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

Theorem pthdepisspth 23645
Description: A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.)
Assertion
Ref Expression
pthdepisspth  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  F ( V SPaths  E ) P )

Proof of Theorem pthdepisspth
Dummy variables  e 
f  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pth 23589 . . . . 5  |- Paths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } )
21brovmpt2ex 6854 . . . 4  |-  ( F ( V Paths  E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 ispth 23639 . . . . 5  |-  ( ( ( 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 )
) ) )  =  (/) ) ) )
4 simp-4l 765 . . . . . . . . 9  |-  ( ( ( ( ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  F ( V Trails  E ) P )
5 trliswlk 23610 . . . . . . . . . . . 12  |-  ( F ( V Trails  E ) P  ->  F ( V Walks  E ) P )
6 2mwlk 23599 . . . . . . . . . . . 12  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
75, 6syl 16 . . . . . . . . . . 11  |-  ( F ( V Trails  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
8 lencl 12370 . . . . . . . . . . . . . 14  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  NN0 )
98ad5antr 733 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( # `  F
)  e.  NN0 )
10 simp-5r 768 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  P :
( 0 ... ( # `
 F ) ) --> V )
11 simp-4r 766 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
12 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )
1310, 11, 123jca 1168 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) ) )
14 simpllr 758 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( ( P " { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )
15 injresinj 11759 . . . . . . . . . . . . 13  |-  ( (
# `  F )  e.  NN0  ->  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  (
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/)  ->  Fun  `' P
) ) )
169, 13, 14, 15syl3c 61 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  Fun  `' P
)
1716exp31 604 . . . . . . . . . . 11  |-  ( ( ( ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  -> 
( ( P ` 
0 )  =/=  ( P `  ( # `  F
) )  ->  Fun  `' P ) ) )
187, 17sylanl1 650 . . . . . . . . . 10  |-  ( ( ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  -> 
( ( P ` 
0 )  =/=  ( P `  ( # `  F
) )  ->  Fun  `' P ) ) )
1918imp31 432 . . . . . . . . 9  |-  ( ( ( ( ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  Fun  `' P
)
204, 19jca 532 . . . . . . . 8  |-  ( ( ( ( ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( F
( V Trails  E ) P  /\  Fun  `' P
) )
2120exp31 604 . . . . . . 7  |-  ( ( ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  -> 
( ( P ` 
0 )  =/=  ( P `  ( # `  F
) )  ->  ( F ( V Trails  E
) P  /\  Fun  `' P ) ) ) )
22213impa 1183 . . . . . 6  |-  ( ( F ( V Trails  E
) P  /\  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )  ->  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( P `  0 )  =/=  ( P `  ( # `
 F ) )  ->  ( F ( V Trails  E ) P  /\  Fun  `' P
) ) ) )
2322com12 31 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( F ( V Trails  E
) P  /\  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )  ->  ( ( P `  0 )  =/=  ( P `  ( # `  F ) )  ->  ( F
( V Trails  E ) P  /\  Fun  `' P
) ) ) )
243, 23sylbid 215 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Paths  E ) P  ->  ( ( P `
 0 )  =/=  ( P `  ( # `
 F ) )  ->  ( F ( V Trails  E ) P  /\  Fun  `' P
) ) ) )
252, 24mpcom 36 . . 3  |-  ( F ( V Paths  E ) P  ->  ( ( P `  0 )  =/=  ( P `  ( # `
 F ) )  ->  ( F ( V Trails  E ) P  /\  Fun  `' P
) ) )
2625imp 429 . 2  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( F
( V Trails  E ) P  /\  Fun  `' P
) )
272adantr 465 . . 3  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
28 isspth 23640 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V SPaths  E ) P 
<->  ( F ( V Trails  E ) P  /\  Fun  `' P ) ) )
2927, 28syl 16 . 2  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( F
( V SPaths  E ) P 
<->  ( F ( V Trails  E ) P  /\  Fun  `' P ) ) )
3026, 29mpbird 232 1  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  F ( V SPaths  E ) P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078    i^i cin 3438   (/)c0 3748   {cpr 3990   class class class wbr 4403   `'ccnv 4950   dom cdm 4951    |` cres 4953   "cima 4954   Fun wfun 5523   -->wf 5525   ` cfv 5529  (class class class)co 6203   0cc0 9396   1c1 9397   NN0cn0 10693   ...cfz 11557  ..^cfzo 11668   #chash 12223  Word cword 12342   Walks cwalk 23577   Trails ctrail 23578   Paths cpath 23579   SPaths cspath 23580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-hash 12224  df-word 12350  df-wlk 23587  df-trail 23588  df-pth 23589  df-spth 23590
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator