Theorem 0enwwlksnge1 26749

Description: In graphs without edges, there are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 7-May-2021.)

Assertion

Ref	Expression
0enwwlksnge1	|- ( ( (Edg ` G ) = (/) /\ N e. NN ) -> ( N WWalksN G ) = (/) )

Proof of Theorem 0enwwlksnge1

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

Step	Hyp	Ref	Expression
1		nnnn0 11299	. . . 4 |- ( N e. NN -> N e. NN0 )
2		wwlksn 26729	. . . 4 |- ( N e. NN0 -> ( N WWalksN G ) = { w e. (WWalks ` G ) | ( # ` w ) = ( N + 1 ) } )
3	1, 2	syl 17	. . 3 |- ( N e. NN -> ( N WWalksN G ) = { w e. (WWalks ` G ) | ( # ` w ) = ( N + 1 ) } )
4	3	adantl 482	. 2 |- ( ( (Edg ` G ) = (/) /\ N e. NN ) -> ( N WWalksN G ) = { w e. (WWalks ` G ) | ( # ` w ) = ( N + 1 ) } )
5		eqid 2622	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
6		eqid 2622	. . . . . . . 8 |- (Edg ` G ) = (Edg ` G )
7	5, 6	iswwlks 26728	. . . . . . 7 |- ( 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		nncn 11028	. . . . . . . . . . . . . . . . 17 |- ( N e. NN -> N e. CC )
9		pncan1 10454	. . . . . . . . . . . . . . . . 17 |- ( N e. CC -> ( ( N + 1 ) - 1 ) = N )
10	8, 9	syl 17	. . . . . . . . . . . . . . . 16 |- ( N e. NN -> ( ( N + 1 ) - 1 ) = N )
11		id 22	. . . . . . . . . . . . . . . 16 |- ( N e. NN -> N e. NN )
12	10, 11	eqeltrd 2701	. . . . . . . . . . . . . . 15 |- ( N e. NN -> ( ( N + 1 ) - 1 ) e. NN )
13	12	adantl 482	. . . . . . . . . . . . . 14 |- ( ( (Edg ` G ) = (/) /\ N e. NN ) -> ( ( N + 1 ) - 1 ) e. NN )
14	13	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( # ` w ) = ( N + 1 ) /\ ( (Edg ` G ) = (/) /\ N e. NN ) ) -> ( ( N + 1 ) - 1 ) e. NN )
15		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( ( # ` w ) = ( N + 1 ) -> ( ( # ` w ) - 1 ) = ( ( N + 1 ) - 1 ) )
16	15	eleq1d 2686	. . . . . . . . . . . . . 14 |- ( ( # ` w ) = ( N + 1 ) -> ( ( ( # ` w ) - 1 ) e. NN <-> ( ( N + 1 ) - 1 ) e. NN ) )
17	16	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( # ` w ) = ( N + 1 ) /\ ( (Edg ` G ) = (/) /\ N e. NN ) ) -> ( ( ( # ` w ) - 1 ) e. NN <-> ( ( N + 1 ) - 1 ) e. NN ) )
18	14, 17	mpbird 247	. . . . . . . . . . . 12 |- ( ( ( # ` w ) = ( N + 1 ) /\ ( (Edg ` G ) = (/) /\ N e. NN ) ) -> ( ( # ` w ) - 1 ) e. NN )
19		lbfzo0 12507	. . . . . . . . . . . 12 |- ( 0 e. ( 0..^ ( ( # ` w ) - 1 ) ) <-> ( ( # ` w ) - 1 ) e. NN )
20	18, 19	sylibr 224	. . . . . . . . . . 11 |- ( ( ( # ` w ) = ( N + 1 ) /\ ( (Edg ` G ) = (/) /\ N e. NN ) ) -> 0 e. ( 0..^ ( ( # ` w ) - 1 ) ) )
21		fveq2 6191	. . . . . . . . . . . . . 14 |- ( i = 0 -> ( w ` i ) = ( w ` 0 ) )
22		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( i = 0 -> ( i + 1 ) = ( 0 + 1 ) )
23		0p1e1 11132	. . . . . . . . . . . . . . . 16 |- ( 0 + 1 ) = 1
24	22, 23	syl6eq 2672	. . . . . . . . . . . . . . 15 |- ( i = 0 -> ( i + 1 ) = 1 )
25	24	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( i = 0 -> ( w ` ( i + 1 ) ) = ( w ` 1 ) )
26	21, 25	preq12d 4276	. . . . . . . . . . . . 13 |- ( i = 0 -> { ( w ` i ) , ( w ` ( i + 1 ) ) } = { ( w ` 0 ) , ( w ` 1 ) } )
27	26	eleq1d 2686	. . . . . . . . . . . 12 |- ( i = 0 -> ( { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
