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Theorem eupares 21697
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 21688 . . . . 5  |-  ( G ( V EulPaths  E ) P  ->  V UMGrph  E )
31, 2syl 16 . . . 4  |-  ( ph  ->  V UMGrph  E )
4 umgrares 21359 . . . 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 4252 . 2  |-  ( ph  ->  V UMGrph  F )
8 eupares.n . . . 4  |-  ( ph  ->  N  e.  ( 0 ... ( # `  G
) ) )
9 elfznn0 11083 . . . 4  |-  ( N  e.  ( 0 ... ( # `  G
) )  ->  N  e.  NN0 )
108, 9syl 16 . . 3  |-  ( ph  ->  N  e.  NN0 )
11 umgraf2 21352 . . . . . . . . 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 5591 . . . . . . . 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 21692 . . . . . . 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 5673 . . . . . 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 11056 . . . . . . 7  |-  ( N  e.  ( 0 ... ( # `  G
) )  ->  ( # `
 G )  e.  ( ZZ>= `  N )
)
208, 19syl 16 . . . . . 6  |-  ( ph  ->  ( # `  G
)  e.  ( ZZ>= `  N ) )
21 fzss2 11092 . . . . . 6  |-  ( (
# `  G )  e.  ( ZZ>= `  N )  ->  ( 1 ... N
)  C_  ( 1 ... ( # `  G
) ) )
2220, 21syl 16 . . . . 5  |-  ( ph  ->  ( 1 ... N
)  C_  ( 1 ... ( # `  G
) ) )
23 f1ores 5689 . . . . 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 5665 . . . . 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 21694 . . . . . 6  |-  ( G ( V EulPaths  E ) P  ->  P : ( 0 ... ( # `  G ) ) --> V )
301, 29syl 16 . . . . 5  |-  ( ph  ->  P : ( 0 ... ( # `  G
) ) --> V )
31 fzss2 11092 . . . . . 6  |-  ( (
# `  G )  e.  ( ZZ>= `  N )  ->  ( 0 ... N
)  C_  ( 0 ... ( # `  G
) ) )
3220, 31syl 16 . . . . 5  |-  ( ph  ->  ( 0 ... N
)  C_  ( 0 ... ( # `  G
) ) )
33 fssres 5610 . . . . 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 5585 . . . 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 3348 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  ( 1 ... ( # `
 G ) ) )
40 eupaseg 21695 . . . . . 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 5729 . . . . . . . 8  |-  ( H `
 k )  =  ( ( G  |`  ( 1 ... N
) ) `  k
)
43 fvres 5745 . . . . . . . . 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 2480 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( H `  k )  =  ( G `  k ) )
4645fveq2d 5732 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( H `  k ) )  =  ( F `  ( G `  k )
) )
476fveq1i 5729 . . . . . . 7  |-  ( F `
 ( G `  k ) )  =  ( ( E  |`  ( G " ( 1 ... N ) ) ) `  ( G `
 k ) )
48 f1ofun 5676 . . . . . . . . . . 11  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  Fun  G )
4916, 48syl 16 . . . . . . . . . 10  |-  ( ph  ->  Fun  G )
50 f1of 5674 . . . . . . . . . . . 12  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  G : ( 1 ... ( # `  G
) ) --> dom  E
)
51 fdm 5595 . . . . . . . . . . . 12  |-  ( G : ( 1 ... ( # `  G
) ) --> dom  E  ->  dom  G  =  ( 1 ... ( # `  G ) ) )
5216, 50, 513syl 19 . . . . . . . . . . 11  |-  ( ph  ->  dom  G  =  ( 1 ... ( # `  G ) ) )
5322, 52sseqtr4d 3385 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... N
)  C_  dom  G )
54 funfvima2 5974 . . . . . . . . . 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 5745 . . . . . . . 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 2480 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( G `  k ) )  =  ( E `  ( G `  k )
) )
6046, 59eqtrd 2468 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( H `  k ) )  =  ( E `  ( G `  k )
) )
61 elfznn 11080 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
6261adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  NN )
63 nnm1nn0 10261 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
6462, 63syl 16 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  NN0 )
65 nn0uz 10520 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
6664, 65syl6eleq 2526 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  ( ZZ>= `  0
) )
6762nncnd 10016 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  CC )
68 ax-1cn 9048 . . . . . . . . . 10  |-  1  e.  CC
