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Theorem eupares 24183
Description: The restriction of an Eulerian path to an initial segment of the path forms an Eulerian path on the subgraph consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
Hypotheses
Ref Expression
eupares.g  |-  ( ph  ->  G ( V EulPaths  E ) P )
eupares.n  |-  ( ph  ->  N  e.  ( 0 ... ( # `  G
) ) )
eupares.f  |-  F  =  ( E  |`  ( G " ( 1 ... N ) ) )
eupares.h  |-  H  =  ( G  |`  (
1 ... N ) )
eupares.q  |-  Q  =  ( P  |`  (
0 ... N ) )
Assertion
Ref Expression
eupares  |-  ( ph  ->  H ( V EulPaths  F ) Q )

Proof of Theorem eupares
Dummy variables  k  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eupares.g . . . . 5  |-  ( ph  ->  G ( V EulPaths  E ) P )
2 eupagra 24166 . . . . 5  |-  ( G ( V EulPaths  E ) P  ->  V UMGrph  E )
31, 2syl 15 . . . 4  |-  ( ph  ->  V UMGrph  E )
4 umgrares 24160 . . . 4  |-  ( V UMGrph  E  ->  V UMGrph  ( E  |`  ( G " (
1 ... N ) ) ) )
53, 4syl 15 . . 3  |-  ( ph  ->  V UMGrph  ( E  |`  ( G " ( 1 ... N ) ) ) )
6 eupares.f . . 3  |-  F  =  ( E  |`  ( G " ( 1 ... N ) ) )
75, 6syl6breqr 4100 . 2  |-  ( ph  ->  V UMGrph  F )
8 eupares.n . . . 4  |-  ( ph  ->  N  e.  ( 0 ... ( # `  G
) ) )
9 elfznn0 10869 . . . 4  |-  ( N  e.  ( 0 ... ( # `  G
) )  ->  N  e.  NN0 )
108, 9syl 15 . . 3  |-  ( ph  ->  N  e.  NN0 )
11 umgraf2 24153 . . . . . . . . 9  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
123, 11syl 15 . . . . . . . 8  |-  ( ph  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
13 ffn 5427 . . . . . . . 8  |-  ( E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  ->  E  Fn  dom  E
)
1412, 13syl 15 . . . . . . 7  |-  ( ph  ->  E  Fn  dom  E
)
15 eupaf1o 24169 . . . . . . 7  |-  ( ( G ( V EulPaths  E ) P  /\  E  Fn  dom  E )  ->  G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E )
161, 14, 15syl2anc 642 . . . . . 6  |-  ( ph  ->  G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E )
17 f1of1 5509 . . . . . 6  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  G : ( 1 ... ( # `  G
) ) -1-1-> dom  E
)
1816, 17syl 15 . . . . 5  |-  ( ph  ->  G : ( 1 ... ( # `  G
) ) -1-1-> dom  E
)
19 elfzuz3 10842 . . . . . . 7  |-  ( N  e.  ( 0 ... ( # `  G
) )  ->  ( # `
 G )  e.  ( ZZ>= `  N )
)
208, 19syl 15 . . . . . 6  |-  ( ph  ->  ( # `  G
)  e.  ( ZZ>= `  N ) )
21 fzss2 10878 . . . . . 6  |-  ( (
# `  G )  e.  ( ZZ>= `  N )  ->  ( 1 ... N
)  C_  ( 1 ... ( # `  G
) ) )
2220, 21syl 15 . . . . 5  |-  ( ph  ->  ( 1 ... N
)  C_  ( 1 ... ( # `  G
) ) )
23 f1ores 5525 . . . . 5  |-  ( ( G : ( 1 ... ( # `  G
) ) -1-1-> dom  E  /\  ( 1 ... N
)  C_  ( 1 ... ( # `  G
) ) )  -> 
( G  |`  (
1 ... N ) ) : ( 1 ... N ) -1-1-onto-> ( G " (
1 ... N ) ) )
2418, 22, 23syl2anc 642 . . . 4  |-  ( ph  ->  ( G  |`  (
1 ... N ) ) : ( 1 ... N ) -1-1-onto-> ( G " (
1 ... N ) ) )
25 eupares.h . . . . 5  |-  H  =  ( G  |`  (
