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Theorem eupares 21650
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 21641 . . . . 5  |-  ( G ( V EulPaths  E ) P  ->  V UMGrph  E )
31, 2syl 16 . . . 4  |-  ( ph  ->  V UMGrph  E )
4 umgrares 21312 . . . 4  |-  ( V UMGrph  E  ->  V UMGrph  ( E  |`  ( G " (
1 ... N ) ) ) )
53, 4syl 16 . . 3  |-  ( ph  ->  V UMGrph  ( E  |`  ( G " ( 1 ... N ) ) ) )
6 eupares.f . . 3  |-  F  =  ( E  |`  ( G " ( 1 ... N ) ) )
75, 6syl6breqr 4212 . 2  |-  ( ph  ->  V UMGrph  F )
8 eupares.n . . . 4  |-  ( ph  ->  N  e.  ( 0 ... ( # `  G
) ) )
9 elfznn0 11039 . . . 4  |-  ( N  e.  ( 0 ... ( # `  G
) )  ->  N  e.  NN0 )
108, 9syl 16 . . 3  |-  ( ph  ->  N  e.  NN0 )
11 umgraf2 21305 . . . . . . . . 9  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
123, 11syl 16 . . . . . . . 8  |-  ( ph  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
13 ffn 5550 . . . . . . . 8  |-  ( E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  ->  E  Fn  dom  E
)
1412, 13syl 16 . . . . . . 7  |-  ( ph  ->  E  Fn  dom  E
)
15 eupaf1o 21645 . . . . . . 7  |-  ( ( G ( V EulPaths  E ) P  /\  E  Fn  dom  E )  ->  G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E )
161, 14, 15syl2anc 643 . . . . . 6  |-  ( ph  ->  G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E )
17 f1of1 5632 . . . . . 6  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  G : ( 1 ... ( # `  G
) ) -1-1-> dom  E
)
1816, 17syl 16 . . . . 5  |-  ( ph  ->  G : ( 1 ... ( # `  G
) ) -1-1-> dom  E
)
19 elfzuz3 11012 . . . . . . 7  |-  ( N  e.  ( 0 ... ( # `  G
) )  ->  ( # `
 G )  e.  ( ZZ>= `  N )
)
208, 19syl 16 . . . . . 6  |-  ( ph  ->  ( # `  G
)  e.  ( ZZ>= `  N ) )
21 fzss2 11048 . . . . . 6  |-  ( (
# `  G )  e.  ( ZZ>= `  N )  ->  ( 1 ... N
)  C_  ( 1 ... ( # `  G
) ) )
2220, 21syl 16 . . . . 5  |-  ( ph  ->  ( 1 ... N
)  C_  ( 1 ... ( # `  G
) ) )
23 f1ores 5648 . . . . 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 643 . . . 4  |-  ( ph  ->  ( G  |`  (
1 ... N ) ) : ( 1 ... N ) -1-1-onto-> ( G " (
1 ... N ) ) )
25 eupares.h . . . . 5  |-  H  =  ( G  |`  (
1 ... N ) )
26 f1oeq1 5624 . . . . 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 204 . . 3  |-  ( ph  ->  H : ( 1 ... N ) -1-1-onto-> ( G
" ( 1 ... N ) ) )
29 eupapf 21647 . . . . . 6  |-  ( G ( V EulPaths  E ) P  ->  P : ( 0 ... ( # `  G ) ) --> V )
301, 29syl 16 . . . . 5  |-  ( ph  ->  P : ( 0 ... ( # `  G
) ) --> V )
31 fzss2 11048 . . . . . 6  |-  ( (
# `  G )  e.  ( ZZ>= `  N )  ->  ( 0 ... N
)  C_  ( 0 ... ( # `  G
) ) )
3220, 31syl 16 . . . . 5  |-  ( ph  ->  ( 0 ... N
)  C_  ( 0 ... ( # `  G
) ) )
33 fssres 5569 . . . . 5  |-  ( ( P : ( 0 ... ( # `  G
) ) --> V  /\  ( 0 ... N
)  C_  ( 0 ... ( # `  G
) ) )  -> 
( P  |`  (
0 ... N ) ) : ( 0 ... N ) --> V )
3430, 32, 33syl2anc 643 . . . 4  |-  ( ph  ->  ( P  |`  (
0 ... N ) ) : ( 0 ... N ) --> V )
35 eupares.q . . . . 5  |-  Q  =  ( P  |`  (
0 ... N ) )
3635feq1i 5544 . . . 4  |-  ( Q : ( 0 ... N ) --> V  <->  ( P  |`  ( 0 ... N
) ) : ( 0 ... N ) --> V )
3734, 36sylibr 204 . . 3  |-  ( ph  ->  Q : ( 0 ... N ) --> V )
381adantr 452 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  G
( V EulPaths  E ) P )
3922sselda 3308 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  ( 1 ... ( # `
 G ) ) )
40 eupaseg 21648 . . . . . 6  |-  ( ( G ( V EulPaths  E ) P  /\  k  e.  ( 1 ... ( # `
 G ) ) )  ->  ( E `  ( G `  k
) )  =  {
( P `  (
k  -  1 ) ) ,  ( P `
 k ) } )
4138, 39, 40syl2anc 643 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( E `  ( G `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } )
4225fveq1i 5688 . . . . . . . 8  |-  ( H `
 k )  =  ( ( G  |`  ( 1 ... N
) ) `  k
)
43 fvres 5704 . . . . . . . . 9  |-  ( k  e.  ( 1 ... N )  ->  (
