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Theorem eupath 23905
Description: A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.)
Assertion
Ref Expression
eupath  |-  ( ( V EulPaths  E )  =/=  (/)  ->  ( # `
 { x  e.  V  |  -.  2  ||  ( ( V VDeg  E
) `  x ) } )  e.  {
0 ,  2 } )
Distinct variable groups:    x, E    x, V

Proof of Theorem eupath
Dummy variables  f  p  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 releupa 23880 . . . . 5  |-  Rel  ( V EulPaths  E )
2 reldm0 4896 . . . . 5  |-  ( Rel  ( V EulPaths  E )  ->  ( ( V EulPaths  E )  =  (/)  <->  dom  ( V EulPaths  E )  =  (/) ) )
31, 2ax-mp 8 . . . 4  |-  ( ( V EulPaths  E )  =  (/)  <->  dom  ( V EulPaths  E )  =  (/) )
43necon3bii 2478 . . 3  |-  ( ( V EulPaths  E )  =/=  (/)  <->  dom  ( V EulPaths  E )  =/=  (/) )
5 n0 3464 . . 3  |-  ( dom  ( V EulPaths  E )  =/=  (/)  <->  E. f  f  e. 
dom  ( V EulPaths  E ) )
64, 5bitri 240 . 2  |-  ( ( V EulPaths  E )  =/=  (/)  <->  E. f 
f  e.  dom  ( V EulPaths  E ) )
7 vex 2791 . . . . 5  |-  f  e. 
_V
87eldm 4876 . . . 4  |-  ( f  e.  dom  ( V EulPaths  E )  <->  E. p  f ( V EulPaths  E ) p )
9 eupagra 23882 . . . . . . . . 9  |-  ( f ( V EulPaths  E )
p  ->  V UMGrph  E )
10 umgraf2 23869 . . . . . . . . 9  |-  ( V UMGrph  E  ->  E : dom  E --> { y  e.  ( ~P V  \  { (/)
} )  |  (
# `  y )  <_  2 } )
11 ffn 5389 . . . . . . . . 9  |-  ( E : dom  E --> { y  e.  ( ~P V  \  { (/) } )  |  ( # `  y
)  <_  2 }  ->  E  Fn  dom  E
)
129, 10, 113syl 18 . . . . . . . 8  |-  ( f ( V EulPaths  E )
p  ->  E  Fn  dom  E )
13 id 19 . . . . . . . 8  |-  ( f ( V EulPaths  E )
p  ->  f ( V EulPaths  E ) p )
1412, 13eupath2 23904 . . . . . . 7  |-  ( f ( V EulPaths  E )
p  ->  { x  e.  V  |  -.  2  ||  ( ( V VDeg 
E ) `  x
) }  =  if ( ( p ` 
0 )  =  ( p `  ( # `  f ) ) ,  (/) ,  { ( p `
 0 ) ,  ( p `  ( # `
 f ) ) } ) )
1514fveq2d 5529 . . . . . 6  |-  ( f ( V EulPaths  E )
p  ->  ( # `  {
x  e.  V  |  -.  2  ||  ( ( V VDeg  E ) `  x ) } )  =  ( # `  if ( ( p ` 
0 )  =  ( p `  ( # `  f ) ) ,  (/) ,  { ( p `
 0 ) ,  ( p `  ( # `
 f ) ) } ) ) )
16 fveq2 5525 . . . . . . . 8  |-  ( (/)  =  if ( ( p `
 0 )  =  ( p `  ( # `
 f ) ) ,  (/) ,  { ( p `  0 ) ,  ( p `  ( # `  f ) ) } )  -> 
( # `  (/) )  =  ( # `  if ( ( p ` 
0 )  =  ( p `  ( # `  f ) ) ,  (/) ,  { ( p `
 0 ) ,  ( p `  ( # `
 f ) ) } ) ) )
1716eleq1d 2349 . . . . . . 7  |-  ( (/)  =  if ( ( p `
 0 )  =  ( p `  ( # `
 f ) ) ,  (/) ,  { ( p `  0 ) ,  ( p `  ( # `  f ) ) } )  -> 
( ( # `  (/) )  e. 
{ 0 ,  2 }  <->  ( # `  if ( ( p ` 
0 )  =  ( p `  ( # `  f ) ) ,  (/) ,  { ( p `
 0 ) ,  ( p `  ( # `
 f ) ) } ) )  e. 
{ 0 ,  2 } ) )
18 fveq2 5525 . . . . . . . 8  |-  ( { ( p `  0
) ,  ( p `
 ( # `  f
) ) }  =  if ( ( p ` 
0 )  =  ( p `  ( # `  f ) ) ,  (/) ,  { ( p `
 0 ) ,  ( p `  ( # `
 f ) ) } )  ->  ( # `
 { ( p `
 0 ) ,  ( p `  ( # `
 f ) ) } )  =  (
# `  if (
( p `  0
)  =  ( p `
 ( # `  f
) ) ,  (/) ,  { ( p ` 
0 ) ,  ( p `  ( # `  f ) ) } ) ) )
1918eleq1d 2349 . . . . . . 7  |-  ( { ( p `  0
) ,  ( p `
 ( # `  f
) ) }  =  if ( ( p ` 
0 )  =  ( p `  ( # `  f ) ) ,  (/) ,  { ( p `
 0 ) ,  ( p `  ( # `
 f ) ) } )  ->  (
( # `  { ( p `  0 ) ,  ( p `  ( # `  f ) ) } )  e. 
