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

Theorem eupai 23593
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 5515 . . . . 5  |-  ( E  Fn  A  ->  dom  E  =  A )
2 iseupa 23591 . . . . 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 16 . . . 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 486 . . 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 463 . 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 5647 . . . . . . . . . . . . . 14  |-  ( F : ( 1 ... n ) -1-1-onto-> A  ->  F  Fn  ( 1 ... n
) )
76ad2antll 728 . . . . . . . . . . . . 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 11800 . . . . . . . . . . . . 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 7391 . . . . . . . . . . . . 13  |-  ( ( F  Fn  ( 1 ... n )  /\  ( 1 ... n
)  e.  Fin )  ->  ( 1 ... n
)  ~~  F )
107, 8, 9syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 1 ... n )  ~~  F )
11 enfi 7534 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... n ) 
~~  F  ->  (
( 1 ... n
)  e.  Fin  <->  F  e.  Fin ) )
1210, 11syl 16 . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . 13  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  F  e.  Fin )
14 hashen 12123 . . . . . . . . . . . . 13  |-  ( ( ( 1 ... n
)  e.  Fin  /\  F  e.  Fin )  ->  ( ( # `  (
1 ... n ) )  =  ( # `  F
)  <->  ( 1 ... n )  ~~  F
) )
158, 13, 14syl2anc 661 . . . . . . . . . . . 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 232 . . . . . . . . . . 11  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  (
1 ... n ) )  =  ( # `  F
) )
17 hashfz1 12122 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  ( # `  ( 1 ... n
) )  =  n )
1817ad2antrl 727 . . . . . . . . . . 11  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  (
1 ... n ) )  =  n )
1916, 18eqtr3d 2477 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  F
)  =  n )
20 simprl 755 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  n  e.  NN0 )
2119, 20eqeltrd 2517 . . . . . . . . 9  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  F
)  e.  NN0 )
2221a1d 25 . . . . . . . 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 756 . . . . . . . . . 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 6112 . . . . . . . . . . 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 5638 . . . . . . . . . . 11  |-  ( ( 1 ... ( # `  F ) )  =  ( 1 ... n
)  ->  ( F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  <->  F : ( 1 ... n ) -1-1-onto-> A ) )
2624, 25syl 16 . . . . . . . . . 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 232 . . . . . . . . 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 25 . . . . . . . 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 6112 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 0 ... ( # `  F
) )  =  ( 0 ... n ) )
3029feq2d 5552 . . . . . . . . 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 223 . . . . . . . 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 1169 . . . . . . 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 2928 . . . . . . . 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 223 . . . . . . 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 563 . . . . . 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 ) } ) ) )
3635expd 436 . . . . 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 615 . . . 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 1201 . . 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 2846 . 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 15 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   {cpr 3884   class class class wbr 4297   dom cdm 4845    Fn wfn 5418   -->wf 5419   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096    ~~ cen 7312   Fincfn 7315   0cc0 9287   1c1 9288    - cmin 9600   NN0cn0 10584   ...cfz 11442   #chash 12108   UMGrph cumg 23251   EulPaths ceup 23588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-er 7106  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-hash 12109  df-umgra 23252  df-eupa 23589
This theorem is referenced by:  eupacl  23595  eupaf1o  23596  eupapf  23598  eupaseg  23599
  Copyright terms: Public domain W3C validator