Theorem wspthsnwspthsnon 26811

Description: A simple path of fixed length is a simple path of fixed length between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-May-2021.)

Hypothesis

Ref	Expression
wwlksnwwlksnon.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
wspthsnwspthsnon	|- ( ( N e. NN0 /\ G e. U ) -> ( W e. ( N WSPathsN G ) <-> E. a e. V E. b e. V W e. ( a ( N WSPathsNOn G ) b ) ) )

Distinct variable groups:   G, a, b   N, a, b   U, a, b   V, a, b   W, a, b

Proof of Theorem wspthsnwspthsnon

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iswspthn 26736	. . 3 |- ( W e. ( N WSPathsN G ) <-> ( W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) )
2	1	a1i 11	. 2 |- ( ( N e. NN0 /\ G e. U ) -> ( W e. ( N WSPathsN G ) <-> ( W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) ) )
3		wwlksnwwlksnon.v	. . . . 5 |- V = (Vtx ` G )
4	3	wwlksnwwlksnon 26810	. . . 4 |- ( ( N e. NN0 /\ G e. U ) -> ( W e. ( N WWalksN G ) <-> E. a e. V E. b e. V W e. ( a ( N WWalksNOn G ) b ) ) )
5	4	anbi1d 741	. . 3 |- ( ( N e. NN0 /\ G e. U ) -> ( ( W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) <-> ( E. a e. V E. b e. V W e. ( a ( N WWalksNOn G ) b ) /\ E. f f (SPaths ` G ) W ) ) )
6		r19.41vv 3091	. . 3 |- ( E. a e. V E. b e. V ( W e. ( a ( N WWalksNOn G ) b ) /\ E. f f (SPaths ` G ) W ) <-> ( E. a e. V E. b e. V W e. ( a ( N WWalksNOn G ) b ) /\ E. f f (SPaths ` G ) W ) )
7	5, 6	syl6bbr 278	. 2 |- ( ( N e. NN0 /\ G e. U ) -> ( ( W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) <-> E. a e. V E. b e. V ( W e. ( a ( N WWalksNOn G ) b ) /\ E. f f (SPaths ` G ) W ) ) )
8		3anass 1042	. . . . . . . 8 |- ( ( f (SPaths ` G ) W /\ ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) <-> ( f (SPaths ` G ) W /\ ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) )
9	8	a1i 11	. . . . . . 7 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) /\ W e. ( a ( N WWalksNOn G ) b ) ) -> ( ( f (SPaths ` G ) W /\ ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) <-> ( f (SPaths ` G ) W /\ ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) ) )
10		simpr 477	. . . . . . . 8 |- ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) -> ( a e. V /\ b e. V ) )
11		vex 3203	. . . . . . . . 9 |- f e. _V
12	11	jctl 564	. . . . . . . 8 |- ( W e. ( a ( N WWalksNOn G ) b ) -> ( f e. _V /\ W e. ( a ( N WWalksNOn G ) b ) ) )
13	3	isspthonpth 26645	. . . . . . . 8 |- ( ( ( a e. V /\ b e. V ) /\ ( f e. _V /\ W e. ( a ( N WWalksNOn G ) b ) ) ) -> ( f ( a (SPathsOn ` G ) b ) W <-> ( f (SPaths ` G ) W /\ ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) )
14	10, 12, 13	syl2an 494	. . . . . . 7 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) /\ W e. ( a ( N WWalksNOn G ) b ) ) -> ( f ( a (SPathsOn ` G ) b ) W <-> ( f (SPaths ` G ) W /\ ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) )
15		spthiswlk 26624	. . . . . . . . . 10 |- ( f (SPaths ` G ) W -> f (Walks ` G ) W )
16		wlklenvm1 26517	. . . . . . . . . 10 |- ( f (Walks ` G ) W -> ( # ` f ) = ( ( # ` W ) - 1 ) )
17	3	wwlknon 26742	. . . . . . . . . . . . . 14 |- ( ( a e. V /\ b e. V ) -> ( W e. ( a ( N WWalksNOn G ) b ) <-> ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) ) )
18	17	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) -> ( W e. ( a ( N WWalksNOn G ) b ) <-> ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) ) )
19		simpl2 1065	. . . . . . . . . . . . . . . . . 18 |- ( ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) /\ ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) ) -> ( W ` 0 ) = a )
20		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) -> ( # ` f ) = ( ( # ` W ) - 1 ) )
21	20	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) /\ ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) ) -> ( # ` f ) = ( ( # ` W ) - 1 ) )
22		wwlknbp2 26752	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( W e. ( N WWalksN G ) -> ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) )
23		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( # ` W ) = ( N + 1 ) -> ( ( # ` W ) - 1 ) = ( ( N + 1 ) - 1 ) )
24	23	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( ( # ` W ) - 1 ) = ( ( N + 1 ) - 1 ) )
25		nn0cn 11302	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. NN0 -> N e. CC )
26		pncan1 10454	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. CC -> ( ( N + 1 ) - 1 ) = N )
27	25, 26	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. NN0 -> ( ( N + 1 ) - 1 ) = N )
28	27	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) -> ( ( N + 1 ) - 1 ) = N )
29	24, 28	sylan9eq 2676	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) ) -> ( ( # ` W ) - 1 ) = N )
30	29	ex 450	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) -> ( ( # ` W ) - 1 ) = N ) )
31	22, 30	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( W e. ( N WWalksN G ) -> ( ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) -> ( ( # ` W ) - 1 ) = N ) )
32	31	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) -> ( ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) -> ( ( # ` W ) - 1 ) = N ) )
33	32	imp 445	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) /\ ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) ) -> ( ( # ` W ) - 1 ) = N )
