Theorem 2pthon 25380

Description: A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)

Assertion

Ref	Expression
2pthon	|- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) -> { <. 0 , i >. , <. 1 , j >. } ( A ( V PathOn E ) C ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ) )

Proof of Theorem 2pthon

Step	Hyp	Ref	Expression
1		vex 3059	. . . . . . . . 9 |- i e. _V
2		vex 3059	. . . . . . . . 9 |- j e. _V
3	1, 2	pm3.2i 461	. . . . . . . 8 |- ( i e. _V /\ j e. _V )
4		eqid 2461	. . . . . . . 8 |- { <. 0 , i >. , <. 1 , j >. } = { <. 0 , i >. , <. 1 , j >. }
5		eqid 2461	. . . . . . . 8 |- { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } = { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. }
6	3, 4, 5	constr2trl 25377	. . . . . . 7 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) -> { <. 0 , i >. , <. 1 , j >. } ( V Trails E ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ) )
7	6	3adant3 1034	. . . . . 6 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) -> { <. 0 , i >. , <. 1 , j >. } ( V Trails E ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ) )
8	7	imp 435	. . . . 5 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> { <. 0 , i >. , <. 1 , j >. } ( V Trails E ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } )
9		trliswlk 25317	. . . . 5 |- ( { <. 0 , i >. , <. 1 , j >. } ( V Trails E ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } -> { <. 0 , i >. , <. 1 , j >. } ( V Walks E ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } )
10	8, 9	syl 17	. . . 4 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> { <. 0 , i >. , <. 1 , j >. } ( V Walks E ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } )
11		c0ex 9662	. . . . . . . . 9 |- 0 e. _V
12	11	jctl 548	. . . . . . . 8 |- ( A e. V -> ( 0 e. _V /\ A e. V ) )
13	12	3ad2ant1 1035	. . . . . . 7 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( 0 e. _V /\ A e. V ) )
14		0ne1 10704	. . . . . . . 8 |- 0 =/= 1
15		0ne2 10849	. . . . . . . 8 |- 0 =/= 2
16	14, 15	pm3.2i 461	. . . . . . 7 |- ( 0 =/= 1 /\ 0 =/= 2 )
17		fvtp1g 6137	. . . . . . 7 |- ( ( ( 0 e. _V /\ A e. V ) /\ ( 0 =/= 1 /\ 0 =/= 2 ) ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 0 ) = A )
18	13, 16, 17	sylancl 673	. . . . . 6 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 0 ) = A )
19	18	3ad2ant2 1036	. . . . 5 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 0 ) = A )
20	19	adantr 471	. . . 4 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 0 ) = A )
21		ax-1ne0 9633	. . . . . . . . 9 |- 1 =/= 0
22	21	nesymi 2692	. . . . . . . 8 |- -. 0 = 1
23	11, 1	opth1 4688	. . . . . . . . 9 |- ( <. 0 , i >. = <. 1 , j >. -> 0 = 1 )
24	23	necon3bi 2661	. . . . . . . 8 |- ( -. 0 = 1 -> <. 0 , i >. =/= <. 1 , j >. )
25	22, 24	ax-mp 5	. . . . . . 7 |- <. 0 , i >. =/= <. 1 , j >.
26		opex 4677	. . . . . . . 8 |- <. 0 , i >. e. _V
27		opex 4677	. . . . . . . 8 |- <. 1 , j >. e. _V
28		hashprg 12603	. . . . . . . 8 |- ( ( <. 0 , i >. e. _V /\ <. 1 , j >. e. _V ) -> ( <. 0 , i >. =/= <. 1 , j >. <-> ( # ` { <. 0 , i >. , <. 1 , j >. } ) = 2 ) )
29	26, 27, 28	mp2an 683	. . . . . . 7 |- ( <. 0 , i >. =/= <. 1 , j >. <-> ( # ` { <. 0 , i >. , <. 1 , j >. } ) = 2 )
30	25, 29	mpbi 213	. . . . . 6 |- ( # ` { <. 0 , i >. , <. 1 , j >. } ) = 2
31	30	fveq2i 5890	. . . . 5 |- ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` ( # ` { <. 0 , i >. , <. 1 , j >. } ) ) = ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 2 )
32		2z 10997	. . . . . . . . . 10 |- 2 e. ZZ
33	32	jctl 548	. . . . . . . . 9 |- ( C e. V -> ( 2 e. ZZ /\ C e. V ) )
34	33	3ad2ant3 1037	. . . . . . . 8 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( 2 e. ZZ /\ C e. V ) )
35		1ne2 10850	. . . . . . . . 9 |- 1 =/= 2
36	15, 35	pm3.2i 461	. . . . . . . 8 |- ( 0 =/= 2 /\ 1 =/= 2 )
37		fvtp3g 6139	. . . . . . . 8 |- ( ( ( 2 e. ZZ /\ C e. V ) /\ ( 0 =/= 2 /\ 1 =/= 2 ) ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 2 ) = C )
38	34, 36, 37	sylancl 673	. . . . . . 7 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 2 ) = C )
39	38	3ad2ant2 1036	. . . . . 6 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 2 ) = C )
40	39	adantr 471	. . . . 5 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 2 ) = C )
41	31, 40	syl5eq 2507	. . . 4 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` ( # ` { <. 0 , i >. , <. 1 , j >. } ) ) = C )
42		simpl1 1017	. . . . . 6 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> ( V e. X /\ E e. Y ) )
43		prex 4655	. . . . . . . 8 |- { <. 0 , i >. , <. 1 , j >. } e. _V
