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Theorem eulerpartlemmf 29208
Description: Lemma for eulerpart 29215. (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 14419 . . . . 5  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
2 f1of 5814 . . . . 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 29202 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  <->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
1413biimpi 198 . . . . . . 7  |-  ( A  e.  ( T  i^i  R )  ->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
1514simp1d 1020 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  A  e.  ( NN0  ^m  NN ) )
16 nn0ex 10875 . . . . . . 7  |-  NN0  e.  _V
17 nnex 10615 . . . . . . 7  |-  NN  e.  _V
1816, 17elmap 7500 . . . . . 6  |-  ( A  e.  ( NN0  ^m  NN )  <->  A : NN --> NN0 )
1915, 18sylib 200 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  A : NN
--> NN0 )
20 ssrab2 3514 . . . . . 6  |-  { z  e.  NN  |  -.  2  ||  z }  C_  NN
217, 20eqsstri 3462 . . . . 5  |-  J  C_  NN
22 fssres 5749 . . . . 5  |-  ( ( A : NN --> NN0  /\  J  C_  NN )  -> 
( A  |`  J ) : J --> NN0 )
2319, 21, 22sylancl 668 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( A  |`  J ) : J --> NN0 )
24 fco2 5740 . . . 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 669 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) ) : J --> ( ~P
NN0  i^i  Fin )
)
2616pwex 4586 . . . . 5  |-  ~P NN0  e.  _V
2726inex1 4544 . . . 4  |-  ( ~P
NN0  i^i  Fin )  e.  _V
2817, 21ssexi 4548 . . . 4  |-  J  e. 
_V
2927, 28elmap 7500 . . 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 216 . 2  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  ( ( ~P NN0  i^i  Fin )  ^m  J ) )
3114simp2d 1021 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " NN )  e. 
Fin )
32 0nn0 10884 . . . . . . . . 9  |-  0  e.  NN0
33 suppimacnv 6925 . . . . . . . . 9  |-  ( ( A  e.  ( T  i^i  R )  /\  0  e.  NN0 )  -> 
( A supp  0 )  =  ( `' A " ( _V  \  {
0 } ) ) )
3432, 33mpan2 677 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  ( A supp  0 )  =  ( `' A " ( _V 
\  { 0 } ) ) )
35 frnsuppeq 6926 . . . . . . . . . 10  |-  ( ( NN  e.  _V  /\  0  e.  NN0 )  -> 
( A : NN --> NN0  ->  ( A supp  0
)  =  ( `' A " ( NN0  \  { 0 } ) ) ) )
3617, 32, 35mp2an 678 . . . . . . . . 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 2487 . . . . . . 7  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " ( _V  \  { 0 } ) )  =  ( `' A " ( NN0  \  { 0 } ) ) )
3938eleq1d 2513 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( ( `' A " ( _V 
\  { 0 } ) )  e.  Fin  <->  ( `' A " ( NN0  \  { 0 } ) )  e.  Fin )
)
40 dfn2 10882 . . . . . . . 8  |-  NN  =  ( NN0  \  { 0 } )
4140imaeq2i 5166 . . . . . . 7  |-  ( `' A " NN )  =  ( `' A " ( NN0  \  {
0 } ) )
4241eleq1i 2520 . . . . . 6  |-  ( ( `' A " NN )  e.  Fin  <->  ( `' A " ( NN0  \  {
0 } ) )  e.  Fin )
4339, 42syl6bbr 267 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( ( `' A " ( _V 
\  { 0 } ) )  e.  Fin  <->  ( `' A " NN )  e.  Fin ) )
4431, 43mpbird 236 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " ( _V  \  { 0 } ) )  e.  Fin )
45 resss 5128 . . . . 5  |-  ( A  |`  J )  C_  A
46 cnvss 5007 . . . . 5  |-  ( ( A  |`  J )  C_  A  ->  `' ( A  |`  J )  C_  `' A )
47 imass1 5203 . . . . 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 7792 . . . 4  |-  ( ( ( `' A "
( _V  \  {
0 } ) )  e.  Fin  /\  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  C_  ( `' A " ( _V 
\  { 0 } ) ) )  -> 
( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  e.  Fin )