( G  |`  (
1 ... N ) ) `
 k )  =  ( G `  k
) )
4443adantl 453 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
( G  |`  (
1 ... N ) ) `
 k )  =  ( G `  k
) )
4542, 44syl5eq 2448 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( H `  k )  =  ( G `  k ) )
4645fveq2d 5691 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( H `  k ) )  =  ( F `  ( G `  k )
) )
476fveq1i 5688 . . . . . . 7  |-  ( F `
 ( G `  k ) )  =  ( ( E  |`  ( G " ( 1 ... N ) ) ) `  ( G `
 k ) )
48 f1ofun 5635 . . . . . . . . . . 11  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  Fun  G )
4916, 48syl 16 . . . . . . . . . 10  |-  ( ph  ->  Fun  G )
50 f1of 5633 . . . . . . . . . . . 12  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  G : ( 1 ... ( # `  G
) ) --> dom  E
)
51 fdm 5554 . . . . . . . . . . . 12  |-  ( G : ( 1 ... ( # `  G
) ) --> dom  E  ->  dom  G  =  ( 1 ... ( # `  G ) ) )
5216, 50, 513syl 19 . . . . . . . . . . 11  |-  ( ph  ->  dom  G  =  ( 1 ... ( # `  G ) ) )
5322, 52sseqtr4d 3345 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... N
)  C_  dom  G )
54 funfvima2 5933 . . . . . . . . . 10  |-  ( ( Fun  G  /\  (
1 ... N )  C_  dom  G )  ->  (
k  e.  ( 1 ... N )  -> 
( G `  k
)  e.  ( G
" ( 1 ... N ) ) ) )
5549, 53, 54syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( k  e.  ( 1 ... N )  ->  ( G `  k )  e.  ( G " ( 1 ... N ) ) ) )
5655imp 419 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( G `  k )  e.  ( G " (
1 ... N ) ) )
57 fvres 5704 . . . . . . . 8  |-  ( ( G `  k )  e.  ( G "
( 1 ... N
) )  ->  (
( E  |`  ( G " ( 1 ... N ) ) ) `
 ( G `  k ) )  =  ( E `  ( G `  k )
) )
5856, 57syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
( E  |`  ( G " ( 1 ... N ) ) ) `
 ( G `  k ) )  =  ( E `  ( G `  k )
) )
5947, 58syl5eq 2448 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( G `  k ) )  =  ( E `  ( G `  k )
) )
6046, 59eqtrd 2436 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( H `  k ) )  =  ( E `  ( G `  k )
) )
61 elfznn 11036 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
6261adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  NN )
63 nnm1nn0 10217 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
6462, 63syl 16 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  NN0 )
65 nn0uz 10476 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
6664, 65syl6eleq 2494 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  ( ZZ>= `  0
) )
6762nncnd 9972 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  CC )
68 ax-1cn 9004 . . . . . . . . . 10  |-  1  e.  CC
69 npcan 9270 . . . . . . . . . 10  |-  ( ( k  e.  CC  /\  1  e.  CC )  ->  ( ( k  - 
1 )  +  1 )  =  k )
7067, 68, 69sylancl 644 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
( k  -  1 )  +  1 )  =  k )
71 1e0p1 10366 . . . . . . . . . . . 12  |-  1  =  ( 0  +  1 )
7271oveq1i 6050 . . . . . . . . . . 11  |-  ( 1 ... N )  =  ( ( 0  +  1 ) ... N
)
73 0z 10249 . . . . . . . . . . . 12  |-  0  e.  ZZ
74 fzp1ss 11054 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  (
( 0  +  1 ) ... N ) 
C_  ( 0 ... N ) )
7573, 74mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 0  +  1 ) ... N
)  C_  ( 0 ... N ) )
7672, 75syl5eqss 3352 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... N
)  C_  ( 0 ... N ) )
7776sselda 3308 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  ( 0 ... N
) )
7870, 77eqeltrd 2478 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
( k  -  1 )  +  1 )  e.  ( 0 ... N ) )
79 peano2fzr 11025 . . . . . . . 8  |-  ( ( ( k  -  1 )  e.  ( ZZ>= ` 
0 )  /\  (
( k  -  1 )  +  1 )  e.  ( 0 ... N ) )  -> 
( k  -  1 )  e.  ( 0 ... N ) )
8066, 78, 79syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  ( 0 ... N ) )
