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Theorem eupares 23899
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 23882 . . . . 5  |-  ( G ( V EulPaths  E ) P  ->  V UMGrph  E )
31, 2syl 15 . . . 4  |-  ( ph  ->  V UMGrph  E )
4 umgrares 23876 . . . 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 4063 . 2  |-  ( ph  ->  V UMGrph  F )
8 eupares.n . . . 4  |-  ( ph  ->  N  e.  ( 0 ... ( # `  G
) ) )
9 elfznn0 10822 . . . 4  |-  ( N  e.  ( 0 ... ( # `  G
) )  ->  N  e.  NN0 )
108, 9syl 15 . . 3  |-  ( ph  ->  N  e.  NN0 )
11 umgraf2 23869 . . . . . . . . 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 5389 . . . . . . . 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 23885 . . . . . . 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 5471 . . . . . 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 10795 . . . . . . 7  |-  ( N  e.  ( 0 ... ( # `  G
) )  ->  ( # `
 G )  e.  ( ZZ>= `  N )
)
208, 19syl 15 . . . . . 6  |-  ( ph  ->  ( # `  G
)  e.  ( ZZ>= `  N ) )
21 fzss2 10831 . . . . . 6  |-  ( (
# `  G )  e.  ( ZZ>= `  N )  ->  ( 1 ... N
)  C_  ( 1 ... ( # `  G
) ) )
2220, 21syl 15 . . . . 5  |-  ( ph  ->  ( 1 ... N
)  C_  ( 1 ... ( # `  G
) ) )
23 f1ores 5487 . . . . 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 5463 . . . . 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 23887 . . . . . 6  |-  ( G ( V EulPaths  E ) P  ->  P : ( 0 ... ( # `  G ) ) --> V )
301, 29syl 15 . . . . 5  |-  ( ph  ->  P : ( 0 ... ( # `  G
) ) --> V )
31 fzss2 10831 . . . . . 6  |-  ( (
# `  G )  e.  ( ZZ>= `  N )  ->  ( 0 ... N
)  C_  ( 0 ... ( # `  G
) ) )
3220, 31syl 15 . . . . 5  |-  ( ph  ->  ( 0 ... N
)  C_  ( 0 ... ( # `  G
) ) )
33 fssres 5408 . . . . 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 5383 . . . 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 3180 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  ( 1 ... ( # `
 G ) ) )
40 eupaseg 23888 . . . . . 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 5526 . . . . . . . 8  |-  ( H `
 k )  =  ( ( G  |`  ( 1 ... N
) ) `  k
)
43 fvres 5542 . . . . . . . . 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 2327 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( H `  k )  =  ( G `  k ) )
4645fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( H `  k ) )  =  ( F `  ( G `  k )
) )
476fveq1i 5526 . . . . . . 7  |-  ( F `
 ( G `  k ) )  =  ( ( E  |`  ( G " ( 1 ... N ) ) ) `  ( G `
 k ) )
48 f1ofun 5474 . . . . . . . . . . 11  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  Fun  G )
4916, 48syl 15 . . . . . . . . . 10  |-  ( ph  ->  Fun  G )
50 f1of 5472 . . . . . . . . . . . 12  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  G : ( 1 ... ( # `  G
) ) --> dom  E
)
51 fdm 5393 . . . . . . . . . . . 12  |-  ( G : ( 1 ... ( # `  G
) ) --> dom  E  ->  dom  G  =  ( 1 ... ( # `  G ) ) )
5216, 50, 513syl 18 . . . . . . . . . . 11  |-  ( ph  ->  dom  G  =  ( 1 ... ( # `  G ) ) )
5322, 52sseqtr4d 3215 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... N
)  C_  dom  G )
54 funfvima2 5754 . . . . . . . . . 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 5542 . . . . . . . 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 2327 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( G `  k ) )  =  ( E `  ( G `  k )
) )
6046, 59eqtrd 2315 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( H `  k ) )  =  ( E `  ( G `  k )
) )
61 elfznn 10819 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
6261adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  NN )
63 nnm1nn0 10005 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
6462, 63syl 15 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  NN0 )
65 nn0uz 10262 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
6664, 65syl6eleq 2373 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  ( ZZ>= `  0
) )
6762nncnd 9762 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  CC )
68 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
