Theorem wwlksnfi 26801

Description: The number of walks represented by words of fixed length is finite if the number of vertices is finite (in the graph). (Contributed by Alexander van der Vekens, 30-Jul-2018.) (Revised by AV, 19-Apr-2021.)

Assertion

Ref	Expression
wwlksnfi	|- ( (Vtx ` G ) e. Fin -> ( N WWalksN G ) e. Fin )

Proof of Theorem wwlksnfi

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

Step	Hyp	Ref	Expression
1		wwlksn 26729	. . . . . . . 8 |- ( N e. NN0 -> ( N WWalksN G ) = { w e. (WWalks ` G ) | ( # ` w ) = ( N + 1 ) } )
2		df-rab 2921	. . . . . . . 8 |- { w e. (WWalks ` G ) | ( # ` w ) = ( N + 1 ) } = { w | ( w e. (WWalks ` G ) /\ ( # ` w ) = ( N + 1 ) ) }
3	1, 2	syl6eq 2672	. . . . . . 7 |- ( N e. NN0 -> ( N WWalksN G ) = { w | ( w e. (WWalks ` G ) /\ ( # ` w ) = ( N + 1 ) ) } )
4	3	adantl 482	. . . . . 6 |- ( ( G e. _V /\ N e. NN0 ) -> ( N WWalksN G ) = { w | ( w e. (WWalks ` G ) /\ ( # ` w ) = ( N + 1 ) ) } )
5		eqid 2622	. . . . . . . . . . 11 |- (Vtx ` G ) = (Vtx ` G )
6		eqid 2622	. . . . . . . . . . 11 |- (Edg ` G ) = (Edg ` G )
7	5, 6	iswwlks 26728	. . . . . . . . . 10 |- ( w e. (WWalks ` G ) <-> ( w =/= (/) /\ w e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) )
8	7	a1i 11	. . . . . . . . 9 |- ( ( G e. _V /\ N e. NN0 ) -> ( w e. (WWalks ` G ) <-> ( w =/= (/) /\ w e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) ) )
9	8	anbi1d 741	. . . . . . . 8 |- ( ( G e. _V /\ N e. NN0 ) -> ( ( w e. (WWalks ` G ) /\ ( # ` w ) = ( N + 1 ) ) <-> ( ( w =/= (/) /\ w e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) ) )
10	9	abbidv 2741	. . . . . . 7 |- ( ( G e. _V /\ N e. NN0 ) -> { w | ( w e. (WWalks ` G ) /\ ( # ` w ) = ( N + 1 ) ) } = { w | ( ( w =/= (/) /\ w e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) } )
11		3anan12 1051	. . . . . . . . . . 11 |- ( ( w =/= (/) /\ w e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) <-> ( w e. Word (Vtx ` G ) /\ ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) ) )
12	11	anbi1i 731	. . . . . . . . . 10 |- ( ( ( w =/= (/) /\ w e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) <-> ( ( w e. Word (Vtx ` G ) /\ ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ ( # ` w ) = ( N + 1 ) ) )
13		anass 681	. . . . . . . . . 10 |- ( ( ( w e. Word (Vtx ` G ) /\ ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ ( # ` w ) = ( N + 1 ) ) <-> ( w e. Word (Vtx ` G ) /\ ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) ) )
14	12, 13	bitri 264	. . . . . . . . 9 |- ( ( ( w =/= (/) /\ w e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) <-> ( w e. Word (Vtx ` G ) /\ ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) ) )
15	14	abbii 2739	. . . . . . . 8 |- { w | ( ( w =/= (/) /\ w e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) } = { w | ( w e. Word (Vtx ` G ) /\ ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) ) }
16		df-rab 2921	. . . . . . . 8 |- { w e. Word (Vtx ` G ) | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) } = { w | ( w e. Word (Vtx ` G ) /\ ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) ) }
17	15, 16	eqtr4i 2647	. . . . . . 7 |- { w | ( ( w =/= (/) /\ w e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) } = { w e. Word (Vtx ` G ) | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) }
18	10, 17	syl6eq 2672	. . . . . 6 |- ( ( G e. _V /\ N e. NN0 ) -> { w | ( w e. (WWalks ` G ) /\ ( # ` w ) = ( N + 1 ) ) } = { w e. Word (Vtx ` G ) | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) } )
19	4, 18	eqtrd 2656	. . . . 5 |- ( ( G e. _V /\ N e. NN0 ) -> ( N WWalksN G ) = { w e. Word (Vtx ` G ) | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) } )
20	19	adantr 481	. . . 4 |- ( ( ( G e. _V /\ N e. NN0 ) /\ (Vtx ` G ) e. Fin ) -> ( N WWalksN G ) = { w e. Word (Vtx ` G ) | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) } )
