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Theorem constr2pth 24730
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.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
Hypotheses
Ref Expression
2trlY.i  |-  ( I  e.  U  /\  J  e.  W )
2trlY.f  |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }
2trlY.p  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
Assertion
Ref Expression
constr2pth  |-  ( ( ( 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 } )  ->  F ( V Paths  E
) P ) )

Proof of Theorem constr2pth
StepHypRef Expression
1 2trlY.i . . . . . 6  |-  ( I  e.  U  /\  J  e.  W )
2 2trlY.f . . . . . 6  |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }
3 2trlY.p . . . . . 6  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
41, 2, 3constr2trl 24728 . . . . 5  |-  ( ( ( 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 } )  ->  F ( V Trails  E
) P ) )
543adant3 1016 . . . 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 } )  ->  F ( V Trails  E
) P ) )
65imp 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 } ) )  ->  F ( V Trails  E
) P )
732pthlem1 24724 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  Fun  `' ( P  |`  ( 1..^ 2 ) ) )
81, 22trllemA 24679 . . . . . . 7  |-  ( # `  F )  =  2
9 oveq2 6304 . . . . . . . . . 10  |-  ( (
# `  F )  =  2  ->  (
1..^ ( # `  F
) )  =  ( 1..^ 2 ) )
109reseq2d 5283 . . . . . . . . 9  |-  ( (
# `  F )  =  2  ->  ( P  |`  ( 1..^ (
# `  F )
) )  =  ( P  |`  ( 1..^ 2 ) ) )
1110cnveqd 5188 . . . . . . . 8  |-  ( (
# `  F )  =  2  ->  `' ( P  |`  ( 1..^ ( # `  F
) ) )  =  `' ( P  |`  ( 1..^ 2 ) ) )
1211funeqd 5615 . . . . . . 7  |-  ( (
# `  F )  =  2  ->  ( Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  <->  Fun  `' ( P  |`  ( 1..^ 2 ) ) ) )
138, 12ax-mp 5 . . . . . 6  |-  ( Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  <->  Fun  `' ( P  |`  ( 1..^ 2 ) ) )
147, 13sylibr 212 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F ) ) ) )
15143ad2ant2 1018 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
1615adantr 465 . . 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 } ) )  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
1732pthlem2 24725 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C ) )  -> 
( ( P " { 0 ,  2 } )  i^i  ( P " ( 1..^ 2 ) ) )  =  (/) )
18 preq2 4112 . . . . . . . . . . 11  |-  ( (
# `  F )  =  2  ->  { 0 ,  ( # `  F
) }  =  {
0 ,  2 } )
1918imaeq2d 5347 . . . . . . . . . 10  |-  ( (
# `  F )  =  2  ->  ( P " { 0 ,  ( # `  F
) } )  =  ( P " {
0 ,  2 } ) )
209imaeq2d 5347 . . . . . . . . . 10  |-  ( (
# `  F )  =  2  ->  ( P " ( 1..^ (
# `  F )
) )  =  ( P " ( 1..^ 2 ) ) )
2119, 20ineq12d 3697 . . . . . . . . 9  |-  ( (
# `  F )  =  2  ->  (
( P " {
0 ,  ( # `  F ) } )  i^i  ( P "
( 1..^ ( # `  F ) ) ) )  =  ( ( P " { 0 ,  2 } )  i^i  ( P "
( 1..^ 2 ) ) ) )
2221eqeq1d 2459 . . . . . . . 8  |-  ( (
# `  F )  =  2  ->  (
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) 
<->  ( ( P " { 0 ,  2 } )  i^i  ( P " ( 1..^ 2 ) ) )  =  (/) ) )
238, 22ax-mp 5 . . . . . . 7  |-  ( ( ( P " {
0 ,  ( # `  F ) } )  i^i  ( P "
( 1..^ ( # `  F ) ) ) )  =  (/)  <->  ( ( P " { 0 ,  2 } )  i^i  ( P " (
1..^ 2 ) ) )  =  (/) )
2417, 23sylibr 212 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C ) )  -> 
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )
25243adantr2 1156 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  (
( P " {
0 ,  ( # `  F ) } )  i^i  ( P "
( 1..^ ( # `  F ) ) ) )  =  (/) )
26253adant1 1014 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )
2726adantr 465 . . 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 } ) )  -> 
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )
28 prex 4698 . . . . . . . . 9  |-  { <. 0 ,  I >. , 
<. 1 ,  J >. }  e.  _V
292, 28eqeltri 2541 . . . . . . . 8  |-  F  e. 
_V
30 tpex 6598 . . . . . . . . 9  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V
313, 30eqeltri 2541 . . . . . . . 8  |-  P  e. 
_V
3229, 31pm3.2i 455 . . . . . . 7  |-  ( F  e.  _V  /\  P  e.  _V )
3332jctr 542 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
34333ad2ant1 1017 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3534adantr 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 } ) )  -> 
( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
36 ispth 24697 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  _V  /\  P  e. 
_V ) )  -> 
( F ( V Paths 
E ) P  <->  ( F
( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
3735, 36syl 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 } ) )  -> 
( F ( V Paths 
E ) P  <->  ( F
( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
386, 16, 27, 37mpbir3and 1179 . 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 } ) )  ->  F ( V Paths  E
) P )
3938ex 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 } )  ->  F ( V Paths  E
) P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    i^i cin 3470   (/)c0 3793   {cpr 4034   {ctp 4036   <.cop 4038   class class class wbr 4456   `'ccnv 5007    |` cres 5010   "cima 5011   Fun wfun 5588   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510   2c2 10606  ..^cfzo 11821   #chash 12408   Trails ctrail 24626   Paths cpath 24627
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-wlk 24635  df-trail 24636  df-pth 24637
This theorem is referenced by:  2pthon  24731
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