8135fveq1i 5688 . . . . . . . 8  |-  ( Q `
 ( k  - 
1 ) )  =  ( ( P  |`  ( 0 ... N
) ) `  (
k  -  1 ) )
82 fvres 5704 . . . . . . . 8  |-  ( ( k  -  1 )  e.  ( 0 ... N )  ->  (
( P  |`  (
0 ... N ) ) `
 ( k  - 
1 ) )  =  ( P `  (
k  -  1 ) ) )
8381, 82syl5eq 2448 . . . . . . 7  |-  ( ( k  -  1 )  e.  ( 0 ... N )  ->  ( Q `  ( k  -  1 ) )  =  ( P `  ( k  -  1 ) ) )
8480, 83syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( Q `  ( k  -  1 ) )  =  ( P `  ( k  -  1 ) ) )
8535fveq1i 5688 . . . . . . . 8  |-  ( Q `
 k )  =  ( ( P  |`  ( 0 ... N
) ) `  k
)
86 fvres 5704 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  (
( P  |`  (
0 ... N ) ) `
 k )  =  ( P `  k
) )
8785, 86syl5eq 2448 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  ( Q `  k )  =  ( P `  k ) )
8877, 87syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( Q `  k )  =  ( P `  k ) )
8984, 88preq12d 3851 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) }  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } )
9041, 60, 893eqtr4d 2446 . . . 4  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( H `  k ) )  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } )
9190ralrimiva 2749 . . 3  |-  ( ph  ->  A. k  e.  ( 1 ... N ) ( F `  ( H `  k )
)  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } )
92 oveq2 6048 . . . . . 6  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
93 f1oeq2 5625 . . . . . 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 16 . . . . 5  |-  ( n  =  N  ->  ( H : ( 1 ... n ) -1-1-onto-> ( G " (
1 ... N ) )  <-> 
H : ( 1 ... N ) -1-1-onto-> ( G
" ( 1 ... N ) ) ) )
95 oveq2 6048 . . . . . 6  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
9695feq2d 5540 . . . . 5  |-  ( n  =  N  ->  ( Q : ( 0 ... n ) --> V  <->  Q :
( 0 ... N
) --> V ) )
9792raleqdv 2870 . . . . 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 1254 . . . 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 3012 . . 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 1186 . 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 5030 . . . . 5  |-  dom  F  =  dom  ( E  |`  ( G " ( 1 ... N ) ) )
102 dmres 5126 . . . . 5  |-  dom  ( E  |`  ( G "
( 1 ... N
) ) )  =  ( ( G "
( 1 ... N
) )  i^i  dom  E )
103101, 102eqtri 2424 . . . 4  |-  dom  F  =  ( ( G
" ( 1 ... N ) )  i^i 
dom  E )
104 imassrn 5175 . . . . . 6  |-  ( G
" ( 1 ... N ) )  C_  ran  G
105 f1ofo 5640 . . . . . . 7  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  G : ( 1 ... ( # `  G
) ) -onto-> dom  E
)
106 forn 5615 . . . . . . 7  |-  ( G : ( 1 ... ( # `  G
) ) -onto-> dom  E  ->  ran  G  =  dom  E )
10716, 105, 1063syl 19 . . . . . 6  |-  ( ph  ->  ran  G  =  dom  E )
108104, 107syl5sseq 3356 . . . . 5  |-  ( ph  ->  ( G " (
1 ... N ) ) 
C_  dom  E )
109 df-ss 3294 . . . . 5  |-  ( ( G " ( 1 ... N ) ) 
C_  dom  E  <->  ( ( G " ( 1 ... N ) )  i^i 
dom  E )  =  ( G " (
1 ... N ) ) )
110108, 109sylib 189 . . . 4  |-  ( ph  ->  ( ( G "
( 1 ... N
) )  i^i  dom  E )  =  ( G
" ( 1 ... N ) ) )
111103, 110syl5eq 2448 . . 3  |-  ( ph  ->  dom  F  =  ( G " ( 1 ... N ) ) )
112 iseupa 21640 . . 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 16 . 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 889 1  |-  ( ph  ->  H ( V EulPaths  F ) Q )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   {crab 2670    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   {cpr 3775   class class class wbr 4172   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    <_ cle 9077    - cmin 9247   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   #chash 11573   UMGrph cumg 21300   EulPaths ceup 21637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574  df-umgra 21301  df-eupa 21638
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