Theorem isclwwlk 16549

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 16547	. . . 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 5782	. . 3 |- ( W e. (ClWWalks ` G ) -> G e. _V )
3		fstwrdne 11321	. . . . 5 |- ( ( W e. Word V /\ W =/= (/) ) -> ( W ` 0 ) e. V )
4		clwwlk.v	. . . . . 6 |- V = (Vtx ` G )
5	4	1vgrex 16175	. . . . 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 1193	. . 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 16548	. . . . 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 2308	. . . 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 2433	. . . . . 6 |- ( w = W -> ( w =/= (/) <-> W =/= (/) ) )
12		fveq2 5690	. . . . . . . . 9 |- ( w = W -> (♯ ` w ) = (♯ ` W ) )
13	12	oveq1d 6090	. . . . . . . 8 |- ( w = W -> ( (♯ ` w ) - 1 ) = ( (♯ ` W ) - 1 ) )
14	13	oveq2d 6091	. . . . . . 7 |- ( w = W -> ( 0..^ ( (♯ ` w ) - 1 ) ) = ( 0..^ ( (♯ ` W ) - 1 ) ) )
15		fveq1 5689	. . . . . . . . 9 |- ( w = W -> ( w ` i ) = ( W ` i ) )
16		fveq1 5689	. . . . . . . . 9 |- ( w = W -> ( w ` ( i + 1 ) ) = ( W ` ( i + 1 ) ) )
17	15, 16	preq12d 3792	. . . . . . . 8 |- ( w = W -> { ( w ` i ) , ( w ` ( i + 1 ) ) } = { ( W ` i ) , ( W ` ( i + 1 ) ) } )
18	17	eleq1d 2307	. . . . . . 7 |- ( w = W -> ( { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E <-> { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) )
19	14, 18	raleqbidv 2765	. . . . . 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 5690	. . . . . . . 8 |- ( w = W -> (lastS ` w ) = (lastS ` W ) )
21		fveq1 5689	. . . . . . . 8 |- ( w = W -> ( w ` 0 ) = ( W ` 0 ) )
22	20, 21	preq12d 3792	. . . . . . 7 |- ( w = W -> { (lastS ` w ) , ( w ` 0 ) } = { (lastS ` W ) , ( W ` 0 ) } )
23	22	eleq1d 2307	. . . . . 6 |- ( w = W -> ( { (lastS ` w ) , ( w ` 0 ) } e. E <-> { (lastS ` W ) , ( W ` 0 ) } e. E ) )
24	11, 19, 23	3anbi123d 1353	. . . . 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 2982	. . . 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 716	. 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 1013	. . 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 405	. . 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 1013	. . . . 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 461	. . 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 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   {crab 2532   _Vcvv 2821   (/)c0 3520   {cpr 3706   ` cfv 5372  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   - cmin 8487  ..^cfzo 10527  ♯chash 11192  Word cword 11282  lastSclsw 11327  Vtxcvtx 16167  Edgcedg 16212  ClWWalkscclwwlk 16546

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-map 6914  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-ndx 13333  df-slot 13334  df-base 13336  df-vtx 16169  df-clwwlk 16547

This theorem is referenced by:  clwwlkbp  16550  clwwlksswrd  16552  clwwlk1loop  16554  clwwlkccat  16556  isclwwlknx  16571  clwwlknonel  16587