1 ... N ) )
26 f1oeq1 5501 . . . . 5  |-  ( H  =  ( G  |`  ( 1 ... N
) )  ->  ( H : ( 1 ... N ) -1-1-onto-> ( G " (
1 ... N ) )  <-> 
( G  |`  (
1 ... N ) ) : ( 1 ... N ) -1-1-onto-> ( G " (
1 ... N ) ) ) )
2725, 26ax-mp 8 . . . 4  |-  ( H : ( 1 ... N ) -1-1-onto-> ( G " (
1 ... N ) )  <-> 
( G  |`  (
1 ... N ) ) : ( 1 ... N ) -1-1-onto-> ( G " (
1 ... N ) ) )
2824, 27sylibr 203 . . 3  |-  ( ph  ->  H : ( 1 ... N ) -1-1-onto-> ( G
" ( 1 ... N ) ) )
29 eupapf 24171 . . . . . 6  |-  ( G ( V EulPaths  E ) P  ->  P : ( 0 ... ( # `  G ) ) --> V )
301, 29syl 15 . . . . 5  |-  ( ph  ->  P : ( 0 ... ( # `  G
) ) --> V )
31 fzss2 10878 . . . . . 6  |-  ( (
# `  G )  e.  ( ZZ>= `  N )  ->  ( 0 ... N
)  C_  ( 0 ... ( # `  G
) ) )
3220, 31syl 15 . . . . 5  |-  ( ph  ->  ( 0 ... N
)  C_  ( 0 ... ( # `  G
) ) )
33 fssres 5446 . . . . 5  |-  ( ( P : ( 0 ... ( # `  G
) ) --> V  /\  ( 0 ... N
)  C_  ( 0 ... ( # `  G
) ) )  -> 
( P  |`  (
0 ... N ) ) : ( 0 ... N ) --> V )
3430, 32, 33syl2anc 642 . . . 4  |-  ( ph  ->  ( P  |`  (
0 ... N ) ) : ( 0 ... N ) --> V )
35 eupares.q . . . . 5  |-  Q  =  ( P  |`  (
0 ... N ) )
3635feq1i 5421 . . . 4  |-  ( Q : ( 0 ... N ) --> V  <->  ( P  |`  ( 0 ... N
) ) : ( 0 ... N ) --> V )
3734, 36sylibr 203 . . 3  |-  ( ph  ->  Q : ( 0 ... N ) --> V )
381adantr 451 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  G
( V EulPaths  E ) P )
3922sselda 3214 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  ( 1 ... ( # `
 G ) ) )
40 eupaseg 24172 . . . . . 6  |-  ( ( G ( V EulPaths  E ) P  /\  k  e.  ( 1 ... ( # `
 G ) ) )  ->  ( E `  ( G `  k
) )  =  {
( P `  (
k  -  1 ) ) ,  ( P `
 k ) } )
4138, 39, 40syl2anc 642 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( E `  ( G `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } )
4225fveq1i 5564 . . . . . . . 8  |-  ( H `
 k )  =  ( ( G  |`  ( 1 ... N
) ) `  k
)
43 fvres 5580 . . . . . . . . 9  |-  ( k  e.  ( 1 ... N )  ->  (
( G  |`  (
1 ... N ) ) `
 k )  =  ( G `  k
) )
4443adantl 452 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
( G  |`  (
1 ... N ) ) `
 k )  =  ( G `  k
) )
4542, 44syl5eq 2360 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( H `  k )  =  ( G `  k ) )
4645fveq2d 5567 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( H `  k ) )  =  ( F `  ( G `  k )
) )
476fveq1i 5564 . . . . . . 7  |-  ( F `
 ( G `  k ) )  =  ( ( E  |`  ( G " ( 1 ... N ) ) ) `  ( G `
 k ) )
48 f1ofun 5512 . . . . . . . . . . 11  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  Fun  G )
4916, 48syl 15 . . . . . . . . . 10  |-  ( ph  ->  Fun  G )
50 f1of 5510 . . . . . . . . . . . 12  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  G : ( 1 ... ( # `  G
) ) --> dom  E
)
51 fdm 5431 . . . . . . . . . . . 12  |-  ( G : ( 1 ... ( # `  G
) ) --> dom  E  ->  dom  G  =  ( 1 ... ( # `  G ) ) )
