Theorem clwwlknonel 16302

Description: Characterization of a word over the set of vertices representing a closed walk on vertex X of (nonzero) length N in a graph G. This theorem would not hold for N = 0 if W = X = (/). (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 24-Mar-2022.)

Hypotheses

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

Assertion

Ref	Expression
clwwlknonel	|- ( N =/= 0 -> ( W e. ( X (ClWWalksNOn ` G ) 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 /\ ( W ` 0 ) = X ) ) )

Distinct variable groups:   i, G   i, W

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

Proof of Theorem clwwlknonel

Step	Hyp	Ref	Expression
1		clwwlknonel.v	. . . . . . 7 |- V = (Vtx ` G )
2		clwwlknonel.e	. . . . . . 7 |- E = (Edg ` G )
3	1, 2	isclwwlk 16264	. . . . . 6 |- ( 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 ) )
4		simpl 109	. . . . . . . . . . . . 13 |- ( ( (♯ ` W ) = N /\ W = (/) ) -> (♯ ` W ) = N )
5		fveq2 5639	. . . . . . . . . . . . . . 15 |- ( W = (/) -> (♯ ` W ) = (♯ ` (/) ) )
6		hash0 11059	. . . . . . . . . . . . . . 15 |- (♯ ` (/) ) = 0
7	5, 6	eqtrdi 2280	. . . . . . . . . . . . . 14 |- ( W = (/) -> (♯ ` W ) = 0 )
8	7	adantl 277	. . . . . . . . . . . . 13 |- ( ( (♯ ` W ) = N /\ W = (/) ) -> (♯ ` W ) = 0 )
9	4, 8	eqtr3d 2266	. . . . . . . . . . . 12 |- ( ( (♯ ` W ) = N /\ W = (/) ) -> N = 0 )
10	9	ex 115	. . . . . . . . . . 11 |- ( (♯ ` W ) = N -> ( W = (/) -> N = 0 ) )
11	10	necon3d 2446	. . . . . . . . . 10 |- ( (♯ ` W ) = N -> ( N =/= 0 -> W =/= (/) ) )
12	11	impcom 125	. . . . . . . . 9 |- ( ( N =/= 0 /\ (♯ ` W ) = N ) -> W =/= (/) )
13	12	biantrud 304	. . . . . . . 8 |- ( ( N =/= 0 /\ (♯ ` W ) = N ) -> ( W e. Word V <-> ( W e. Word V /\ W =/= (/) ) ) )
14	13	bicomd 141	. . . . . . 7 |- ( ( N =/= 0 /\ (♯ ` W ) = N ) -> ( ( W e. Word V /\ W =/= (/) ) <-> W e. Word V ) )
15	14	3anbi1d 1352	. . . . . 6 |- ( ( N =/= 0 /\ (♯ ` 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 ) ) )
16	3, 15	bitrid 192	. . . . 5 |- ( ( N =/= 0 /\ (♯ ` W ) = N ) -> ( W e. (ClWWalks ` G ) <-> ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) )
17	16	a1d 22	. . . 4 |- ( ( N =/= 0 /\ (♯ ` W ) = N ) -> ( ( W ` 0 ) = X -> ( W e. (ClWWalks ` G ) <-> ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) )
18	17	expimpd 363	. . 3 |- ( N =/= 0 -> ( ( (♯ ` W ) = N /\ ( W ` 0 ) = X ) -> ( W e. (ClWWalks ` G ) <-> ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) )
19	18	pm5.32rd 451	. 2 |- ( N =/= 0 -> ( ( W e. (ClWWalks ` G ) /\ ( (♯ ` W ) = N /\ ( W ` 0 ) = X ) ) <-> ( ( 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 ` 0 ) = X ) ) ) )
20		isclwwlknon 16300	. . 3 |- ( W e. ( X (ClWWalksNOn ` G ) N ) <-> ( W e. ( N ClWWalksN G ) /\ ( W ` 0 ) = X ) )
21		isclwwlkn 16283	. . . 4 |- ( W e. ( N ClWWalksN G ) <-> ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) )
22	21	anbi1i 458	. . 3 |- ( ( W e. ( N ClWWalksN G ) /\ ( W ` 0 ) = X ) <-> ( ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) /\ ( W ` 0 ) = X ) )
23		anass 401	. . 3 |- ( ( ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) /\ ( W ` 0 ) = X ) <-> ( W e. (ClWWalks ` G ) /\ ( (♯ ` W ) = N /\ ( W ` 0 ) = X ) ) )
24	20, 22, 23	3bitri 206	. 2 |- ( W e. ( X (ClWWalksNOn ` G ) N ) <-> ( W e. (ClWWalks ` G ) /\ ( (♯ ` W ) = N /\ ( W ` 0 ) = X ) ) )
25		3anass 1008	. 2 |- ( ( ( 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 ` 0 ) = X ) <-> ( ( 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 ` 0 ) = X ) ) )
26	19, 24, 25	3bitr4g 223	1 |- ( N =/= 0 -> ( W e. ( X (ClWWalksNOn ` G ) 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 /\ ( W ` 0 ) = X ) ) )

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 6018   0cc0 8032   1c1 8033   + caddc 8035   - cmin 8350  ..^cfzo 10377  ♯chash 11038  Word cword 11117  lastSclsw 11162  Vtxcvtx 15882  Edgcedg 15927  ClWWalkscclwwlk 16261   ClWWalksN cclwwlkn 16273  ClWWalksNOncclwwlknon 16296

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 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149

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 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-map 6819  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378  df-ihash 11039  df-word 11118  df-ndx 13103  df-slot 13104  df-base 13106  df-vtx 15884  df-clwwlk 16262  df-clwwlkn 16274  df-clwwlknon 16297

This theorem is referenced by:  clwwlknonex2  16309