Theorem isclwwlk 16515

Description: Properties of a word to represent a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)

Hypotheses

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

Assertion

Ref	Expression
isclwwlk	|- ( 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 ) )

Distinct variable groups:   i, G   i, W

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

Proof of Theorem isclwwlk

Dummy variables g w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clwwlk 16513	. . . 4 |- ClWWalks = ( g e. _V |-> { w e. Word (Vtx ` g ) | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) /\ { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) ) } )
2	1	mptrcl 5765	. . 3 |- ( W e. (ClWWalks ` G ) -> G e. _V )
3		fstwrdne 11288	. . . . 5 |- ( ( W e. Word V /\ W =/= (/) ) -> ( W ` 0 ) e. V )
4		clwwlk.v	. . . . . 6 |- V = (Vtx ` G )
5	4	1vgrex 16141	. . . . 5 |- ( ( W ` 0 ) e. V -> G e. _V )
6	3, 5	syl 14	. . . 4 |- ( ( W e. Word V /\ W =/= (/) ) -> G e. _V )
7	6	3ad2antr1 1189	. . 3 |- ( ( W e. Word V /\ ( W =/= (/) /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) -> G e. _V )
8		clwwlk.e	. . . . . 6 |- E = (Edg ` G )
9	4, 8	clwwlkg 16514	. . . . 5 |- ( G e. _V -> (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 ) } )
10	9	eleq2d 2304	. . . 4 |- ( G e. _V -> ( W e. (ClWWalks ` G ) <-> W 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 ) } ) )
11		neeq1 2427	. . . . . 6 |- ( w = W -> ( w =/= (/) <-> W =/= (/) ) )
12		fveq2 5675	. . . . . . . . 9 |- ( w = W -> (♯ ` w ) = (♯ ` W ) )
13	12	oveq1d 6073	. . . . . . . 8 |- ( w = W -> ( (♯ ` w ) - 1 ) = ( (♯ ` W ) - 1 ) )
14	13	oveq2d 6074	. . . . . . 7 |- ( w = W -> ( 0..^ ( (♯ ` w ) - 1 ) ) = ( 0..^ ( (♯ ` W ) - 1 ) ) )
15		fveq1 5674	. . . . . . . . 9 |- ( w = W -> ( w ` i ) = ( W ` i ) )
16		fveq1 5674	. . . . . . . . 9 |- ( w = W -> ( w ` ( i + 1 ) ) = ( W ` ( i + 1 ) ) )
17	15, 16	preq12d 3781	. . . . . . . 8 |- ( w = W -> { ( w ` i ) , ( w ` ( i + 1 ) ) } = { ( W ` i ) , ( W ` ( i + 1 ) ) } )
18	17	eleq1d 2303	. . . . . . 7 |- ( w = W -> ( { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E <-> { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) )
19	14, 18	raleqbidv 2759	. . . . . 6 |- ( w = W -> ( A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E <-> A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) )
20		fveq2 5675	. . . . . . . 8 |- ( w = W -> (lastS ` w ) = (lastS ` W ) )
21		fveq1 5674	. . . . . . . 8 |- ( w = W -> ( w ` 0 ) = ( W ` 0 ) )
22	20, 21	preq12d 3781	. . . . . . 7 |- ( w = W -> { (lastS ` w ) , ( w ` 0 ) } = { (lastS ` W ) , ( W ` 0 ) } )
23	22	eleq1d 2303	. . . . . 6 |- ( w = W -> ( { (lastS ` w ) , ( w ` 0 ) } e. E <-> { (lastS ` W ) , ( W ` 0 ) } e. E ) )
24	11, 19, 23	3anbi123d 1349	. . . . 5 |- ( w = W -> ( ( w =/= (/) /\ 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 ) ) )
25	24	elrab 2976	. . . 4 |- ( W 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 ) } <-> ( W e. Word V /\ ( W =/= (/) /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) )
26	10, 25	bitrdi 196	. . 3 |- ( G e. _V -> ( 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 ) ) ) )
27	2, 7, 26	pm5.21nii 712	. 2 |- ( 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 ) ) )
28		3anass 1009	. . 3 |- ( ( ( 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 ) ) )
29		anass 401	. . 3 |- ( ( ( 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 ) ) ) )
30		3anass 1009	. . . . 5 |- ( ( W =/= (/) /\ 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 ) ) )
31	30	bicomi 132	. . . 4 |- ( ( W =/= (/) /\ ( 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 ) )
32	31	anbi2i 457	. . 3 |- ( ( 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 ) ) )
33	28, 29, 32	3bitri 206	. 2 |- ( ( ( 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 ) ) )
34	27, 33	bitr4i 187	1 |- ( 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 ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   {crab 2526   _Vcvv 2815   (/)c0 3512   {cpr 3695   ` cfv 5357  (class class class)co 6058   0cc0 8143   1c1 8144   + caddc 8146   - cmin 8460  ..^cfzo 10498  ♯chash 11163  Word cword 11249  lastSclsw 11294  Vtxcvtx 16133  Edgcedg 16178  ClWWalkscclwwlk 16512

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-ndx 13299  df-slot 13300  df-base 13302  df-vtx 16135  df-clwwlk 16513

This theorem is referenced by:  clwwlkbp  16516  clwwlksswrd  16518  clwwlk1loop  16520  clwwlkccat  16522  isclwwlknx  16537  clwwlknonel  16553