69 npcan 9314 . . . . . . . . . 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 10410 . . . . . . . . . . . 12  |-  1  =  ( 0  +  1 )
7271oveq1i 6091 . . . . . . . . . . 11  |-  ( 1 ... N )  =  ( ( 0  +  1 ) ... N
)
73 0z 10293 . . . . . . . . . . . 12  |-  0  e.  ZZ
74 fzp1ss 11098 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  (
( 0  +  1 ) ... N ) 
C_  ( 0 ... N ) )
7573, 74mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 0  +  1 ) ... N
)  C_  ( 0 ... N ) )
7672, 75syl5eqss 3392 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... N
)  C_  ( 0 ... N ) )
7776sselda 3348 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  k  e.  ( 0 ... N
) )
7870, 77eqeltrd 2510 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  (
( k  -  1 )  +  1 )  e.  ( 0 ... N ) )
79 peano2fzr 11069 . . . . . . . 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 5729 . . . . . . . 8  |-  ( Q `
 ( k  - 
1 ) )  =  ( ( P  |`  ( 0 ... N
) ) `  (
k  -  1 ) )
82 fvres 5745 . . . . . . . 8  |-  ( ( k  -  1 )  e.  ( 0 ... N )  ->  (
( P  |`  (
0 ... N ) ) `
 ( k  - 
1 ) )  =  ( P `  (
k  -  1 ) ) )
8381, 82syl5eq 2480 . . . . . . 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 5729 . . . . . . . 8  |-  ( Q `
 k )  =  ( ( P  |`  ( 0 ... N
) ) `  k
)
86 fvres 5745 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  (
( P  |`  (
0 ... N ) ) `
 k )  =  ( P `  k
) )
8785, 86syl5eq 2480 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  ( Q `  k )  =  ( P `  k ) )
8877, 87syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( Q `  k )  =  ( P `  k ) )
8984, 88preq12d 3891 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) }  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } )
9041, 60, 893eqtr4d 2478 . . . 4  |-  ( (
ph  /\  k  e.  ( 1 ... N
) )  ->  ( F `  ( H `  k ) )  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } )
9190ralrimiva 2789 . . 3  |-  ( ph  ->  A. k  e.  ( 1 ... N ) ( F `  ( H `  k )
)  =  { ( Q `  ( k  -  1 ) ) ,  ( Q `  k ) } )
92 oveq2 6089 . . . . . 6  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
93 f1oeq2 5666 . . . . . 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 6089 . . . . . 6  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
9695feq2d 5581 . . . . 5  |-  ( n  =  N  ->  ( Q : ( 0 ... n ) --> V  <->  Q :
( 0 ... N
) --> V ) )
9792raleqdv 2910 . . . . 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 3052 . . 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 5071 . . . . 5  |-  dom  F  =  dom  ( E  |`  ( G " ( 1 ... N ) ) )
102 dmres 5167 . . . . 5  |-  dom  ( E  |`  ( G "
( 1 ... N
) ) )  =  ( ( G "
( 1 ... N
) )  i^i  dom  E )
103101, 102eqtri 2456 . . . 4  |-  dom  F  =  ( ( G
" ( 1 ... N ) )  i^i 
dom  E )
104 imassrn 5216 . . . . . 6  |-  ( G
" ( 1 ... N ) )  C_  ran  G
105 f1ofo 5681 . . . . . . 7  |-  ( G : ( 1 ... ( # `  G
) ) -1-1-onto-> dom  E  ->  G : ( 1 ... ( # `  G
) ) -onto-> dom  E
)
106 forn 5656 . . . . . . 7  |-  ( G : ( 1 ... ( # `  G
) ) -onto-> dom  E  ->  ran  G  =  dom  E )
10716, 105, 1063syl 19 . . . . . 6  |-  ( ph  ->  ran  G  =  dom  E )
108104, 107syl5sseq 3396 . . . . 5  |-  ( ph  ->  ( G " (
1 ... N ) ) 
C_  dom  E )
109 df-ss 3334 . . . . 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 2480 . . 3  |-  ( ph  ->  dom  F  =  ( G " ( 1 ... N ) ) )
112 iseupa 21687 . . 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 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   {crab 2709    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   {cpr 3815   class class class wbr 4212   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881   Fun wfun 5448    Fn wfn 5449   -->wf 5450   -1-1->wf1 5451   -onto->wfo 5452   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    <_ cle 9121    - cmin 9291   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043   #chash 11618   UMGrph cumg 21347   EulPaths ceup 21684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-hash 11619  df-umgra 21348  df-eupa 21685
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