Theorem clwwlks 26879

Description: The set of closed walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)

Hypotheses

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

Assertion

Ref	Expression
clwwlks	|- (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, w   w, V

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

Proof of Theorem clwwlks

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clwwlks 26877	. . . 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	a1i 11	. . 3 |- ( G e. _V -> 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 ) ) } ) )
3		fveq2 6191	. . . . . . 7 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
4		clwwlks.v	. . . . . . 7 |- V = (Vtx ` G )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( g = G -> (Vtx ` g ) = V )
6		wrdeq 13327	. . . . . 6 |- ( (Vtx ` g ) = V -> Word (Vtx ` g ) = Word V )
7	5, 6	syl 17	. . . . 5 |- ( g = G -> Word (Vtx ` g ) = Word V )
8		fveq2 6191	. . . . . . . . 9 |- ( g = G -> (Edg ` g ) = (Edg ` G ) )
9		clwwlks.e	. . . . . . . . 9 |- E = (Edg ` G )
10	8, 9	syl6eqr 2674	. . . . . . . 8 |- ( g = G -> (Edg ` g ) = E )
11	10	eleq2d 2687	. . . . . . 7 |- ( g = G -> ( { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) <-> { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E ) )
12	11	ralbidv 2986	. . . . . 6 |- ( g = G -> ( A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) <-> A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E ) )
13	10	eleq2d 2687	. . . . . 6 |- ( g = G -> ( { ( lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) <-> { ( lastS ` w ) , ( w ` 0 ) } e. E ) )
14	12, 13	3anbi23d 1402	. . . . 5 |- ( g = G -> ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) /\ { ( lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) ) <-> ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) ) )
15	7, 14	rabeqbidv 3195	. . . 4 |- ( g = G -> { 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 ) ) } = { w e. Word V | ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } )
16	15	adantl 482	. . 3 |- ( ( G e. _V /\ g = G ) -> { 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 ) ) } = { w e. Word V | ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } )
17		id 22	. . 3 |- ( G e. _V -> G e. _V )
18		fvex 6201	. . . . . 6 |- (Vtx ` G ) e. _V
19	4, 18	eqeltri 2697	. . . . 5 |- V e. _V
20	19	a1i 11	. . . 4 |- ( G e. _V -> V e. _V )
21		wrdexg 13315	. . . 4 |- ( V e. _V -> Word V e. _V )
22		rabexg 4812	. . . 4 |- (Word V e. _V -> { w e. Word V | ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } e. _V )
23	20, 21, 22	3syl 18	. . 3 |- ( G e. _V -> { w e. Word V | ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } e. _V )
24	2, 16, 17, 23	fvmptd 6288	. 2 |- ( 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 ) } )
25		fvprc 6185	. . 3 |- ( -. G e. _V -> (ClWWalks ` G ) = (/) )
26		noel 3919	. . . . . . . 8 |- -. { ( lastS ` w ) , ( w ` 0 ) } e. (/)
27		fvprc 6185	. . . . . . . . . 10 |- ( -. G e. _V -> (Edg ` G ) = (/) )
28	9, 27	syl5eq 2668	. . . . . . . . 9 |- ( -. G e. _V -> E = (/) )
29	28	eleq2d 2687	. . . . . . . 8 |- ( -. G e. _V -> ( { ( lastS ` w ) , ( w ` 0 ) } e. E <-> { ( lastS ` w ) , ( w ` 0 ) } e. (/) ) )
30	26, 29	mtbiri 317	. . . . . . 7 |- ( -. G e. _V -> -. { ( lastS ` w ) , ( w ` 0 ) } e. E )
31	30	adantr 481	. . . . . 6 |- ( ( -. G e. _V /\ w e. Word V ) -> -. { ( lastS ` w ) , ( w ` 0 ) } e. E )
32	31	intn3an3d 1444	. . . . 5 |- ( ( -. G e. _V /\ 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	32	ralrimiva 2966	. . . 4 |- ( -. G e. _V -> A. 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		rabeq0 3957	. . . 4 |- ( { 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. w e. Word V -. ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) )
35	33, 34	sylibr 224	. . 3 |- ( -. G e. _V -> { w e. Word V | ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } = (/) )
36	25, 35	eqtr4d 2659	. 2 |- ( -. 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 ) } )
37	24, 36	pm2.61i 176	1 |- (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:   -. wn 3   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   (/)c0 3915   {cpr 4179   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292  Vtxcvtx 25874  Edgcedg 25939  ClWWalkscclwwlks 26875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-clwwlks 26877

This theorem is referenced by:  isclwwlks  26880  clwwlkssswrd  26911