5216, 50, 513syl 18 . . . . . . . . . . 11  |-  ( ph  ->  dom  G  =  ( 1 ... ( # `  G ) ) )
5322, 52sseqtr4d 3249 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... N
)  C_  dom  G )
54 funfvima2 5795 . . . . . . . . . 10  |-  ( ( Fun  G  /\  (
1 ... N )  C_  dom  G )  ->  (
k  e.  ( 1 ... N )  -> 
( G `  k
)  e.  ( G
" ( 1 ... N ) ) ) )
5549, 53, 54syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( k  e.  ( 1 ... N )  ->  ( G `  k )  e.  ( G " ( 1 ... N ) ) ) )
5655imp 418 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( G `  k )  e.  ( G " (
1 ... N ) ) )
57 fvres 5580 . . . . . . . 8  |-  ( ( G `  k )  e.  ( G "
( 1 ... N
) )  ->  (
( E  |`  ( G " ( 1 ... N ) ) ) `
 ( G `  k ) )  =  ( E `  ( G `  k )
) )
5856, 57syl 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
( E  |`  ( G " ( 1 ... N ) ) ) `
 ( G `  k ) )  =  ( E `  ( G `  k )
) )
5947, 58syl5eq 2360 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( G `  k ) )  =  ( E `  ( G `  k )
) )
6046, 59eqtrd 2348 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( H `  k ) )  =  ( E `  ( G `  k )
) )
61 elfznn 10866 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
6261adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  NN )
63 nnm1nn0 10052 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
6462, 63syl 15 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  NN0 )
65 nn0uz 10309 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
6664, 65syl6eleq 2406 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  ( ZZ>= `  0
) )
6762nncnd 9807 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  CC )
68 ax-1cn 8840 . . . . . . . . . 10  |-  1  e.  CC
69 npcan 9105 . . . . . . . . . 10  |-  ( ( k  e.  CC  /\  1  e.  CC )  ->  ( ( k  - 
1 )  +  1 )  =  k )
7067, 68, 69sylancl 643 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
( k  -  1 )  +  1 )  =  k )
71 1e0p1 10199 . . . . . . . . . . . 12  |-  1  =  ( 0  +  1 )
7271oveq1i 5910 . . . . . . . . . . 11  |-  ( 1 ... N )  =  ( ( 0  +  1 ) ... N
)
73 0z 10082 . . . . . . . . . . . 12  |-  0  e.  ZZ
74 fzp1ss 10884 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  (
( 0  +  1 ) ... N ) 
C_  ( 0 ... N ) )
7573, 74mp1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 0  +  1 ) ... N
)  C_  ( 0 ... N ) )
7672, 75syl5eqss 3256 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... N
)  C_  ( 0 ... N ) )
7776sselda 3214 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  ( 0 ... N
) )
7870, 77eqeltrd 2390 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
( k  -  1 )  +  1 )  e.  ( 0 ... N ) )
79 peano2fzr 10855 . . . . . . . 8  |-  ( ( ( k  -  1 )  e.  ( ZZ>= ` 
0 )  /\  (
( k  -  1 )  +  1 )  e.  ( 0 ... N ) )  -> 
( k  -  1 )  e.  ( 0 ... N ) )
8066, 78, 79syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  ( 0 ... N ) )
8135fveq1i 5564 . . . . . . . 8  |-  ( Q `
 ( k  - 
1 ) )  =  ( ( P  |`  ( 0 ... N
) ) `  (
