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Theorem 2pthwlkonot 25012
Description: For two different vertices, a walk of length 2 between these vertices as ordered triple is a simple path of length 2 between these vertices as ordered triple in an undirected simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
Assertion
Ref Expression
2pthwlkonot  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( A
( V 2SPathOnOt  E ) B )  =  ( A ( V 2WalksOnOt  E ) B ) )

Proof of Theorem 2pthwlkonot
Dummy variables  f  p  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr1 1002 . . . . . . . . . . 11  |-  ( ( ( # `  f
)  =  2  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B ) )  ->  V USGrph  E )
2 simpl 457 . . . . . . . . . . 11  |-  ( ( ( # `  f
)  =  2  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B ) )  ->  ( # `
 f )  =  2 )
3 simpr3 1004 . . . . . . . . . . 11  |-  ( ( ( # `  f
)  =  2  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B ) )  ->  A  =/=  B )
4 usgra2wlkspth 24748 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( # `
 f )  =  2  /\  A  =/= 
B )  ->  (
f ( A ( V WalkOn  E ) B ) p  <->  f ( A ( V SPathOn  E
) B ) p ) )
51, 2, 3, 4syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( # `  f
)  =  2  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B ) )  ->  (
f ( A ( V WalkOn  E ) B ) p  <->  f ( A ( V SPathOn  E
) B ) p ) )
65bicomd 201 . . . . . . . . 9  |-  ( ( ( # `  f
)  =  2  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B ) )  ->  (
f ( A ( V SPathOn  E ) B ) p  <->  f ( A ( V WalkOn  E ) B ) p ) )
76ex 434 . . . . . . . 8  |-  ( (
# `  f )  =  2  ->  (
( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  ( f
( A ( V SPathOn  E ) B ) p  <->  f ( A ( V WalkOn  E ) B ) p ) ) )
87adantr 465 . . . . . . 7  |-  ( ( ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) )  ->  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B )  ->  (
f ( A ( V SPathOn  E ) B ) p  <->  f ( A ( V WalkOn  E ) B ) p ) ) )
98com12 31 . . . . . 6  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( (
( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) )  ->  ( f
( A ( V SPathOn  E ) B ) p  <->  f ( A ( V WalkOn  E ) B ) p ) ) )
109pm5.32rd 640 . . . . 5  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( (
f ( A ( V SPathOn  E ) B ) p  /\  ( (
# `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) )  <->  ( f
( A ( V WalkOn  E ) B ) p  /\  ( (
# `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) ) ) )
11 3anass 977 . . . . 5  |-  ( ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) )  <->  ( f
( A ( V SPathOn  E ) B ) p  /\  ( (
# `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) ) )
12 3anass 977 . . . . 5  |-  ( ( f ( A ( V WalkOn  E ) B ) p  /\  ( # `
 f )  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) )  <->  ( f ( A ( V WalkOn  E
) B ) p  /\  ( ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) ) )
1310, 11, 123bitr4g 288 . . . 4  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( (
f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) )  <->  ( f
( A ( V WalkOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) ) )
14132exbidv 1717 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) )  <->  E. f E. p ( f ( A ( V WalkOn  E
) B ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) ) )
1514rabbidv 3101 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) B ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) }  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V WalkOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) } )
16 usgrav 24465 . . . 4  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
17 2spthonot 24993 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( A ( V 2SPathOnOt  E ) B )  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) } )
1816, 17sylan 471 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( A
( V 2SPathOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) } )
19183adant3 1016 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( A
( V 2SPathOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) } )
20 2wlkonot 24992 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( A ( V 2WalksOnOt  E ) B )  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V WalkOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) } )
2116, 20sylan 471 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( A
( V 2WalksOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V WalkOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) } )
22213adant3 1016 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( A
( V 2WalksOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V WalkOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) } )
2315, 19, 223eqtr4d 2508 1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( A
( V 2SPathOnOt  E ) B )  =  ( A ( V 2WalksOnOt  E ) B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   {crab 2811   _Vcvv 3109   class class class wbr 4456    X. cxp 5006   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   1c1 9510   2c2 10606   #chash 12408   USGrph cusg 24457   WalkOn cwlkon 24629   SPathOn cspthon 24632   2WalksOnOt c2wlkonot 24982   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-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:  usg2spthonot  25015  usg2spthonot0  25016  frg2spot1  25185
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