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Theorem eulerpartlemmf 28511
Description: Lemma for eulerpart 28518 (Contributed by Thierry Arnoux, 30-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)
Hypotheses
Ref Expression
eulerpart.p  |-  P  =  { f  e.  ( NN0  ^m  NN )  |  ( ( `' f " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( f `  k
)  x.  k )  =  N ) }
eulerpart.o  |-  O  =  { g  e.  P  |  A. n  e.  ( `' g " NN )  -.  2  ||  n }
eulerpart.d  |-  D  =  { g  e.  P  |  A. n  e.  NN  ( g `  n
)  <_  1 }
eulerpart.j  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
eulerpart.f  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
eulerpart.h  |-  H  =  { r  e.  ( ( ~P NN0  i^i  Fin )  ^m  J )  |  ( r supp  (/) )  e. 
Fin }
eulerpart.m  |-  M  =  ( r  e.  H  |->  { <. x ,  y
>.  |  ( x  e.  J  /\  y  e.  ( r `  x
) ) } )
eulerpart.r  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
eulerpart.t  |-  T  =  { f  e.  ( NN0  ^m  NN )  |  ( `' f
" NN )  C_  J }
eulerpart.g  |-  G  =  ( o  e.  ( T  i^i  R ) 
|->  ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( o  |`  J ) ) ) ) ) )
Assertion
Ref Expression
eulerpartlemmf  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  H )
Distinct variable groups:    f, k, n, x, y, z    f,
o, r, A    o, F    H, r    f, J   
n, o, r, J, x, y    o, M   
f, N    g, n, P    R, o    T, o
Allowed substitution hints:    A( x, y, z, g, k, n)    D( x, y, z, f, g, k, n, o, r)    P( x, y, z, f, k, o, r)    R( x, y, z, f, g, k, n, r)    T( x, y, z, f, g, k, n, r)    F( x, y, z, f, g, k, n, r)    G( x, y, z, f, g, k, n, o, r)    H( x, y, z, f, g, k, n, o)    J( z, g, k)    M( x, y, z, f, g, k, n, r)    N( x, y, z, g, k, n, o, r)    O( x, y, z, f, g, k, n, o, r)

Proof of Theorem eulerpartlemmf
StepHypRef Expression
1 bitsf1o 14107 . . . . 5  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
2 f1of 5822 . . . . 5  |-  ( (bits  |`  NN0 ) : NN0 -1-1-onto-> ( ~P NN0  i^i  Fin )  ->  (bits  |`  NN0 ) : NN0 --> ( ~P NN0  i^i 
Fin ) )
31, 2ax-mp 5 . . . 4  |-  (bits  |`  NN0 ) : NN0 --> ( ~P NN0  i^i 
Fin )
4 eulerpart.p . . . . . . . . 9  |-  P  =  { f  e.  ( NN0  ^m  NN )  |  ( ( `' f " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( f `  k
)  x.  k )  =  N ) }
5 eulerpart.o . . . . . . . . 9  |-  O  =  { g  e.  P  |  A. n  e.  ( `' g " NN )  -.  2  ||  n }
6 eulerpart.d . . . . . . . . 9  |-  D  =  { g  e.  P  |  A. n  e.  NN  ( g `  n
)  <_  1 }
7 eulerpart.j . . . . . . . . 9  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
8 eulerpart.f . . . . . . . . 9  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
9 eulerpart.h . . . . . . . . 9  |-  H  =  { r  e.  ( ( ~P NN0  i^i  Fin )  ^m  J )  |  ( r supp  (/) )  e. 
