 |- ( W e. Word V -> ( W =/= (/) <-> (♯ ` W ) e. NN ) )
3	2	biimprcd 160	. . . . . . . . . 10 |- ( (♯ ` W ) e. NN -> ( W e. Word V -> W =/= (/) ) )
4	1, 3	biimtrrdi 164	. . . . . . . . 9 |- ( (♯ ` W ) = N -> ( N e. NN -> ( W e. Word V -> W =/= (/) ) ) )
5	4	impcom 125	. . . . . . . 8 |- ( ( N e. NN /\ (♯ ` W ) = N ) -> ( W e. Word V -> W =/= (/) ) )
6	5	imp 124	. . . . . . 7 |- ( ( ( N e. NN /\ (♯ ` W ) = N ) /\ W e. Word V ) -> W =/= (/) )
7	6	biantrurd 305	. . . . . 6 |- ( ( ( N e. NN /\ (♯ ` W ) = N ) /\ W e. Word V ) -> ( ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) <-> ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) )
8	7	bicomd 141	. . . . 5 |- ( ( ( N e. NN /\ (♯ ` W ) = N ) /\ W e. Word V ) -> ( ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) <-> ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) )
9	8	pm5.32da 452	. . . 4 |- ( ( N e. NN /\ (♯ ` W ) = N ) -> ( ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) <-> ( W e. Word V /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) )
10	9	ex 115	. . 3 |- ( N e. NN -> ( (♯ ` W ) = N -> ( ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) <-> ( W e. Word V /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) ) )
11	10	pm5.32rd 451	. 2 |- ( N e. NN -> ( ( ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) /\ (♯ ` W ) = N ) <-> ( ( W e. Word V /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) /\ (♯ ` W ) = N ) ) )
12		nnnn0 9408	. . . 4 |- ( N e. NN -> N e. NN0 )
13		isclwwlkng 16256	. . . 4 |- ( N e. NN0 -> ( W e. ( N ClWWalksN G ) <-> ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) ) )
14	12, 13	syl 14	. . 3 |- ( N e. NN -> ( W e. ( N ClWWalksN G ) <-> ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) ) )
15		isclwwlknx.v	. . . . . 6 |- V = (Vtx ` G )
16		isclwwlknx.e	. . . . . 6 |- E = (Edg ` G )
17	15, 16	isclwwlk 16244	. . . . 5 |- ( W e. (ClWWalks ` G ) <-> ( ( W e. Word V /\ W =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) )
18		3anass 1008	. . . . 5 |- ( ( ( W e. Word V /\ W =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) <-> ( ( W e. Word V /\ W =/= (/) ) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) )
19		anass 401	. . . . 5 |- ( ( ( W e. Word V /\ W =/= (/) ) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) <-> ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) )
20	17, 18, 19	3bitri 206	. . . 4 |- ( W e. (ClWWalks ` G ) <-> ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) )
21	20	anbi1i 458	. . 3 |- ( ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) <-> ( ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) /\ (♯ ` W ) = N ) )
22	14, 21	bitrdi 196	. 2 |- ( N e. NN -> ( W e. ( N ClWWalksN G ) <-> ( ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) /\ (♯ ` W ) = N ) ) )
23		3anass 1008	. . . 4 |- ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) <-> ( W e. Word V /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) )
24	23	anbi1i 458	. . 3 |- ( ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ (♯ ` W ) = N ) <-> ( ( W e. Word V /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) /\ (♯ ` W ) = N ) )
25	24	a1i 9	. 2 |- ( N e. NN -> ( ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ (♯ ` W ) = N ) <-> ( ( W e. Word V /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) /\ (♯ ` W ) = N ) ) )
26	11, 22, 25	3bitr4d 220	1 |- ( N e. NN -> ( W e. ( N ClWWalksN G ) <-> ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ (♯ ` W ) = N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   A.wral 2510   (/)c0 3494   {cpr 3670   ` cfv 5326  (class class class)co 6017   0cc0 8031   1c1 8032   + caddc 8034   - cmin 8349   NNcn 9142   NN0cn0 9401  ..^cfzo 10376  ♯chash 11036  Word cword 11112  lastSclsw 11157  Vtxcvtx 15862  Edgcedg 15907  ClWWalkscclwwlk 16241   ClWWalksN cclwwlkn 16253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-map 6818  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-ndx 13084  df-slot 13085  df-base 13087  df-vtx 15864  df-clwwlk 16242  df-clwwlkn 16254

This theorem is referenced by:  clwwlknp  16267  clwwlkn1  16268  clwwlkn2  16271  clwwlkext2edg  16272  clwwlknonex2  16289