5044, 48, 49sylancl 668 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( A  |`  J )
" ( _V  \  { 0 } ) )  e.  Fin )
51 cnvco 5020 . . . . . 6  |-  `' (bits 
o.  ( A  |`  J ) )  =  ( `' ( A  |`  J )  o.  `'bits )
5251imaeq1i 5165 . . . . 5  |-  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  =  ( ( `' ( A  |`  J )  o.  `'bits ) " ( _V 
\  { (/) } ) )
53 imaco 5340 . . . . 5  |-  ( ( `' ( A  |`  J )  o.  `'bits ) " ( _V  \  { (/) } ) )  =  ( `' ( A  |`  J ) " ( `'bits " ( _V  \  { (/) } ) ) )
5452, 53eqtri 2473 . . . 4  |-  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  =  ( `' ( A  |`  J ) " ( `'bits " ( _V  \  { (/) } ) ) )
55 ffun 5731 . . . . . 6  |-  ( A : NN --> NN0  ->  Fun 
A )
56 funres 5621 . . . . . 6  |-  ( Fun 
A  ->  Fun  ( A  |`  J ) )
5719, 55, 563syl 18 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  Fun  ( A  |`  J ) )
58 ssv 3452 . . . . . . 7  |-  ( `'bits " _V )  C_  _V
59 ssdif 3568 . . . . . . 7  |-  ( ( `'bits " _V )  C_  _V  ->  ( ( `'bits " _V )  \  ( `'bits " { (/) } ) )  C_  ( _V  \  ( `'bits " { (/)
} ) ) )
6058, 59ax-mp 5 . . . . . 6  |-  ( ( `'bits " _V )  \ 
( `'bits " { (/) } ) )  C_  ( _V  \  ( `'bits " { (/)
} ) )
61 bitsf 14400 . . . . . . 7  |- bits : ZZ --> ~P NN0
62 ffun 5731 . . . . . . 7  |-  (bits : ZZ
--> ~P NN0  ->  Fun bits )
63 difpreima 6008 . . . . . . 7  |-  ( Fun bits  ->  ( `'bits " ( _V  \  { (/) } ) )  =  ( ( `'bits " _V )  \ 
( `'bits " { (/) } ) ) )
6461, 62, 63mp2b 10 . . . . . 6  |-  ( `'bits " ( _V  \  { (/) } ) )  =  ( ( `'bits " _V )  \  ( `'bits " { (/) } ) )
65 bitsf1 14420 . . . . . . . . 9  |- bits : ZZ -1-1-> ~P
NN0
66 0z 10948 . . . . . . . . . 10  |-  0  e.  ZZ
67 snssi 4116 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  { 0 }  C_  ZZ )
6866, 67ax-mp 5 . . . . . . . . 9  |-  { 0 }  C_  ZZ
69 f1imacnv 5830 . . . . . . . . 9  |-  ( (bits
: ZZ -1-1-> ~P NN0  /\ 
{ 0 }  C_  ZZ )  ->  ( `'bits " (bits " { 0 } ) )  =  {
0 } )
7065, 68, 69mp2an 678 . . . . . . . 8  |-  ( `'bits " (bits " { 0 } ) )  =  {
0 }
71 ffn 5728 . . . . . . . . . . . 12  |-  (bits : ZZ
--> ~P NN0  -> bits  Fn  ZZ )
7261, 71ax-mp 5 . . . . . . . . . . 11  |- bits  Fn  ZZ
73 fnsnfv 5925 . . . . . . . . . . 11  |-  ( (bits 
Fn  ZZ  /\  0  e.  ZZ )  ->  { (bits `  0 ) }  =  (bits " {
0 } ) )
7472, 66, 73mp2an 678 . . . . . . . . . 10  |-  { (bits `  0 ) }  =  (bits " {
0 } )
75 0bits 14413 . . . . . . . . . . 11  |-  (bits ` 
0 )  =  (/)
7675sneqi 3979 . . . . . . . . . 10  |-  { (bits `  0 ) }  =  { (/) }
7774, 76eqtr3i 2475 . . . . . . . . 9  |-  (bits " { 0 } )  =  { (/) }
7877imaeq2i 5166 . . . . . . . 8  |-  ( `'bits " (bits " { 0 } ) )  =  ( `'bits " { (/) } )
7970, 78eqtr3i 2475 . . . . . . 7  |-  { 0 }  =  ( `'bits " { (/) } )
8079difeq2i 3548 . . . . . 6  |-  ( _V 
\  { 0 } )  =  ( _V 
\  ( `'bits " { (/)
} ) )
8160, 64, 803sstr4i 3471 . . . . 5  |-  ( `'bits " ( _V  \  { (/) } ) ) 
C_  ( _V  \  { 0 } )
82 sspreima 28246 . . . . 5  |-  ( ( Fun  ( A  |`  J )  /\  ( `'bits " ( _V  \  { (/) } ) ) 
C_  ( _V  \  { 0 } ) )  ->  ( `' ( A  |`  J )
" ( `'bits " ( _V  \  { (/) } ) ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
8357, 81, 82sylancl 668 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( A  |`  J )
" ( `'bits " ( _V  \  { (/) } ) ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
8454, 83syl5eqss 3476 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
85 ssfi 7792 . . 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 667 . 2  |-  ( A  e.  ( T  i^i  R )  ->  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  e. 