44		tpex 6616	. . . . . . . 8 |- { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V
45	43, 44	pm3.2i 461	. . . . . . 7 |- ( { <. 0 , i >. , <. 1 , j >. } e. _V /\ { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V )
46	45	a1i 11	. . . . . 6 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> ( { <. 0 , i >. , <. 1 , j >. } e. _V /\ { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V ) )
47		3simpb 1012	. . . . . . . 8 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( A e. V /\ C e. V ) )
48	47	3ad2ant2 1036	. . . . . . 7 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( A e. V /\ C e. V ) )
49	48	adantr 471	. . . . . 6 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> ( A e. V /\ C e. V ) )
50	42, 46, 49	3jca 1194	. . . . 5 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> ( ( V e. X /\ E e. Y ) /\ ( { <. 0 , i >. , <. 1 , j >. } e. _V /\ { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V ) /\ ( A e. V /\ C e. V ) ) )
51		iswlkon 25310	. . . . 5 |- ( ( ( V e. X /\ E e. Y ) /\ ( { <. 0 , i >. , <. 1 , j >. } e. _V /\ { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( { <. 0 , i >. , <. 1 , j >. } ( A ( V WalkOn E ) C ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } <-> ( { <. 0 , i >. , <. 1 , j >. } ( V Walks E ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } /\ ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 0 ) = A /\ ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` ( # ` { <. 0 , i >. , <. 1 , j >. } ) ) = C ) ) )
52	50, 51	syl 17	. . . 4 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> ( { <. 0 , i >. , <. 1 , j >. } ( A ( V WalkOn E ) C ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } <-> ( { <. 0 , i >. , <. 1 , j >. } ( V Walks E ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } /\ ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 0 ) = A /\ ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` ( # ` { <. 0 , i >. , <. 1 , j >. } ) ) = C ) ) )
53	10, 20, 41, 52	mpbir3and 1197	. . 3 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> { <. 0 , i >. , <. 1 , j >. } ( A ( V WalkOn E ) C ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } )
54	3, 4, 5	constr2pth 25379	. . . 4 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) -> { <. 0 , i >. , <. 1 , j >. } ( V Paths E ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ) )
55	54	imp 435	. . 3 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> { <. 0 , i >. , <. 1 , j >. } ( V Paths E ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } )
56		ispthon 25354	. . . 4 |- ( ( ( V e. X /\ E e. Y ) /\ ( { <. 0 , i >. , <. 1 , j >. } e. _V /\ { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( { <. 0 , i >. , <. 1 , j >. } ( A ( V PathOn E ) C ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } <-> ( { <. 0 , i >. , <. 1 , j >. } ( A ( V WalkOn E ) C ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } /\ { <. 0 , i >. , <. 1 , j >. } ( V Paths E ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ) ) )
57	50, 56	syl 17	. . 3 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> ( { <. 0 , i >. , <. 1 , j >. } ( A ( V PathOn E ) C ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } <-> ( { <. 0 , i >. , <. 1 , j >. } ( A ( V WalkOn E ) C ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } /\ { <. 0 , i >. , <. 1 , j >. } ( V Paths E ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ) ) )
58	53, 55, 57	mpbir2and 938	. 2 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) ) -> { <. 0 , i >. , <. 1 , j >. } ( A ( V PathOn E ) C ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } )
59	58	ex 440	1 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( i =/= j /\ ( E ` i ) = { A , B } /\ ( E ` j ) = { B , C } ) -> { <. 0 , i >. , <. 1 , j >. } ( A ( V PathOn E ) C ) { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1454   e. wcel 1897   =/= wne 2632   _Vcvv 3056   {cpr 3981   {ctp 3983   <.cop 3985   class class class wbr 4415   ` cfv 5600  (class class class)co 6314   0cc0 9564   1c1 9565   2c2 10686   ZZcz 10965   #chash 12546   Walks cwalk 25274   Trails ctrail 25275   Paths cpath 25276   WalkOn cwlkon 25278   PathOn cpthon 25280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-pm 7500  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-card 8398  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-fzo 11946  df-hash 12547  df-word 12696  df-wlk 25284  df-trail 25285  df-pth 25286  df-wlkon 25290  df-pthon 25292

This theorem is referenced by:  2pthoncl  25381