Theorem 2pthon 24277

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		simp2 997	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> A =/= C )
2		vex 3116	. . . . . . . . . 10 |- i e. _V
3		vex 3116	. . . . . . . . . 10 |- j e. _V
4	2, 3	pm3.2i 455	. . . . . . . . 9 |- ( i e. _V /\ j e. _V )
5		eqid 2467	. . . . . . . . 9 |- { <. 0 , i >. , <. 1 , j >. } = { <. 0 , i >. , <. 1 , j >. }
6		eqid 2467	. . . . . . . . 9 |- { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } = { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. }
7	4, 5, 6	constr2trl 24274	. . . . . . . 8 |- ( ( ( 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 >. } ) )
8	7	3adant3 1016	. . . . . . 7 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V /\ C e. V ) /\ A =/= 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	1, 8	syl3an3 1263	. . . . . 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 >. } ) )
10	9	imp 429	. . . . 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 >. } )
11		trliswlk 24214	. . . . 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 >. } )
12	10, 11	syl 16	. . . 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 >. } )
13		c0ex 9586	. . . . . . . . 9 |- 0 e. _V
14	13	jctl 541	. . . . . . . 8 |- ( A e. V -> ( 0 e. _V /\ A e. V ) )
15	14	3ad2ant1 1017	. . . . . . 7 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( 0 e. _V /\ A e. V ) )
16		0ne1 10599	. . . . . . . 8 |- 0 =/= 1
17		0ne2 10743	. . . . . . . 8 |- 0 =/= 2
18	16, 17	pm3.2i 455	. . . . . . 7 |- ( 0 =/= 1 /\ 0 =/= 2 )
19		fvtp1g 6109	. . . . . . 7 |- ( ( ( 0 e. _V /\ A e. V ) /\ ( 0 =/= 1 /\ 0 =/= 2 ) ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 0 ) = A )
20	15, 18, 19	sylancl 662	. . . . . 6 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 0 ) = A )
21	20	3ad2ant2 1018	. . . . 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 )
22	21	adantr 465	. . . 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 )
23		ax-1ne0 9557	. . . . . . . . 9 |- 1 =/= 0
24	23	nesymi 2740	. . . . . . . 8 |- -. 0 = 1
25	13, 2	opth1 4720	. . . . . . . . 9 |- ( <. 0 , i >. = <. 1 , j >. -> 0 = 1 )
26	25	necon3bi 2696	. . . . . . . 8 |- ( -. 0 = 1 -> <. 0 , i >. =/= <. 1 , j >. )
27	24, 26	ax-mp 5	. . . . . . 7 |- <. 0 , i >. =/= <. 1 , j >.
28		opex 4711	. . . . . . . 8 |- <. 0 , i >. e. _V
29		opex 4711	. . . . . . . 8 |- <. 1 , j >. e. _V
30		hashprg 12422	. . . . . . . 8 |- ( ( <. 0 , i >. e. _V /\ <. 1 , j >. e. _V ) -> ( <. 0 , i >. =/= <. 1 , j >. <-> ( # ` { <. 0 , i >. , <. 1 , j >. } ) = 2 ) )
31	28, 29, 30	mp2an 672	. . . . . . 7 |- ( <. 0 , i >. =/= <. 1 , j >. <-> ( # ` { <. 0 , i >. , <. 1 , j >. } ) = 2 )
32	27, 31	mpbi 208	. . . . . 6 |- ( # ` { <. 0 , i >. , <. 1 , j >. } ) = 2
33	32	fveq2i 5867	. . . . 5 |- ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` ( # ` { <. 0 , i >. , <. 1 , j >. } ) ) = ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 2 )
34		2z 10892	. . . . . . . . . 10 |- 2 e. ZZ
35	34	jctl 541	. . . . . . . . 9 |- ( C e. V -> ( 2 e. ZZ /\ C e. V ) )
36	35	3ad2ant3 1019	. . . . . . . 8 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( 2 e. ZZ /\ C e. V ) )
37		1ne2 10744	. . . . . . . . 9 |- 1 =/= 2
38	17, 37	pm3.2i 455	. . . . . . . 8 |- ( 0 =/= 2 /\ 1 =/= 2 )
39		fvtp3g 6111	. . . . . . . 8 |- ( ( ( 2 e. ZZ /\ C e. V ) /\ ( 0 =/= 2 /\ 1 =/= 2 ) ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 2 ) = C )
40	36, 38, 39	sylancl 662	. . . . . . 7 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } ` 2 ) = C )
41	40	3ad2ant2 1018	. . . . . 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 )
42	41	adantr 465	. . . . 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 )
43	33, 42	syl5eq 2520	. . . 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 )
44		simpl1 999	. . . . . 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 ) )
45		prex 4689	. . . . . . . 8 |- { <. 0 , i >. , <. 1 , j >. } e. _V
46		tpex 6581	. . . . . . . 8 |- { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V
47	45, 46	pm3.2i 455	. . . . . . 7 |- ( { <. 0 , i >. , <. 1 , j >. } e. _V /\ { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V )
48	47	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 ) )
49		3simpb 994	. . . . . . . 8 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( A e. V /\ C e. V ) )
50	49	3ad2ant2 1018	. . . . . . 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 ) )
51	50	adantr 465	. . . . . 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 ) )
52	44, 48, 51	3jca 1176	. . . . 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 ) ) )
53		iswlkon 24207	. . . . 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 ) ) )
54	52, 53	syl 16	. . . 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 ) ) )
55	12, 22, 43, 54	mpbir3and 1179	. . 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 >. } )
56	4, 5, 6	constr2pth 24276	. . . 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 >. } ) )
57	56	imp 429	. . 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 >. } )
58		ispthon 24251	. . . 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 >. } ) ) )
59	52, 58	syl 16	. . 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 >. } ) ) )
60	55, 57, 59	mpbir2and 920	. 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 >. } )
61	60	ex 434	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 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   _Vcvv 3113   {cpr 4029   {ctp 4031   <.cop 4033   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489   2c2 10581   ZZcz 10860   #chash 12367   Walks cwalk 24171   Trails ctrail 24172   Paths cpath 24173   WalkOn cwlkon 24175   PathOn cpthon 24177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-wlk 24181  df-trail 24182  df-pth 24183  df-wlkon 24187  df-pthon 24189

This theorem is referenced by:  2pthoncl  24278