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Theorem eulerpartlemmf 29031
Description: Lemma for eulerpart 29038 (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 14382 . . . . 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 29025 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  <->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
1413biimpi 197 . . . . . . 7  |-  ( A  e.  ( T  i^i  R )  ->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
1514simp1d 1017 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  A  e.  ( NN0  ^m  NN ) )
16 nn0ex 10864 . . . . . . 7  |-  NN0  e.  _V
17 nnex 10604 . . . . . . 7  |-  NN  e.  _V
1816, 17elmap 7499 . . . . . 6  |-  ( A  e.  ( NN0  ^m  NN )  <->  A : NN --> NN0 )
1915, 18sylib 199 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  A : NN
--> NN0 )
20 ssrab2 3543 . . . . . 6  |-  { z  e.  NN  |  -.  2  ||  z }  C_  NN
217, 20eqsstri 3491 . . . . 5  |-  J  C_  NN
22 fssres 5757 . . . . 5  |-  ( ( A : NN --> NN0  /\  J  C_  NN )  -> 
( A  |`  J ) : J --> NN0 )
2319, 21, 22sylancl 666 . . . 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 667 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) ) : J --> ( ~P
NN0  i^i  Fin )
)
2616pwex 4599 . . . . 5  |-  ~P NN0  e.  _V
2726inex1 4557 . . . 4  |-  ( ~P
NN0  i^i  Fin )  e.  _V
2817, 21ssexi 4561 . . . 4  |-  J  e. 
_V
2927, 28elmap 7499 . . 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 215 . 2  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  ( ( ~P NN0  i^i  Fin )  ^m  J ) )
3114simp2d 1018 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " NN )  e. 
Fin )
32 0nn0 10873 . . . . . . . . 9  |-  0  e.  NN0
33 suppimacnv 6927 . . . . . . . . 9  |-  ( ( A  e.  ( T  i^i  R )  /\  0  e.  NN0 )  -> 
( A supp  0 )  =  ( `' A " ( _V  \  {
0 } ) ) )
3432, 33mpan2 675 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  ( A supp  0 )  =  ( `' A " ( _V 
\  { 0 } ) ) )
35 frnsuppeq 6928 . . . . . . . . . 10  |-  ( ( NN  e.  _V  /\  0  e.  NN0 )  -> 
( A : NN --> NN0  ->  ( A supp  0
)  =  ( `' A " ( NN0  \  { 0 } ) ) ) )
3617, 32, 35mp2an 676 . . . . . . . . 9  |-  ( A : NN --> NN0  ->  ( A supp  0 )  =  ( `' A "
( NN0  \  { 0 } ) ) )
3719, 36syl 17 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  ( A supp  0 )  =  ( `' A " ( NN0  \  { 0 } ) ) )
3834, 37eqtr3d 2463 . . . . . . 7  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " ( _V  \  { 0 } ) )  =  ( `' A " ( NN0  \  { 0 } ) ) )
3938eleq1d 2489 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( ( `' A " ( _V 
\  { 0 } ) )  e.  Fin  <->  ( `' A " ( NN0  \  { 0 } ) )  e.  Fin )
)
40 dfn2 10871 . . . . . . . 8  |-  NN  =  ( NN0  \  { 0 } )
4140imaeq2i 5177 . . . . . . 7  |-  ( `' A " NN )  =  ( `' A " ( NN0  \  {
0 } ) )
4241eleq1i 2497 . . . . . 6  |-  ( ( `' A " NN )  e.  Fin  <->  ( `' A " ( NN0  \  {
0 } ) )  e.  Fin )
4339, 42syl6bbr 266 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( ( `' A " ( _V 
\  { 0 } ) )  e.  Fin  <->  ( `' A " NN )  e.  Fin ) )
4431, 43mpbird 235 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " ( _V  \  { 0 } ) )  e.  Fin )
45 resss 5139 . . . . 5  |-  ( A  |`  J )  C_  A
46 cnvss 5018 . . . . 5  |-  ( ( A  |`  J )  C_  A  ->  `' ( A  |`  J )  C_  `' A )
47 imass1 5214 . . . . 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 7789 . . . 4  |-  ( ( ( `' A "
( _V  \  {
0 } ) )  e.  Fin  /\  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  C_  ( `' A " ( _V 