69 npcan 9060 . . . . . . . . . 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 10152 . . . . . . . . . . . 12  |-  1  =  ( 0  +  1 )
7271oveq1i 5868 . . . . . . . . . . 11  |-  ( 1 ... N )  =  ( ( 0  +  1 ) ... N
)
73 0z 10035 . . . . . . . . . . . 12  |-  0  e.  ZZ
74 fzp1ss 10837 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  (
( 0  +  1 ) ... N ) 
C_  ( 0 ... N ) )
7573, 74mp1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 0  +  1 ) ... N
)  C_  ( 0 ... N ) )
7672, 75syl5eqss 3222 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... N
)  C_  ( 0 ... N ) )
7776sselda 3180 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  ( 0 ... N
) )
7870, 77eqeltrd 2357 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
( k  -  1 )  +  1 )  e.  ( 0 ... N ) )
79 peano2fzr 10808 . . . . . . . 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 5526 . . . . . . . 8  |-  ( Q `
 ( k  - 
1 ) )  =  ( ( P  |`  ( 0 ... N
) ) `  (
k  -  1 ) )
82 fvres 5542 . . . . . . . 8  |-  ( ( k  -  1 )  e.  ( 0 ... N )  ->  (
( P  |`  (
0 ... N ) ) `
 ( k  - 
1 ) )  =  ( P `  (
k  -  1 ) ) )
8381, 82syl5eq 2327 . . . . . . 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 5526 . . . . . . . 8  |-  ( Q `
 k )  =  ( ( P  |`  ( 0 ... N
) ) `  k
)
86 fvres 5542 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  (
( P  |`  (
0 ... N ) ) `
 k )  =  ( P `  k
) )
8785, 86syl5eq 2327 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  ( Q `  k )  =  ( P `  k ) )
8877, 87syl 15 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( Q `  k )  =  ( P `  k ) )
8984, 88preq12d 3714 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) }  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } )
9041, 60, 893eqtr4d 2325 . . . 4  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( H `  k ) )  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } )
9190ralrimiva 2626 . . 3  |-  ( ph  ->  A. k  e.  ( 1 ... N ) ( F `  ( H `  k )
)  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } )
92 oveq2 5866 . . . . . 6  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
93 f1oeq2 5464 . . . . . 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 5866 . . . . . 6  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
9695feq2d 5380 . . . . 5  |-  ( n  =  N  ->  ( Q : ( 0 ... n ) --> V  <->  Q :
( 0 ... N
) --> V ) )
9792raleqdv 2742 . . . . 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 2884 . . 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 4880 . . . . 5  |-  dom  F  =  dom  ( E  |`  ( G " ( 1 ... N ) ) )
102 dmres 4976 . . . . 5  |-  dom  ( E  |`  ( G "
( 1 ... N
) ) )  =  ( ( G "
( 1 ... N
) )  i^i  dom  E )
103101, 102eqtri 2303 . . . 4  |-  dom  F  =  ( ( G
" ( 1 ... N ) )  i^i 
dom  E )
104 imassrn 5025 . . . . . 6  |-  ( G
" ( 1 ... N ) )  C_  ran  G
105 f1ofo 5479 . . . . . . 7  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  G : ( 1 ... ( # `  G
) ) -onto-> dom  E
)
106 forn 5454 . . . . . . 7  |-  ( G : ( 1 ... ( # `  G
) ) -onto-> dom  E  ->  ran  G  =  dom  E )
10716, 105, 1063syl 18 . . . . . 6  |-  ( ph  ->  ran  G  =  dom  E )
108104, 107syl5sseq 3226 . . . . 5  |-  ( ph  ->  ( G " (
1 ... N ) ) 
C_  dom  E )
109 df-ss 3166 . . . . 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 2327 . . 3  |-  ( ph  ->  dom  F  =  ( G " ( 1 ... N ) ) )
112 iseupa 23881 . . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   {cpr 3641   class class class wbr 4023   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    <_ cle 8868    - cmin 9037   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   #chash 11337   UMGrph cumg 23860   EulPaths ceup 23861
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
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-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-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-hash 11338  df-umgra 23863  df-eupa 23864
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