k  -  1 ) )
82 fvres 5580 . . . . . . . 8  |-  ( ( k  -  1 )  e.  ( 0 ... N )  ->  (
( P  |`  (
0 ... N ) ) `
 ( k  - 
1 ) )  =  ( P `  (
k  -  1 ) ) )
8381, 82syl5eq 2360 . . . . . . 7  |-  ( ( k  -  1 )  e.  ( 0 ... N )  ->  ( Q `  ( k  -  1 ) )  =  ( P `  ( k  -  1 ) ) )
8480, 83syl 15 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( Q `  ( k  -  1 ) )  =  ( P `  ( k  -  1 ) ) )
8535fveq1i 5564 . . . . . . . 8  |-  ( Q `
 k )  =  ( ( P  |`  ( 0 ... N
) ) `  k
)
86 fvres 5580 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  (
( P  |`  (
0 ... N ) ) `
 k )  =  ( P `  k
) )
8785, 86syl5eq 2360 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  ( Q `  k )  =  ( P `  k ) )
8877, 87syl 15 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( Q `  k )  =  ( P `  k ) )
8984, 88preq12d 3748 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) }  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } )
9041, 60, 893eqtr4d 2358 . . . 4  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( H `  k ) )  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } )
9190ralrimiva 2660 . . 3  |-  ( ph  ->  A. k  e.  ( 1 ... N ) ( F `  ( H `  k )
)  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } )
92 oveq2 5908 . . . . . 6  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
93 f1oeq2 5502 . . . . . 6  |-  ( ( 1 ... n )  =  ( 1 ... N )  ->  ( H : ( 1 ... n ) -1-1-onto-> ( G " (
1 ... N ) )  <-> 
H : ( 1 ... N ) -1-1-onto-> ( G
" ( 1 ... N ) ) ) )
9492, 93syl 15 . . . . 5  |-  ( n  =  N  ->  ( H : ( 1 ... n ) -1-1-onto-> ( G " (
1 ... N ) )  <-> 
H : ( 1 ... N ) -1-1-onto-> ( G
" ( 1 ... N ) ) ) )
95 oveq2 5908 . . . . . 6  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
9695feq2d 5417 . . . . 5  |-  ( n  =  N  ->  ( Q : ( 0 ... n ) --> V  <->  Q :
( 0 ... N
) --> V ) )
9792raleqdv 2776 . . . . 5  |-  ( n  =  N  ->  ( A. k  e.  (
1 ... n ) ( F `  ( H `
 k ) )  =  { ( Q `
 ( k  - 
1 ) ) ,  ( Q `  k
) }  <->  A. k  e.  ( 1 ... N
) ( F `  ( H `  k ) )  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } ) )
9894, 96, 973anbi123d 1252 . . . 4  |-  ( n  =  N  ->  (
( H : ( 1 ... n ) -1-1-onto-> ( G " ( 1 ... N ) )  /\  Q : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n
) ( F `  ( H `  k ) )  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } )  <-> 
( H : ( 1 ... N ) -1-1-onto-> ( G " ( 1 ... N ) )  /\  Q : ( 0 ... N ) --> V  /\  A. k  e.  ( 1 ... N
) ( F `  ( H `  k ) )  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } ) ) )
9998rspcev 2918 . . 3  |-  ( ( N  e.  NN0  /\  ( H : ( 1 ... N ) -1-1-onto-> ( G
" ( 1 ... N ) )  /\  Q : ( 0 ... N ) --> V  /\  A. k  e.  ( 1 ... N ) ( F `  ( H `
 k ) )  =  { ( Q `
 ( k  - 
1 ) ) ,  ( Q `  k
) } ) )  ->  E. n  e.  NN0  ( H : ( 1 ... n ) -1-1-onto-> ( G
" ( 1 ... N ) )  /\  Q : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n ) ( F `  ( H `