Fin }
10 eulerpart.m . . . . . . . . 9  |-  M  =  ( r  e.  H  |->  { <. x ,  y
>.  |  ( x  e.  J  /\  y  e.  ( r `  x
) ) } )
11 eulerpart.r . . . . . . . . 9  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
12 eulerpart.t . . . . . . . . 9  |-  T  =  { f  e.  ( NN0  ^m  NN )  |  ( `' f
" NN )  C_  J }
134, 5, 6, 7, 8, 9, 10, 11, 12eulerpartlemt0 28505 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  <->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
1413biimpi 194 . . . . . . 7  |-  ( A  e.  ( T  i^i  R )  ->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
1514simp1d 1008 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  A  e.  ( NN0  ^m  NN ) )
16 nn0ex 10822 . . . . . . 7  |-  NN0  e.  _V
17 nnex 10562 . . . . . . 7  |-  NN  e.  _V
1816, 17elmap 7466 . . . . . 6  |-  ( A  e.  ( NN0  ^m  NN )  <->  A : NN --> NN0 )
1915, 18sylib 196 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  A : NN
--> NN0 )
20 ssrab2 3581 . . . . . 6  |-  { z  e.  NN  |  -.  2  ||  z }  C_  NN
217, 20eqsstri 3529 . . . . 5  |-  J  C_  NN
22 fssres 5757 . . . . 5  |-  ( ( A : NN --> NN0  /\  J  C_  NN )  -> 
( A  |`  J ) : J --> NN0 )
2319, 21, 22sylancl 662 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( A  |`  J ) : J --> NN0 )
24 fco2 5748 . . . 4  |-  ( ( (bits  |`  NN0 ) : NN0 --> ( ~P NN0  i^i 
Fin )  /\  ( A  |`  J ) : J --> NN0 )  ->  (bits  o.  ( A  |`  J ) ) : J --> ( ~P
NN0  i^i  Fin )
)
253, 23, 24sylancr 663 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) ) : J --> ( ~P
NN0  i^i  Fin )
)
2616pwex 4639 . . . . 5  |-  ~P NN0  e.  _V
2726inex1 4597 . . . 4  |-  ( ~P
NN0  i^i  Fin )  e.  _V
2817, 21ssexi 4601 . . . 4  |-  J  e. 
_V
2927, 28elmap 7466 . . 3  |-  ( (bits 
o.  ( A  |`  J ) )  e.  ( ( ~P NN0  i^i 
Fin )  ^m  J
)  <->  (bits  o.  ( A  |`  J ) ) : J --> ( ~P
NN0  i^i  Fin )
)
3025, 29sylibr 212 . 2  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  ( ( ~P NN0  i^i  Fin )  ^m  J ) )
3114simp2d 1009 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " NN )  e. 
Fin )
32 0nn0 10831 . . . . . . . . 9  |-  0  e.  NN0
33 suppimacnv 6928 . . . . . . . . 9  |-  ( ( A  e.  ( T  i^i  R )  /\  0  e.  NN0 )  -> 
( A supp  0 )  =  ( `' A " ( _V  \  {
0 } ) ) )
3432, 33mpan2 671 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  ( A supp  0 )  =  ( `' A " ( _V 
\  { 0 } ) ) )
35 frnsuppeq 6929 . . . . . . . . . 10  |-  ( ( NN  e.  _V  /\  0  e.  NN0 )  -> 
( A : NN --> NN0  ->  ( A supp  0
)  =  ( `' A " ( NN0  \  { 0 } ) ) ) )
3617, 32, 35mp2an 672 . . . . . . . . 9  |-  ( A : NN --> NN0  ->  ( A supp  0 )  =  ( `' A "
( NN0  \  { 0 } ) ) )
3719, 36syl 16 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  ( A supp  0 )  =  ( `' A " ( NN0  \  { 0 } ) ) )
