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Theorem constr3pth 25064
Description: Construction of a path from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
Hypotheses
Ref Expression
constr3cycl.f  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. ,  <. 2 ,  ( `' E `  { C ,  A } ) >. }
constr3cycl.p  |-  P  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  A >. } )
Assertion
Ref Expression
constr3pth  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  F
( V Paths  E ) P )

Proof of Theorem constr3pth
StepHypRef Expression
1 usgrav 24742 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 constr3cycl.f . . . . 5  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. ,  <. 2 ,  ( `' E `  { C ,  A } ) >. }
3 constr3cycl.p . . . . 5  |-  P  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  A >. } )
42, 3constr3lem1 25049 . . . 4  |-  ( F  e.  _V  /\  P  e.  _V )
5 simpr 459 . . . . . . . . . . 11  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  V USGrph  E )  ->  V USGrph  E )
65ad2antrr 724 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  V USGrph  E )
7 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )
8 simpr 459 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )
92, 3constr3trl 25063 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  F
( V Trails  E ) P )
106, 7, 8, 9syl3anc 1230 . . . . . . . . 9  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  F ( V Trails  E ) P )
11 3cycl3dv 25046 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( A  =/= 
B  /\  B  =/=  C  /\  C  =/=  A
) )
1211ex 432 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  -> 
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) ) )
1312ad2antlr 725 . . . . . . . . . . . 12  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  -> 
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) ) )
1413imp 427 . . . . . . . . . . 11  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( A  =/= 
B  /\  B  =/=  C  /\  C  =/=  A
) )
1514simp2d 1010 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  B  =/=  C
)
162, 3constr3pthlem2 25060 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  B  =/=  C )  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
177, 15, 16syl2anc 659 . . . . . . . . 9  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  Fun  `' ( P  |`  ( 1..^ (
# `  F )
) ) )
182, 3constr3pthlem3 25061 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C  /\  C  =/=  A ) )  ->  (
( P " {
0 ,  ( # `  F ) } )  i^i  ( P "
( 1..^ ( # `  F ) ) ) )  =  (/) )
197, 14, 18syl2anc 659 . . . . . . . . 9  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )
2010, 17, 193jca 1177 . . . . . . . 8  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) )
2120ex 432 . . . . . . 7  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  -> 
( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
22 ispth 24974 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( 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 )
) ) )  =  (/) ) ) )
2322ad2antrr 724 . . . . . . 7  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( F
( V Paths  E ) P 
<->  ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
2421, 23sylibrd 234 . . . . . 6  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  V USGrph  E )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  ->  F ( V Paths  E
) P ) )
2524ex 432 . . . . 5  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  V USGrph  E )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  ->  F ( V Paths  E
) P ) ) )
2625ex 432 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  (
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  ->  F ( V Paths  E ) P ) ) ) )
274, 26mpan2 669 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V USGrph  E  -> 
( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  (
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  ->  F ( V Paths  E ) P ) ) ) )
281, 27mpcom 34 . 2  |-  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  (
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  ->  F ( V Paths  E ) P ) ) )
29283imp 1191 1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  F
( V Paths  E ) P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3058    u. cun 3411    i^i cin 3412   (/)c0 3737   {cpr 3973   {ctp 3975   <.cop 3977   class class class wbr 4394   `'ccnv 4821   ran crn 4823    |` cres 4824   "cima 4825   Fun wfun 5562   ` cfv 5568  (class class class)co 6277   0cc0 9521   1c1 9522   2c2 10625   3c3 10626  ..^cfzo 11852   #chash 12450   USGrph cusg 24734   Trails ctrail 24903   Paths cpath 24904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-usgra 24737  df-wlk 24912  df-trail 24913  df-pth 24914
This theorem is referenced by:  constr3cycl  25065
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