21		peano2nn0 11333	. . . . . . . . 9 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
22	21	adantl 482	. . . . . . . 8 |- ( ( G e. _V /\ N e. NN0 ) -> ( N + 1 ) e. NN0 )
23	22	anim2i 593	. . . . . . 7 |- ( ( (Vtx ` G ) e. Fin /\ ( G e. _V /\ N e. NN0 ) ) -> ( (Vtx ` G ) e. Fin /\ ( N + 1 ) e. NN0 ) )
24	23	ancoms 469	. . . . . 6 |- ( ( ( G e. _V /\ N e. NN0 ) /\ (Vtx ` G ) e. Fin ) -> ( (Vtx ` G ) e. Fin /\ ( N + 1 ) e. NN0 ) )
25		wrdnfi 13338	. . . . . 6 |- ( ( (Vtx ` G ) e. Fin /\ ( N + 1 ) e. NN0 ) -> { w e. Word (Vtx ` G ) | ( # ` w ) = ( N + 1 ) } e. Fin )
26	24, 25	syl 17	. . . . 5 |- ( ( ( G e. _V /\ N e. NN0 ) /\ (Vtx ` G ) e. Fin ) -> { w e. Word (Vtx ` G ) | ( # ` w ) = ( N + 1 ) } e. Fin )
27		simpr 477	. . . . . . 7 |- ( ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) -> ( # ` w ) = ( N + 1 ) )
28	27	rgenw 2924	. . . . . 6 |- A. w e. Word (Vtx ` G ) ( ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) -> ( # ` w ) = ( N + 1 ) )
29		ss2rab 3678	. . . . . 6 |- ( { w e. Word (Vtx ` G ) | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) } C_ { w e. Word (Vtx ` G ) | ( # ` w ) = ( N + 1 ) } <-> A. w e. Word (Vtx ` G ) ( ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) -> ( # ` w ) = ( N + 1 ) ) )
30	28, 29	mpbir 221	. . . . 5 |- { w e. Word (Vtx ` G ) | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) } C_ { w e. Word (Vtx ` G ) | ( # ` w ) = ( N + 1 ) }
31		ssfi 8180	. . . . 5 |- ( ( { w e. Word (Vtx ` G ) | ( # ` w ) = ( N + 1 ) } e. Fin /\ { w e. Word (Vtx ` G ) | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) } C_ { w e. Word (Vtx ` G ) | ( # ` w ) = ( N + 1 ) } ) -> { w e. Word (Vtx ` G ) | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) } e. Fin )
32	26, 30, 31	sylancl 694	. . . 4 |- ( ( ( G e. _V /\ N e. NN0 ) /\ (Vtx ` G ) e. Fin ) -> { w e. Word (Vtx ` G ) | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( N + 1 ) ) } e. Fin )
33	20, 32	eqeltrd 2701	. . 3 |- ( ( ( G e. _V /\ N e. NN0 ) /\ (Vtx ` G ) e. Fin ) -> ( N WWalksN G ) e. Fin )
34	33	ex 450	. 2 |- ( ( G e. _V /\ N e. NN0 ) -> ( (Vtx ` G ) e. Fin -> ( N WWalksN G ) e. Fin ) )
35		wwlksnndef 26800	. . . . 5 |- ( ( G e/ _V \/ N e/ NN0 ) -> ( N WWalksN G ) = (/) )
36		ioran 511	. . . . . 6 |- ( -. ( G e/ _V \/ N e/ NN0 ) <-> ( -. G e/ _V /\ -. N e/ NN0 ) )
37		nnel 2906	. . . . . . 7 |- ( -. G e/ _V <-> G e. _V )
38		nnel 2906	. . . . . . 7 |- ( -. N e/ NN0 <-> N e. NN0 )
39	37, 38	anbi12i 733	. . . . . 6 |- ( ( -. G e/ _V /\ -. N e/ NN0 ) <-> ( G e. _V /\ N e. NN0 ) )
40	36, 39	sylbb 209	. . . . 5 |- ( -. ( G e/ _V \/ N e/ NN0 ) -> ( G e. _V /\ N e. NN0 ) )
41	35, 40	nsyl4 156	. . . 4 |- ( -. ( G e. _V /\ N e. NN0 ) -> ( N WWalksN G ) = (/) )
42		0fin 8188	. . . . 5 |- (/) e. Fin
43	42	a1i 11	. . . 4 |- ( -. ( G e. _V /\ N e. NN0 ) -> (/) e. Fin )
44	41, 43	eqeltrd 2701	. . 3 |- ( -. ( G e. _V /\ N e. NN0 ) -> ( N WWalksN G ) e. Fin )
45	44	a1d 25	. 2 |- ( -. ( G e. _V /\ N e. NN0 ) -> ( (Vtx ` G ) e. Fin -> ( N WWalksN G ) e. Fin ) )
46	34, 45	pm2.61i 176	1 |- ( (Vtx ` G ) e. Fin -> ( N WWalksN G ) e. Fin )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   e/ wnel 2897   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {cpr 4179   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  Edgcedg 25939  WWalkscwwlks 26717   WWalksN cwwlksn 26718

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-rmo 2920  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-2o 7561  df-oadd 7564  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-cda 8990  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-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  wlksnfi  26802  hashwwlksnext  26809  wspthnfi  26815  wwlksnonfi  26816  rusgrnumwwlks  26869  clwwlknclwwlkdifnum  26874