3834, 37eqtr3d 2500 . . . . . . 7  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " ( _V  \  { 0 } ) )  =  ( `' A " ( NN0  \  { 0 } ) ) )
3938eleq1d 2526 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( ( `' A " ( _V 
\  { 0 } ) )  e.  Fin  <->  ( `' A " ( NN0  \  { 0 } ) )  e.  Fin )
)
40 dfn2 10829 . . . . . . . 8  |-  NN  =  ( NN0  \  { 0 } )
4140imaeq2i 5345 . . . . . . 7  |-  ( `' A " NN )  =  ( `' A " ( NN0  \  {
0 } ) )
4241eleq1i 2534 . . . . . 6  |-  ( ( `' A " NN )  e.  Fin  <->  ( `' A " ( NN0  \  {
0 } ) )  e.  Fin )
4339, 42syl6bbr 263 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( ( `' A " ( _V 
\  { 0 } ) )  e.  Fin  <->  ( `' A " NN )  e.  Fin ) )
4431, 43mpbird 232 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " ( _V  \  { 0 } ) )  e.  Fin )
45 resss 5307 . . . . 5  |-  ( A  |`  J )  C_  A
46 cnvss 5185 . . . . 5  |-  ( ( A  |`  J )  C_  A  ->  `' ( A  |`  J )  C_  `' A )
47 imass1 5381 . . . . 5  |-  ( `' ( A  |`  J ) 
C_  `' A  -> 
( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  C_  ( `' A " ( _V 
\  { 0 } ) ) )
4845, 46, 47mp2b 10 . . . 4  |-  ( `' ( A  |`  J )
" ( _V  \  { 0 } ) )  C_  ( `' A " ( _V  \  { 0 } ) )
49 ssfi 7759 . . . 4  |-  ( ( ( `' A "
( _V  \  {
0 } ) )  e.  Fin  /\  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  C_  ( `' A " ( _V 
\  { 0 } ) ) )  -> 
( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  e.  Fin )
5044, 48, 49sylancl 662 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( A  |`  J )
" ( _V  \  { 0 } ) )  e.  Fin )
51 cnvco 5198 . . . . . 6  |-  `' (bits 
o.  ( A  |`  J ) )  =  ( `' ( A  |`  J )  o.  `'bits )
5251imaeq1i 5344 . . . . 5  |-  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  =  ( ( `' ( A  |`  J )  o.  `'bits ) " ( _V 
\  { (/) } ) )
53 imaco 5518 . . . . 5  |-  ( ( `' ( A  |`  J )  o.  `'bits ) " ( _V  \  { (/) } ) )  =  ( `' ( A  |`  J ) " ( `'bits " ( _V  \  { (/) } ) ) )
5452, 53eqtri 2486 . . . 4  |-  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  =  ( `' ( A  |`  J ) " ( `'bits " ( _V  \  { (/) } ) ) )
55 ffun 5739 . . . . . 6  |-  ( A : NN --> NN0  ->  Fun 
A )
56 funres 5633 . . . . . 6  |-  ( Fun 
A  ->  Fun  ( A  |`  J ) )
5719, 55, 563syl 20 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  Fun  ( A  |`  J ) )
58 ssv 3519 . . . . . . 7  |-  ( `'bits " _V )  C_  _V
59 ssdif 3635 . . . . . . 7  |-  ( ( `'bits " _V )  C_  _V  ->  ( ( `'bits " _V )  \  ( `'bits " { (/) } ) )  C_  ( _V  \  ( `'bits " { (/)
} ) ) )
6058, 59ax-mp 5 . . . . . 6  |-  ( ( `'bits " _V )  \ 
( `'bits " { (/) } ) )  C_  ( _V  \  ( `'bits " { (/)
} ) )
61 bitsf 14089 . . . . . . 7  |- bits : ZZ --> ~P NN0
62 ffun 5739 . . . . . . 7  |-  (bits : ZZ
--> ~P NN0  ->  Fun bits )
