Theorem clwwlknonel 16217

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 16179	. . . . . 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 5633	. . . . . . . . . . . . . . 15 |- ( W = (/) -> (♯ ` W ) = (♯ ` (/) ) )
6		hash0 11046	. . . . . . . . . . . . . . 15 |- (♯ ` (/) ) = 0
7	5, 6	eqtrdi 2278	. . . . . . . . . . . . . 14 |- ( W = (/) -> (♯ ` W ) = 0 )
8	7	adantl 277	. . . . . . . . . . . . 13 |- ( ( (♯ ` W ) = N /\ W = (/) ) -> (♯ ` W ) = 0 )
9	4, 8	eqtr3d 2264	. . . . . . . . . . . 12 |- ( ( (♯ ` W ) = N /\ W = (/) ) -> N = 0 )
10	9	ex 115	. . . . . . . . . . 11 |- ( (♯ ` W ) = N -> ( W = (/) -> N = 0 ) )
11	10	necon3d 2444	. . . . . . . . . 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 1350	. . . . . 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 16215	. . 3 |- ( W e. ( X (ClWWalksNOn ` G ) N ) <-> ( W e. ( N ClWWalksN G ) /\ ( W ` 0 ) = X ) )
21		isclwwlkn 16198	. . . 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 1006	. 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 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   (/)c0 3492   {cpr 3668   ` cfv 5322  (class class class)co 6011   0cc0 8020   1c1 8021   + caddc 8023   - cmin 8338  ..^cfzo 10365  ♯chash 11025  Word cword 11100  lastSclsw 11145  Vtxcvtx 15850  Edgcedg 15895  ClWWalkscclwwlk 16176   ClWWalksN cclwwlkn 16188  ClWWalksNOncclwwlknon 16211

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-mulrcl 8119  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-precex 8130  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136  ax-pre-mulgt0 8137

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-frec 6550  df-1o 6575  df-er 6695  df-map 6812  df-en 6903  df-dom 6904  df-fin 6905  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-reap 8743  df-ap 8750  df-inn 9132  df-n0 9391  df-z 9468  df-uz 9744  df-fz 10232  df-fzo 10366  df-ihash 11026  df-word 11101  df-ndx 13072  df-slot 13073  df-base 13075  df-vtx 15852  df-clwwlk 16177  df-clwwlkn 16189  df-clwwlknon 16212

This theorem is referenced by:  clwwlknonex2  16224