34	21, 33	eqtrd 2656	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) /\ ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) ) -> ( # ` f ) = N )
35	34	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) /\ ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) ) -> ( W ` ( # ` f ) ) = ( W ` N ) )
36		simpl3 1066	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) /\ ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) ) -> ( W ` N ) = b )
37	35, 36	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 |- ( ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) /\ ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) ) -> ( W ` ( # ` f ) ) = b )
38	19, 37	jca 554	. . . . . . . . . . . . . . . . 17 |- ( ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) /\ ( N e. NN0 /\ ( # ` f ) = ( ( # ` W ) - 1 ) ) ) -> ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) )
39	38	exp32 631	. . . . . . . . . . . . . . . 16 |- ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) -> ( N e. NN0 -> ( ( # ` f ) = ( ( # ` W ) - 1 ) -> ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) ) )
40	39	com12 32	. . . . . . . . . . . . . . 15 |- ( N e. NN0 -> ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) -> ( ( # ` f ) = ( ( # ` W ) - 1 ) -> ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) ) )
41	40	adantr 481	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ G e. U ) -> ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) -> ( ( # ` f ) = ( ( # ` W ) - 1 ) -> ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) ) )
42	41	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) -> ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) -> ( ( # ` f ) = ( ( # ` W ) - 1 ) -> ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) ) )
43	18, 42	sylbid 230	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) -> ( W e. ( a ( N WWalksNOn G ) b ) -> ( ( # ` f ) = ( ( # ` W ) - 1 ) -> ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) ) )
44	43	imp 445	. . . . . . . . . . 11 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) /\ W e. ( a ( N WWalksNOn G ) b ) ) -> ( ( # ` f ) = ( ( # ` W ) - 1 ) -> ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) )
45	44	com12 32	. . . . . . . . . 10 |- ( ( # ` f ) = ( ( # ` W ) - 1 ) -> ( ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) /\ W e. ( a ( N WWalksNOn G ) b ) ) -> ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) )
46	15, 16, 45	3syl 18	. . . . . . . . 9 |- ( f (SPaths ` G ) W -> ( ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) /\ W e. ( a ( N WWalksNOn G ) b ) ) -> ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) )
47	46	com12 32	. . . . . . . 8 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) /\ W e. ( a ( N WWalksNOn G ) b ) ) -> ( f (SPaths ` G ) W -> ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) )
48	47	pm4.71d 666	. . . . . . 7 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) /\ W e. ( a ( N WWalksNOn G ) b ) ) -> ( f (SPaths ` G ) W <-> ( f (SPaths ` G ) W /\ ( ( W ` 0 ) = a /\ ( W ` ( # ` f ) ) = b ) ) ) )
49	9, 14, 48	3bitr4rd 301	. . . . . 6 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) /\ W e. ( a ( N WWalksNOn G ) b ) ) -> ( f (SPaths ` G ) W <-> f ( a (SPathsOn ` G ) b ) W ) )
50	49	exbidv 1850	. . . . 5 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) /\ W e. ( a ( N WWalksNOn G ) b ) ) -> ( E. f f (SPaths ` G ) W <-> E. f f ( a (SPathsOn ` G ) b ) W ) )
51	50	pm5.32da 673	. . . 4 |- ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) -> ( ( W e. ( a ( N WWalksNOn G ) b ) /\ E. f f (SPaths ` G ) W ) <-> ( W e. ( a ( N WWalksNOn G ) b ) /\ E. f f ( a (SPathsOn ` G ) b ) W ) ) )
52	3	wspthnon 26743	. . . . 5 |- ( ( a e. V /\ b e. V ) -> ( W e. ( a ( N WSPathsNOn G ) b ) <-> ( W e. ( a ( N WWalksNOn G ) b ) /\ E. f f ( a (SPathsOn ` G ) b ) W ) ) )
53	52	adantl 482	. . . 4 |- ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) -> ( W e. ( a ( N WSPathsNOn G ) b ) <-> ( W e. ( a ( N WWalksNOn G ) b ) /\ E. f f ( a (SPathsOn ` G ) b ) W ) ) )
54	51, 53	bitr4d 271	. . 3 |- ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) -> ( ( W e. ( a ( N WWalksNOn G ) b ) /\ E. f f (SPaths ` G ) W ) <-> W e. ( a ( N WSPathsNOn G ) b ) ) )
55	54	2rexbidva 3056	. 2 |- ( ( N e. NN0 /\ G e. U ) -> ( E. a e. V E. b e. V ( W e. ( a ( N WWalksNOn G ) b ) /\ E. f f (SPaths ` G ) W ) <-> E. a e. V E. b e. V W e. ( a ( N WSPathsNOn G ) b ) ) )
56	2, 7, 55	3bitrd 294	1 |- ( ( N e. NN0 /\ G e. U ) -> ( W e. ( N WSPathsN G ) <-> E. a e. V E. b e. V W e. ( a ( N WSPathsNOn G ) b ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NN0cn0 11292   #chash 13117  Word cword 13291  Vtxcvtx 25874  Walkscwlks 26492  SPathscspths 26609  SPathsOncspthson 26611   WWalksN cwwlksn 26718   WWalksNOn cwwlksnon 26719   WSPathsN cwwspthsn 26720   WSPathsNOn cwwspthsnon 26721

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-ifp 1013  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-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-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-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-spthson 26615  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724  df-wspthsn 26725  df-wspthsnon 26726

This theorem is referenced by:  wspniunwspnon  26819  elwspths2spth  26862  fusgr2wsp2nb  27198