63 difpreima 6016 . . . . . . 7  |-  ( Fun bits  ->  ( `'bits " ( _V  \  { (/) } ) )  =  ( ( `'bits " _V )  \ 
( `'bits " { (/) } ) ) )
6461, 62, 63mp2b 10 . . . . . 6  |-  ( `'bits " ( _V  \  { (/) } ) )  =  ( ( `'bits " _V )  \  ( `'bits " { (/) } ) )
65 bitsf1 14108 . . . . . . . . 9  |- bits : ZZ -1-1-> ~P
NN0
66 0z 10896 . . . . . . . . . 10  |-  0  e.  ZZ
67 snssi 4176 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  { 0 }  C_  ZZ )
6866, 67ax-mp 5 . . . . . . . . 9  |-  { 0 }  C_  ZZ
69 f1imacnv 5838 . . . . . . . . 9  |-  ( (bits
: ZZ -1-1-> ~P NN0  /\ 
{ 0 }  C_  ZZ )  ->  ( `'bits " (bits " { 0 } ) )  =  {
0 } )
7065, 68, 69mp2an 672 . . . . . . . 8  |-  ( `'bits " (bits " { 0 } ) )  =  {
0 }
71 ffn 5737 . . . . . . . . . . . 12  |-  (bits : ZZ
--> ~P NN0  -> bits  Fn  ZZ )
7261, 71ax-mp 5 . . . . . . . . . . 11  |- bits  Fn  ZZ
73 fnsnfv 5933 . . . . . . . . . . 11  |-  ( (bits 
Fn  ZZ  /\  0  e.  ZZ )  ->  { (bits `  0 ) }  =  (bits " {
0 } ) )
7472, 66, 73mp2an 672 . . . . . . . . . 10  |-  { (bits `  0 ) }  =  (bits " {
0 } )
75 0bits 14101 . . . . . . . . . . 11  |-  (bits ` 
0 )  =  (/)
7675sneqi 4043 . . . . . . . . . 10  |-  { (bits `  0 ) }  =  { (/) }
7774, 76eqtr3i 2488 . . . . . . . . 9  |-  (bits " { 0 } )  =  { (/) }
7877imaeq2i 5345 . . . . . . . 8  |-  ( `'bits " (bits " { 0 } ) )  =  ( `'bits " { (/) } )
7970, 78eqtr3i 2488 . . . . . . 7  |-  { 0 }  =  ( `'bits " { (/) } )
8079difeq2i 3615 . . . . . 6  |-  ( _V 
\  { 0 } )  =  ( _V 
\  ( `'bits " { (/)
} ) )
8160, 64, 803sstr4i 3538 . . . . 5  |-  ( `'bits " ( _V  \  { (/) } ) ) 
C_  ( _V  \  { 0 } )
82 sspreima 27633 . . . . 5  |-  ( ( Fun  ( A  |`  J )  /\  ( `'bits " ( _V  \  { (/) } ) ) 
C_  ( _V  \  { 0 } ) )  ->  ( `' ( A  |`  J )
" ( `'bits " ( _V  \  { (/) } ) ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
8357, 81, 82sylancl 662 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( A  |`  J )
" ( `'bits " ( _V  \  { (/) } ) ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
8454, 83syl5eqss 3543 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
85 ssfi 7759 . . 3  |-  ( ( ( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  e.  Fin  /\  ( `' (bits  o.  ( A  |`  J ) ) " ( _V 
\  { (/) } ) )  C_  ( `' ( A  |`  J )
" ( _V  \  { 0 } ) ) )  ->  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) )  e.  Fin )
8650, 84, 85syl2anc 661 . 2  |-  ( A  e.  ( T  i^i  R )  ->  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  e. 
Fin )
87 oveq1 6303 . . . . 5  |-  ( r  =  (bits  o.  ( A  |`  J ) )  ->  ( r supp  (/) )  =  ( (bits  o.  ( A  |`  J ) ) supp  (/) ) )
8887eleq1d 2526 . . . 4  |-  ( r  =  (bits  o.  ( A  |`  J ) )  ->  ( ( r supp  (/) )  e.  Fin  <->  (
(bits  o.  ( A  |`  J ) ) supp  (/) )  e. 