\  { 0 } ) ) )  -> 
( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  e.  Fin )
5044, 48, 49sylancl 666 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( A  |`  J )
" ( _V  \  { 0 } ) )  e.  Fin )
51 cnvco 5031 . . . . . 6  |-  `' (bits 
o.  ( A  |`  J ) )  =  ( `' ( A  |`  J )  o.  `'bits )
5251imaeq1i 5176 . . . . 5  |-  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  =  ( ( `' ( A  |`  J )  o.  `'bits ) " ( _V 
\  { (/) } ) )
53 imaco 5351 . . . . 5  |-  ( ( `' ( A  |`  J )  o.  `'bits ) " ( _V  \  { (/) } ) )  =  ( `' ( A  |`  J ) " ( `'bits " ( _V  \  { (/) } ) ) )
5452, 53eqtri 2449 . . . 4  |-  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  =  ( `' ( A  |`  J ) " ( `'bits " ( _V  \  { (/) } ) ) )
55 ffun 5739 . . . . . 6  |-  ( A : NN --> NN0  ->  Fun 
A )
56 funres 5631 . . . . . 6  |-  ( Fun 
A  ->  Fun  ( A  |`  J ) )
5719, 55, 563syl 18 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  Fun  ( A  |`  J ) )
58 ssv 3481 . . . . . . 7  |-  ( `'bits " _V )  C_  _V
59 ssdif 3597 . . . . . . 7  |-  ( ( `'bits " _V )  C_  _V  ->  ( ( `'bits " _V )  \  ( `'bits " { (/) } ) )  C_  ( _V  \  ( `'bits " { (/)
} ) ) )
6058, 59ax-mp 5 . . . . . 6  |-  ( ( `'bits " _V )  \ 
( `'bits " { (/) } ) )  C_  ( _V  \  ( `'bits " { (/)
} ) )
61 bitsf 14364 . . . . . . 7  |- bits : ZZ --> ~P NN0
62 ffun 5739 . . . . . . 7  |-  (bits : ZZ
--> ~P NN0  ->  Fun bits )
63 difpreima 6014 . . . . . . 7  |-  ( Fun bits  ->  ( `'bits " ( _V  \  { (/) } ) )  =  ( ( `'bits " _V )  \ 
( `'bits " { (/) } ) ) )
6461, 62, 63mp2b 10 . . . . . 6  |-  ( `'bits " ( _V  \  { (/) } ) )  =  ( ( `'bits " _V )  \  ( `'bits " { (/) } ) )
65 bitsf1 14383 . . . . . . . . 9  |- bits : ZZ -1-1-> ~P
NN0
66 0z 10937 . . . . . . . . . 10  |-  0  e.  ZZ
67 snssi 4138 . . . . . . . . . 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 676 . . . . . . . 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 5932 . . . . . . . . . . 11  |-  ( (bits 
Fn  ZZ  /\  0  e.  ZZ )  ->  { (bits `  0 ) }  =  (bits " {
0 } ) )
7472, 66, 73mp2an 676 . . . . . . . . . 10  |-  { (bits `  0 ) }  =  (bits " {
0 } )
75 0bits 14376 . . . . . . . . . . 11  |-  (bits ` 
0 )  =  (/)
7675sneqi 4004 . . . . . . . . . 10  |-  { (bits `  0 ) }  =  { (/) }
7774, 76eqtr3i 2451 . . . . . . . . 9  |-  (bits " { 0 } )  =  { (/) }
7877imaeq2i 5177 . . . . . . . 8  |-  ( `'bits " (bits " { 0 } ) )  =  ( `'bits " { (/) } )
7970, 78eqtr3i 2451 . . . . . . 7  |-  { 0 }  =  ( `'bits " { (/) } )
8079difeq2i 3577 . . . . . 6  |-  ( _V 
\  { 0 } )  =  ( _V 
\  ( `'bits " { (/)
} ) )
8160, 64, 803sstr4i 3500 . . . . 5  |-  ( `'bits " ( _V  \  { (/) } ) ) 
C_  ( _V  \  { 0 } )
82 sspreima 28083 . . . . 5  |-  ( ( Fun  ( A  |`  J )  /\  ( `'bits " ( _V  \  { (/) } ) ) 
C_  ( _V  \  { 0 } ) )  ->  ( `' ( A  |`  J )
" ( `'bits " ( _V  \  { (/) } ) ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
8357, 81, 82sylancl 666 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( A  |`  J )
" ( `'bits " ( _V  \  { (/) } ) ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
8454, 83syl5eqss 3505 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
85 ssfi 7789 . . 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 665 . 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 2489 . . . 4  |-  ( r  =  (bits  o.  ( A  |`  J ) )  ->  ( ( r supp  (/) )  e.  Fin  <->  (
(bits  o.  ( A  |`  J ) ) supp  (/) )  e. 