Fin )
87 oveq1 6297 . . . . 5  |-  ( r  =  (bits  o.  ( A  |`  J ) )  ->  ( r supp  (/) )  =  ( (bits  o.  ( A  |`  J ) ) supp  (/) ) )
8887eleq1d 2513 . . . 4  |-  ( r  =  (bits  o.  ( A  |`  J ) )  ->  ( ( r supp  (/) )  e.  Fin  <->  (
(bits  o.  ( A  |`  J ) ) supp  (/) )  e. 
Fin ) )
8988, 9elrab2 3198 . . 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 10946 . . . . . 6  |-  ZZ  e.  _V
91 fex 6138 . . . . . 6  |-  ( (bits
: ZZ --> ~P NN0  /\  ZZ  e.  _V )  -> bits  e.  _V )
9261, 90, 91mp2an 678 . . . . 5  |- bits  e.  _V
93 resexg 5147 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( A  |`  J )  e.  _V )
94 coexg 6744 . . . . 5  |-  ( (bits 
e.  _V  /\  ( A  |`  J )  e. 
_V )  ->  (bits  o.  ( A  |`  J ) )  e.  _V )
9592, 93, 94sylancr 669 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  _V )
96 0ex 4535 . . . . . . 7  |-  (/)  e.  _V
97 suppimacnv 6925 . . . . . . 7  |-  ( ( (bits  o.  ( A  |`  J ) )  e. 
_V  /\  (/)  e.  _V )  ->  ( (bits  o.  ( A  |`  J ) ) supp  (/) )  =  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) ) )
9896, 97mpan2 677 . . . . . 6  |-  ( (bits 
o.  ( A  |`  J ) )  e. 
_V  ->  ( (bits  o.  ( A  |`  J ) ) supp  (/) )  =  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) ) )
9998eleq1d 2513 . . . . 5  |-  ( (bits 
o.  ( A  |`  J ) )  e. 
_V  ->  ( ( (bits 
o.  ( A  |`  J ) ) supp  (/) )  e. 
Fin 
<->  ( `' (bits  o.  ( A  |`  J ) ) " ( _V 
\  { (/) } ) )  e.  Fin )
)
10099anbi2d 710 . . . 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 261 . 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 933 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   {cab 2437   A.wral 2737   {crab 2741   _Vcvv 3045    \ cdif 3401    i^i cin 3403    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   {csn 3968   class class class wbr 4402   {copab 4460    |-> cmpt 4461   `'ccnv 4833    |` cres 4836   "cima 4837    o. ccom 4838   Fun wfun 5576    Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   supp csupp 6914    ^m cmap 7472   Fincfn 7569   0cc0 9539   1c1 9540    x. cmul 9544    <_ cle 9676   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ^cexp 12272   sum_csu 13752    || cdvds 14305  bitscbits 14392  𝟭cind 28832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-dvds 14306  df-bits 14395
This theorem is referenced by:  eulerpartlemgvv  29209  eulerpartlemgf  29212
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