Fin ) )
8988, 9elrab2 3259 . . 3  |-  ( (bits 
o.  ( A  |`  J ) )  e.  H  <->  ( (bits  o.  ( A  |`  J ) )  e.  ( ( ~P NN0  i^i  Fin )  ^m  J )  /\  ( (bits  o.  ( A  |`  J ) ) supp  (/) )  e.  Fin ) )
90 zex 10894 . . . . . 6  |-  ZZ  e.  _V
91 fex 6146 . . . . . 6  |-  ( (bits
: ZZ --> ~P NN0  /\  ZZ  e.  _V )  -> bits  e.  _V )
9261, 90, 91mp2an 672 . . . . 5  |- bits  e.  _V
93 resexg 5326 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( A  |`  J )  e.  _V )
94 coexg 6750 . . . . 5  |-  ( (bits 
e.  _V  /\  ( A  |`  J )  e. 
_V )  ->  (bits  o.  ( A  |`  J ) )  e.  _V )
9592, 93, 94sylancr 663 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  _V )
96 0ex 4587 . . . . . . 7  |-  (/)  e.  _V
97 suppimacnv 6928 . . . . . . 7  |-  ( ( (bits  o.  ( A  |`  J ) )  e. 
_V  /\  (/)  e.  _V )  ->  ( (bits  o.  ( A  |`  J ) ) supp  (/) )  =  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) ) )
9896, 97mpan2 671 . . . . . 6  |-  ( (bits 
o.  ( A  |`  J ) )  e. 
_V  ->  ( (bits  o.  ( A  |`  J ) ) supp  (/) )  =  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) ) )
9998eleq1d 2526 . . . . 5  |-  ( (bits 
o.  ( A  |`  J ) )  e. 
_V  ->  ( ( (bits 
o.  ( A  |`  J ) ) supp  (/) )  e. 
Fin 
<->  ( `' (bits  o.  ( A  |`  J ) ) " ( _V 
\  { (/) } ) )  e.  Fin )
)
10099anbi2d 703 . . . 4  |-  ( (bits 
o.  ( A  |`  J ) )  e. 
_V  ->  ( ( (bits 
o.  ( A  |`  J ) )  e.  ( ( ~P NN0  i^i 
Fin )  ^m  J
)  /\  ( (bits  o.  ( A  |`  J ) ) supp  (/) )  e.  Fin ) 
<->  ( (bits  o.  ( A  |`  J ) )  e.  ( ( ~P
NN0  i^i  Fin )  ^m  J )  /\  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) )  e.  Fin ) ) )
10195, 100syl 16 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( (
(bits  o.  ( A  |`  J ) )  e.  ( ( ~P NN0  i^i 
Fin )  ^m  J
)  /\  ( (bits  o.  ( A  |`  J ) ) supp  (/) )  e.  Fin ) 
<->  ( (bits  o.  ( A  |`  J ) )  e.  ( ( ~P
NN0  i^i  Fin )  ^m  J )  /\  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) )  e.  Fin ) ) )
10289, 101syl5bb 257 . 2  |-  ( A  e.  ( T  i^i  R )  ->  ( (bits  o.  ( A  |`  J ) )  e.  H  <->  ( (bits  o.  ( A  |`  J ) )  e.  ( ( ~P NN0  i^i  Fin )  ^m  J )  /\  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) )  e.  Fin ) ) )
10330, 86, 102mpbir2and 922 1  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  H )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   {crab 2811   _Vcvv 3109    \ cdif 3468    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   class class class wbr 4456   {copab 4514    |-> cmpt 4515   `'ccnv 5007    |` cres 5010   "cima 5011    o. ccom 5012   Fun wfun 5588    Fn wfn 5589   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   supp csupp 6917    ^m cmap 7438   Fincfn 7535   0cc0 9509   1c1 9510    x. cmul 9514    <_ cle 9646   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ^cexp 12169   sum_csu 13520    || cdvds 13998  bitscbits 14081  𝟭cind 28185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-dvds 13999  df-bits 14084
This theorem is referenced by:  eulerpartlemgvv  28512  eulerpartlemgf  28515
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