28	27	adantl 482	. . . . . . . . . . 11 |- ( ( ( ( # ` w ) = ( N + 1 ) /\ ( (Edg ` G ) = (/) /\ N e. NN ) ) /\ i = 0 ) -> ( { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
29	20, 28	rspcdv 3312	. . . . . . . . . 10 |- ( ( ( # ` w ) = ( N + 1 ) /\ ( (Edg ` G ) = (/) /\ N e. NN ) ) -> ( A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) -> { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
30		eleq2 2690	. . . . . . . . . . . . 13 |- ( (Edg ` G ) = (/) -> ( { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) <-> { ( w ` 0 ) , ( w ` 1 ) } e. (/) ) )
31		noel 3919	. . . . . . . . . . . . . 14 |- -. { ( w ` 0 ) , ( w ` 1 ) } e. (/)
32	31	pm2.21i 116	. . . . . . . . . . . . 13 |- ( { ( w ` 0 ) , ( w ` 1 ) } e. (/) -> -. ( # ` w ) = ( N + 1 ) )
33	30, 32	syl6bi 243	. . . . . . . . . . . 12 |- ( (Edg ` G ) = (/) -> ( { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) -> -. ( # ` w ) = ( N + 1 ) ) )
34	33	adantr 481	. . . . . . . . . . 11 |- ( ( (Edg ` G ) = (/) /\ N e. NN ) -> ( { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) -> -. ( # ` w ) = ( N + 1 ) ) )
35	34	adantl 482	. . . . . . . . . 10 |- ( ( ( # ` w ) = ( N + 1 ) /\ ( (Edg ` G ) = (/) /\ N e. NN ) ) -> ( { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) -> -. ( # ` w ) = ( N + 1 ) ) )
36	29, 35	syldc 48	. . . . . . . . 9 |- ( A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) -> ( ( ( # ` w ) = ( N + 1 ) /\ ( (Edg ` G ) = (/) /\ N e. NN ) ) -> -. ( # ` w ) = ( N + 1 ) ) )
37	36	3ad2ant3 1084	. . . . . . . 8 |- ( ( w =/= (/) /\ w e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) -> ( ( ( # ` w ) = ( N + 1 ) /\ ( (Edg ` G ) = (/) /\ N e. NN ) ) -> -. ( # ` w ) = ( N + 1 ) ) )
38	37	com12 32	. . . . . . 7 |- ( ( ( # ` w ) = ( N + 1 ) /\ ( (Edg ` G ) = (/) /\ N e. NN ) ) -> ( ( w =/= (/) /\ w e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) -> -. ( # ` w ) = ( N + 1 ) ) )
39	7, 38	syl5bi 232	. . . . . 6 |- ( ( ( # ` w ) = ( N + 1 ) /\ ( (Edg ` G ) = (/) /\ N e. NN ) ) -> ( w e. (WWalks ` G ) -> -. ( # ` w ) = ( N + 1 ) ) )
40	39	expimpd 629	. . . . 5 |- ( ( # ` w ) = ( N + 1 ) -> ( ( ( (Edg ` G ) = (/) /\ N e. NN ) /\ w e. (WWalks ` G ) ) -> -. ( # ` w ) = ( N + 1 ) ) )
41		ax-1 6	. . . . 5 |- ( -. ( # ` w ) = ( N + 1 ) -> ( ( ( (Edg ` G ) = (/) /\ N e. NN ) /\ w e. (WWalks ` G ) ) -> -. ( # ` w ) = ( N + 1 ) ) )
42	40, 41	pm2.61i 176	. . . 4 |- ( ( ( (Edg ` G ) = (/) /\ N e. NN ) /\ w e. (WWalks ` G ) ) -> -. ( # ` w ) = ( N + 1 ) )
43	42	ralrimiva 2966	. . 3 |- ( ( (Edg ` G ) = (/) /\ N e. NN ) -> A. w e. (WWalks ` G ) -. ( # ` w ) = ( N + 1 ) )
44		rabeq0 3957	. . 3 |- ( { w e. (WWalks ` G ) | ( # ` w ) = ( N + 1 ) } = (/) <-> A. w e. (WWalks ` G ) -. ( # ` w ) = ( N + 1 ) )
45	43, 44	sylibr 224	. 2 |- ( ( (Edg ` G ) = (/) /\ N e. NN ) -> { w e. (WWalks ` G ) | ( # ` w ) = ( N + 1 ) } = (/) )
46	4, 45	eqtrd 2656	1 |- ( ( (Edg ` G ) = (/) /\ N e. NN ) -> ( N WWalksN G ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   (/)c0 3915   {cpr 4179   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   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-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-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  rusgr0edg  26868