MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usg2spthonot1 Structured version   Unicode version

Theorem usg2spthonot1 25017
Description: A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
Assertion
Ref Expression
usg2spthonot1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( ( T  = 
<. A ,  b ,  C >.  /\  A  =/= 
C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
Distinct variable groups:    A, b    C, b    E, b    T, b    V, b

Proof of Theorem usg2spthonot1
StepHypRef Expression
1 usgrav 24465 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 el2spthonot0 24998 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( T  e.  ( A
( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) ) )
31, 2sylan 471 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) ) )
4 simpll 753 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  V USGrph  E )
5 simplrl 761 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  A  e.  V )
6 simpr 461 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  b  e.  V )
7 simplrr 762 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  C  e.  V )
8 usg2spthonot0 25016 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  b  e.  V  /\  C  e.  V )
)  ->  ( <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C )  <-> 
( ( A  =  A  /\  C  =  C  /\  A  =/= 
C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
94, 5, 6, 7, 8syl13anc 1230 . . . . . 6  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  ( <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C )  <->  ( ( A  =  A  /\  C  =  C  /\  A  =/=  C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
10 simp3 998 . . . . . . . 8  |-  ( ( A  =  A  /\  C  =  C  /\  A  =/=  C )  ->  A  =/=  C )
1110anim1i 568 . . . . . . 7  |-  ( ( ( A  =  A  /\  C  =  C  /\  A  =/=  C
)  /\  ( { A ,  b }  e.  ran  E  /\  {
b ,  C }  e.  ran  E ) )  ->  ( A  =/= 
C  /\  ( { A ,  b }  e.  ran  E  /\  {
b ,  C }  e.  ran  E ) ) )
12 eqidd 2458 . . . . . . . . 9  |-  ( A  =/=  C  ->  A  =  A )
13 eqidd 2458 . . . . . . . . 9  |-  ( A  =/=  C  ->  C  =  C )
14 id 22 . . . . . . . . 9  |-  ( A  =/=  C  ->  A  =/=  C )
1512, 13, 143jca 1176 . . . . . . . 8  |-  ( A  =/=  C  ->  ( A  =  A  /\  C  =  C  /\  A  =/=  C ) )
1615anim1i 568 . . . . . . 7  |-  ( ( A  =/=  C  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) )  ->  ( ( A  =  A  /\  C  =  C  /\  A  =/=  C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
1711, 16impbii 188 . . . . . 6  |-  ( ( ( A  =  A  /\  C  =  C  /\  A  =/=  C
)  /\  ( { A ,  b }  e.  ran  E  /\  {
b ,  C }  e.  ran  E ) )  <-> 
( A  =/=  C  /\  ( { A , 
b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
189, 17syl6bb 261 . . . . 5  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  ( <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C )  <->  ( A  =/=  C  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
1918anbi2d 703 . . . 4  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  (
( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) )  <->  ( T  =  <. A ,  b ,  C >.  /\  ( A  =/=  C  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) ) )
20 anass 649 . . . 4  |-  ( ( ( T  =  <. A ,  b ,  C >.  /\  A  =/=  C
)  /\  ( { A ,  b }  e.  ran  E  /\  {
b ,  C }  e.  ran  E ) )  <-> 
( T  =  <. A ,  b ,  C >.  /\  ( A  =/= 
C  /\  ( { A ,  b }  e.  ran  E  /\  {
b ,  C }  e.  ran  E ) ) ) )
2119, 20syl6bbr 263 . . 3  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  (
( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) )  <->  ( ( T  =  <. A , 
b ,  C >.  /\  A  =/=  C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
2221rexbidva 2965 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( E. b  e.  V  ( T  =  <. A , 
b ,  C >.  /\ 
<. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) )  <->  E. b  e.  V  ( ( T  =  <. A , 
b ,  C >.  /\  A  =/=  C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
233, 22bitrd 253 1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( ( T  = 
<. A ,  b ,  C >.  /\  A  =/= 
C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   _Vcvv 3109   {cpr 4034   <.cotp 4040   class class class wbr 4456   ran crn 5009  (class class class)co 6296   USGrph cusg 24457   2SPathOnOt c2pthonot 24984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-usgra 24460  df-wlk 24635  df-trail 24636  df-pth 24637  df-spth 24638  df-wlkon 24641  df-spthon 24644  df-2wlkonot 24985  df-2spthonot 24987
This theorem is referenced by:  usg2spot2nb  25192
  Copyright terms: Public domain W3C validator