{ 0 ,  2 }  <->  ( # `  if ( ( p ` 
0 )  =  ( p `  ( # `  f ) ) ,  (/) ,  { ( p `
 0 ) ,  ( p `  ( # `
 f ) ) } ) )  e. 
{ 0 ,  2 } ) )
20 hash0 11355 . . . . . . . . 9  |-  ( # `  (/) )  =  0
21 c0ex 8832 . . . . . . . . . 10  |-  0  e.  _V
2221prid1 3734 . . . . . . . . 9  |-  0  e.  { 0 ,  2 }
2320, 22eqeltri 2353 . . . . . . . 8  |-  ( # `  (/) )  e.  {
0 ,  2 }
2423a1i 10 . . . . . . 7  |-  ( ( f ( V EulPaths  E ) p  /\  ( p `
 0 )  =  ( p `  ( # `
 f ) ) )  ->  ( # `  (/) )  e. 
{ 0 ,  2 } )
25 simpr 447 . . . . . . . . . 10  |-  ( ( f ( V EulPaths  E ) p  /\  -.  (
p `  0 )  =  ( p `  ( # `  f ) ) )  ->  -.  ( p `  0
)  =  ( p `
 ( # `  f
) ) )
26 df-ne 2448 . . . . . . . . . 10  |-  ( ( p `  0 )  =/=  ( p `  ( # `  f ) )  <->  -.  ( p `  0 )  =  ( p `  ( # `
 f ) ) )
2725, 26sylibr 203 . . . . . . . . 9  |-  ( ( f ( V EulPaths  E ) p  /\  -.  (
p `  0 )  =  ( p `  ( # `  f ) ) )  ->  (
p `  0 )  =/=  ( p `  ( # `
 f ) ) )
28 fvex 5539 . . . . . . . . . 10  |-  ( p `
 0 )  e. 
_V
29 fvex 5539 . . . . . . . . . 10  |-  ( p `
 ( # `  f
) )  e.  _V
30 hashprg 11368 . . . . . . . . . 10  |-  ( ( ( p `  0
)  e.  _V  /\  ( p `  ( # `
 f ) )  e.  _V )  -> 
( ( p ` 
0 )  =/=  (
p `  ( # `  f
) )  <->  ( # `  {
( p `  0
) ,  ( p `
 ( # `  f
) ) } )  =  2 ) )
3128, 29, 30mp2an 653 . . . . . . . . 9  |-  ( ( p `  0 )  =/=  ( p `  ( # `  f ) )  <->  ( # `  {
( p `  0
) ,  ( p `
 ( # `  f
) ) } )  =  2 )
3227, 31sylib 188 . . . . . . . 8  |-  ( ( f ( V EulPaths  E ) p  /\  -.  (
p `  0 )  =  ( p `  ( # `  f ) ) )  ->  ( # `
 { ( p `
 0 ) ,  ( p `  ( # `
 f ) ) } )  =  2 )
33 2cn 9816 . . . . . . . . . 10  |-  2  e.  CC
3433elexi 2797 . . . . . . . . 9  |-  2  e.  _V
3534prid2 3735 . . . . . . . 8  |-  2  e.  { 0 ,  2 }
3632, 35syl6eqel 2371 . . . . . . 7  |-  ( ( f ( V EulPaths  E ) p  /\  -.  (
p `  0 )  =  ( p `  ( # `  f ) ) )  ->  ( # `
 { ( p `
 0 ) ,  ( p `  ( # `
 f ) ) } )  e.  {
0 ,  2 } )
3717, 19, 24, 36ifbothda 3595 . . . . . 6  |-  ( f ( V EulPaths  E )
p  ->  ( # `  if ( ( p ` 
0 )  =  ( p `  ( # `  f ) ) ,  (/) ,  { ( p `
 0 ) ,  ( p `  ( # `
 f ) ) } ) )  e. 
{ 0 ,  2 } )
3815, 37eqeltrd 2357 . . . . 5  |-  ( f ( V EulPaths  E )
p  ->  ( # `  {
x  e.  V  |  -.  2  ||  ( ( V VDeg  E ) `  x ) } )  e.  { 0 ,  2 } )
3938exlimiv 1666 . . . 4  |-  ( E. p  f ( V EulPaths  E ) p  -> 
( # `  { x  e.  V  |  -.  2  ||  ( ( V VDeg 
E ) `  x
) } )  e. 
{ 0 ,  2 } )
408, 39sylbi 187 . . 3  |-  ( f  e.  dom  ( V EulPaths  E )  ->  ( # `
 { x  e.  V  |  -.  2  ||  ( ( V VDeg  E
) `  x ) } )  e.  {
0 ,  2 } )
4140exlimiv 1666 . 2  |-  ( E. f  f  e.  dom  ( V EulPaths  E )  -> 
( # `  { x  e.  V  |  -.  2  ||  ( ( V VDeg 
E ) `  x
) } )  e. 
{ 0 ,  2 } )
426, 41sylbi 187 1  |-  ( ( V EulPaths  E )  =/=  (/)  ->  ( # `
 { x  e.  V  |  -.  2  ||  ( ( V VDeg  E
) `  x ) } )  e.  {
0 ,  2 } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788    \ cdif 3149   (/)c0 3455   ifcif 3565   ~Pcpw 3625   {csn 3640   {cpr 3641   class class class wbr 4023   dom cdm 4689   Rel wrel 4694    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737    <_ cle 8868   2c2 9795   #chash 11337    || cdivides 12531   UMGrph cumg 23860   EulPaths ceup 23861   VDeg cvdg 23862
This theorem is referenced by:  konigsberg  23911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-prm 12759  df-umgra 23863  df-eupa 23864  df-vdgr 23865
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