Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eupai Unicode version

Theorem eupai 23898
Description: Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
eupai  |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  (
( ( # `  F
)  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) )
Distinct variable groups:    A, k    k, E    k, F    P, k    k, V

Proof of Theorem eupai
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 fndm 5359 . . . . 5  |-  ( E  Fn  A  ->  dom  E  =  A )
2 iseupa 23896 . . . . 5  |-  ( dom 
E  =  A  -> 
( F ( V EulPaths  E ) P  <->  ( V UMGrph  E  /\  E. n  e. 
NN0  ( F :
( 1 ... n
)
-1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n
) ( E `  ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) ) )
31, 2syl 15 . . . 4  |-  ( E  Fn  A  ->  ( F ( V EulPaths  E ) P  <->  ( V UMGrph  E  /\  E. n  e.  NN0  ( F : ( 1 ... n ) -1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n ) ( E `  ( F `
 k ) )  =  { ( P `
 ( k  - 
1 ) ) ,  ( P `  k
) } ) ) ) )
43biimpac 472 . . 3  |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  ( V UMGrph  E  /\  E. n  e.  NN0  ( F :
( 1 ... n
)
-1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n
) ( E `  ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) )
54simprd 449 . 2  |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  E. n  e.  NN0  ( F :
( 1 ... n
)
-1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n
) ( E `  ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) )
6 f1ofn 5489 . . . . . . . . . . . . . 14  |-  ( F : ( 1 ... n ) -1-1-onto-> A  ->  F  Fn  ( 1 ... n
) )
76ad2antll 709 . . . . . . . . . . . . 13  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  F  Fn  ( 1 ... n
) )
8 fzfid 11051 . . . . . . . . . . . . 13  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 1 ... n )  e. 
Fin )
9 fndmeng 6953 . . . . . . . . . . . . 13  |-  ( ( F  Fn  ( 1 ... n )  /\  ( 1 ... n
)  e.  Fin )  ->  ( 1 ... n
)  ~~  F )
107, 8, 9syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 1 ... n )  ~~  F )
11 enfi 7095 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... n ) 
~~  F  ->  (
( 1 ... n
)  e.  Fin  <->  F  e.  Fin ) )
1210, 11syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( (
1 ... n )  e. 
Fin 
<->  F  e.  Fin )
)
138, 12mpbid 201 . . . . . . . . . . . . 13  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  F  e.  Fin )
14 hashen 11362 . . . . . . . . . . . . 13  |-  ( ( ( 1 ... n
)  e.  Fin  /\  F  e.  Fin )  ->  ( ( # `  (
1 ... n ) )  =  ( # `  F
)  <->  ( 1 ... n )  ~~  F
) )
158, 13, 14syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( ( # `
 ( 1 ... n ) )  =  ( # `  F
)  <->  ( 1 ... n )  ~~  F
) )
1610, 15mpbird 223 . . . . . . . . . . 11  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  (
1 ... n ) )  =  ( # `  F
) )
17 hashfz1 11361 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  ( # `  ( 1 ... n
) )  =  n )
1817ad2antrl 708 . . . . . . . . . . 11  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  (
1 ... n ) )  =  n )
1916, 18eqtr3d 2330 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  F
)  =  n )
20 simprl 732 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  n  e.  NN0 )
2119, 20eqeltrd 2370 . . . . . . . . 9  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  F
)  e.  NN0 )
2221a1d 22 . . . . . . . 8  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( P : ( 0 ... n ) --> V  -> 
( # `  F )  e.  NN0 ) )
23 simprr 733 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  F :
( 1 ... n
)
-1-1-onto-> A )
2419oveq2d 5890 . . . . . . . . . . 11  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 1 ... ( # `  F
) )  =  ( 1 ... n ) )
25 f1oeq2 5480 . . . . . . . . . . 11  |-  ( ( 1 ... ( # `  F ) )  =  ( 1 ... n
)  ->  ( F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  <->  F : ( 1 ... n ) -1-1-onto-> A ) )
2624, 25syl 15 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  <->  F : ( 1 ... n ) -1-1-onto-> A ) )
2723, 26mpbird 223 . . . . . . . . 9  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  F :
( 1 ... ( # `
 F ) ) -1-1-onto-> A )
2827a1d 22 . . . . . . . 8  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( P : ( 0 ... n ) --> V  ->  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A ) )
2919oveq2d 5890 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 0 ... ( # `  F
) )  =  ( 0 ... n ) )
3029feq2d 5396 . . . . . . . . 9  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... n
) --> V ) )
3130biimprd 214 . . . . . . . 8  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( P : ( 0 ... n ) --> V  ->  P : ( 0 ... ( # `  F
) ) --> V ) )
3222, 28, 313jcad 1133 . . . . . . 7  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( P : ( 0 ... n ) --> V  -> 
( ( # `  F
)  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V ) ) )
3324raleqdv 2755 . . . . . . . 8  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) }  <->  A. k  e.  ( 1 ... n ) ( E `  ( F `  k )
)  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) )
3433biimprd 214 . . . . . . 7  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( A. k  e.  ( 1 ... n ) ( E `  ( F `
 k ) )  =  { ( P `
 ( k  - 
1 ) ) ,  ( P `  k
) }  ->  A. k  e.  ( 1 ... ( # `
 F ) ) ( E `  ( F `  k )
)  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) )
3532, 34anim12d 546 . . . . . 6  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( ( P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n ) ( E `  ( F `
 k ) )  =  { ( P `
 ( k  - 
1 ) ) ,  ( P `  k
) } )  -> 
( ( ( # `  F )  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) )
3635exp3a 425 . . . . 5  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( P : ( 0 ... n ) --> V  -> 
( A. k  e.  ( 1 ... n
) ( E `  ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) }  ->  ( ( ( # `  F
)  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) ) )
3736expr 598 . . . 4  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  n  e.  NN0 )  ->  ( F :
( 1 ... n
)
-1-1-onto-> A  ->  ( P :
( 0 ... n
) --> V  ->  ( A. k  e.  (
1 ... n ) ( E `  ( F `
 k ) )  =  { ( P `
 ( k  - 
1 ) ) ,  ( P `  k
) }  ->  (
( ( # `  F
)  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) ) ) )
38373impd 1165 . . 3  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  n  e.  NN0 )  ->  ( ( F : ( 1 ... n ) -1-1-onto-> A  /\  P :
( 0 ... n
) --> V  /\  A. k  e.  ( 1 ... n ) ( E `  ( F `
 k ) )  =  { ( P `
 ( k  - 
1 ) ) ,  ( P `  k
) } )  -> 
( ( ( # `  F )  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) )
3938rexlimdva 2680 . 2  |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  ( E. n  e.  NN0  ( F : ( 1 ... n ) -1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n ) ( E `  ( F `
 k ) )  =  { ( P `
 ( k  - 
1 ) ) ,  ( P `  k
) } )  -> 
( ( ( # `  F )  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) )
405, 39mpd 14 1  |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  (
( ( # `  F
)  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {cpr 3654   class class class wbr 4039   dom cdm 4705    Fn wfn 5266   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    ~~ cen 6876   Fincfn 6879   0cc0 8753   1c1 8754    - cmin 9053   NN0cn0 9981   ...cfz 10798   #chash 11353   UMGrph cumg 23875   EulPaths ceup 23876
This theorem is referenced by:  eupacl  23899  eupaf1o  23900  eupapf  23902  eupaseg  23903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354  df-umgra 23878  df-eupa 23879
  Copyright terms: Public domain W3C validator