Fin ) )
8988, 9elrab2 3228 . . 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 10935 . . . . . 6  |-  ZZ  e.  _V
91 fex 6144 . . . . . 6  |-  ( (bits
: ZZ --> ~P NN0  /\  ZZ  e.  _V )  -> bits  e.  _V )
9261, 90, 91mp2an 676 . . . . 5  |- bits  e.  _V
93 resexg 5158 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( A  |`  J )  e.  _V )
94 coexg 6749 . . . . 5  |-  ( (bits 
e.  _V  /\  ( A  |`  J )  e. 
_V )  ->  (bits  o.  ( A  |`  J ) )  e.  _V )
9592, 93, 94sylancr 667 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  _V )
96 0ex 4548 . . . . . . 7  |-  (/)  e.  _V
97 suppimacnv 6927 . . . . . . 7  |-  ( ( (bits  o.  ( A  |`  J ) )  e. 
_V  /\  (/)  e.  _V )  ->  ( (bits  o.  ( A  |`  J ) ) supp  (/) )  =  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) ) )
9896, 97mpan2 675 . . . . . 6  |-  ( (bits 
o.  ( A  |`  J ) )  e. 
_V  ->  ( (bits  o.  ( A  |`  J ) ) supp  (/) )  =  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) ) )
9998eleq1d 2489 . . . . 5  |-  ( (bits 
o.  ( A  |`  J ) )  e. 
_V  ->  ( ( (bits 
o.  ( A  |`  J ) ) supp  (/) )  e. 
Fin 
<->  ( `' (bits  o.  ( A  |`  J ) ) " ( _V 
\  { (/) } ) )  e.  Fin )
)
10099anbi2d 708 . . . 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 17 . . 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 260 . 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 930 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   {cab 2405   A.wral 2773   {crab 2777   _Vcvv 3078    \ cdif 3430    i^i cin 3432    C_ wss 3433   (/)c0 3758   ~Pcpw 3976   {csn 3993   class class class wbr 4417   {copab 4474    |-> cmpt 4475   `'ccnv 4844    |` cres 4847   "cima 4848    o. ccom 4849   Fun wfun 5586    Fn wfn 5587   -->wf 5588   -1-1->wf1 5589   -1-1-onto->wf1o 5591   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   supp csupp 6916    ^m cmap 7471   Fincfn 7568   0cc0 9528   1c1 9529    x. cmul 9533    <_ cle 9665   NNcn 10598   2c2 10648   NN0cn0 10858   ZZcz 10926   ^cexp 12258   sum_csu 13719    || cdvds 14272  bitscbits 14356  𝟭cind 28668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-disj 4389  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-oi 8016  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-fz 11772  df-fzo 11903  df-fl 12014  df-mod 12083  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519  df-sum 13720  df-dvds 14273  df-bits 14359
This theorem is referenced by:  eulerpartlemgvv  29032  eulerpartlemgf  29035
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