 k ) )  =  { ( Q `
 ( k  - 
1 ) ) ,  ( Q `  k
) } ) )
10010, 28, 37, 91, 99syl13anc 1184 . 2  |-  ( ph  ->  E. n  e.  NN0  ( H : ( 1 ... n ) -1-1-onto-> ( G
" ( 1 ... N ) )  /\  Q : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n ) ( F `  ( H `
 k ) )  =  { ( Q `
 ( k  - 
1 ) ) ,  ( Q `  k
) } ) )
1016dmeqi 4917 . . . . 5  |-  dom  F  =  dom  ( E  |`  ( G " ( 1 ... N ) ) )
102 dmres 5013 . . . . 5  |-  dom  ( E  |`  ( G "
( 1 ... N
) ) )  =  ( ( G "
( 1 ... N
) )  i^i  dom  E )
103101, 102eqtri 2336 . . . 4  |-  dom  F  =  ( ( G
" ( 1 ... N ) )  i^i 
dom  E )
104 imassrn 5062 . . . . . 6  |-  ( G
" ( 1 ... N ) )  C_  ran  G
105 f1ofo 5517 . . . . . . 7  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  G : ( 1 ... ( # `  G
) ) -onto-> dom  E
)
106 forn 5492 . . . . . . 7  |-  ( G : ( 1 ... ( # `  G
) ) -onto-> dom  E  ->  ran  G  =  dom  E )
10716, 105, 1063syl 18 . . . . . 6  |-  ( ph  ->  ran  G  =  dom  E )
108104, 107syl5sseq 3260 . . . . 5  |-  ( ph  ->  ( G " (
1 ... N ) ) 
C_  dom  E )
109 df-ss 3200 . . . . 5  |-  ( ( G " ( 1 ... N ) ) 
C_  dom  E  <->  ( ( G " ( 1 ... N ) )  i^i 
dom  E )  =  ( G " (
1 ... N ) ) )
110108, 109sylib 188 . . . 4  |-  ( ph  ->  ( ( G "
( 1 ... N
) )  i^i  dom  E )  =  ( G
" ( 1 ... N ) ) )
111103, 110syl5eq 2360 . . 3  |-  ( ph  ->  dom  F  =  ( G " ( 1 ... N ) ) )
112 iseupa 24165 . . 3  |-  ( dom 
F  =  ( G
" ( 1 ... N ) )  -> 
( H ( V EulPaths  F ) Q  <->  ( V UMGrph  F  /\  E. n  e. 
NN0  ( H :
( 1 ... n
)
-1-1-onto-> ( G " ( 1 ... N ) )  /\  Q : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n
) ( F `  ( H `  k ) )  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } ) ) ) )
113111, 112syl 15 . 2  |-  ( ph  ->  ( H ( V EulPaths  F ) Q  <->  ( V UMGrph  F  /\  E. n  e. 
NN0  ( H :
( 1 ... n
)
-1-1-onto-> ( G " ( 1 ... N ) )  /\  Q : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n
) ( F `  ( H `  k ) )  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } ) ) ) )
1147, 100, 113mpbir2and 888 1  |-  ( ph  ->  H ( V EulPaths  F ) Q )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   E.wrex 2578   {crab 2581    \ cdif 3183    i^i cin 3185    C_ wss 3186   (/)c0 3489   ~Pcpw 3659   {csn 3674   {cpr 3675   class class class wbr 4060   dom cdm 4726   ran crn 4727    |` cres 4728   "cima 4729   Fun wfun 5286    Fn wfn 5287   -->wf 5288   -1-1->wf1 5289   -onto->wfo 5290   -1-1-onto->wf1o 5291   ` cfv 5292  (class class class)co 5900   CCcc 8780   0cc0 8782   1c1 8783    + caddc 8785    <_ cle 8913    - cmin 9082   NNcn 9791   2c2 9840   NN0cn0 10012   ZZcz 10071   ZZ>=cuz 10277   ...cfz 10829   #chash 11384   UMGrph cumg 24144   EulPaths ceup 24145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-er 6702  df-pm 6818  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-hash 11385  df-umgra 24147  df-eupa 24148
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