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Theorem eulerpartlemmf 27982
Description: Lemma for eulerpart 27989 (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 13954 . . . . . 6  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
2 f1of 5816 . . . . . 6  |-  ( (bits  |`  NN0 ) : NN0 -1-1-onto-> ( ~P NN0  i^i  Fin )  ->  (bits  |`  NN0 ) : NN0 --> ( ~P NN0  i^i 
Fin ) )
31, 2ax-mp 5 . . . . 5  |-  (bits  |`  NN0 ) : NN0 --> ( ~P NN0  i^i 
Fin )
4 eulerpart.p . . . . . . . . . 10  |-  P  =  { f  e.  ( NN0  ^m  NN )  |  ( ( `' f " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( f `  k
)  x.  k )  =  N ) }
5 eulerpart.o . . . . . . . . . 10  |-  O  =  { g  e.  P  |  A. n  e.  ( `' g " NN )  -.  2  ||  n }
6 eulerpart.d . . . . . . . . . 10  |-  D  =  { g  e.  P  |  A. n  e.  NN  ( g `  n
)  <_  1 }
7 eulerpart.j . . . . . . . . . 10  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
8 eulerpart.f . . . . . . . . . 10  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
9 eulerpart.h . . . . . . . . . 10  |-  H  =  { r  e.  ( ( ~P NN0  i^i  Fin )  ^m  J )  |  ( r supp  (/) )  e. 
Fin }
10 eulerpart.m . . . . . . . . . 10  |-  M  =  ( r  e.  H  |->  { <. x ,  y
>.  |  ( x  e.  J  /\  y  e.  ( r `  x
) ) } )
11 eulerpart.r . . . . . . . . . 10  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
12 eulerpart.t . . . . . . . . . 10  |-  T  =  { f  e.  ( NN0  ^m  NN )  |  ( `' f
" NN )  C_  J }
134, 5, 6, 7, 8, 9, 10, 11, 12eulerpartlemt0 27976 . . . . . . . . 9  |-  ( A  e.  ( T  i^i  R )  <->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
1413biimpi 194 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
1514simp1d 1008 . . . . . . 7  |-  ( A  e.  ( T  i^i  R )  ->  A  e.  ( NN0  ^m  NN ) )
16 nn0ex 10801 . . . . . . . 8  |-  NN0  e.  _V
17 nnex 10542 . . . . . . . 8  |-  NN  e.  _V
1816, 17elmap 7447 . . . . . . 7  |-  ( A  e.  ( NN0  ^m  NN )  <->  A : NN --> NN0 )
1915, 18sylib 196 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  A : NN
--> NN0 )
20 ssrab2 3585 . . . . . . 7  |-  { z  e.  NN  |  -.  2  ||  z }  C_  NN
217, 20eqsstri 3534 . . . . . 6  |-  J  C_  NN
22 fssres 5751 . . . . . 6  |-  ( ( A : NN --> NN0  /\  J  C_  NN )  -> 
( A  |`  J ) : J --> NN0 )
2319, 21, 22sylancl 662 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( A  |`  J ) : J --> NN0 )
24 fco2 5742 . . . . 5  |-  ( ( (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 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) ) : J --> ( ~P
NN0  i^i  Fin )
)
2616pwex 4630 . . . . . 6  |-  ~P NN0  e.  _V
2726inex1 4588 . . . . 5  |-  ( ~P
NN0  i^i  Fin )  e.  _V
2817, 21ssexi 4592 . . . . 5  |-  J  e. 
_V
2927, 28elmap 7447 . . . 4  |-  ( (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 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  ( ( ~P NN0  i^i  Fin )  ^m  J ) )
3114simp2d 1009 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " NN )  e. 
Fin )
32 0nn0 10810 . . . . . . . . . 10  |-  0  e.  NN0
33 suppimacnv 6912 . . . . . . . . . 10  |-  ( ( A  e.  ( T  i^i  R )  /\  0  e.  NN0 )  -> 
( A supp  0 )  =  ( `' A " ( _V  \  {
0 } ) ) )
3432, 33mpan2 671 . . . . . . . . 9  |-  ( A  e.  ( T  i^i  R )  ->  ( A supp  0 )  =  ( `' A " ( _V 
\  { 0 } ) ) )
35 frnsuppeq 6913 . . . . . . . . . . 11  |-  ( ( NN  e.  _V  /\  0  e.  NN0 )  -> 
( A : NN --> NN0  ->  ( A supp  0
)  =  ( `' A " ( NN0  \  { 0 } ) ) ) )
3617, 32, 35mp2an 672 . . . . . . . . . 10  |-  ( A : NN --> NN0  ->  ( A supp  0 )  =  ( `' A "
( NN0  \  { 0 } ) ) )
3719, 36syl 16 . . . . . . . . 9  |-  ( A  e.  ( T  i^i  R )  ->  ( A supp  0 )  =  ( `' A " ( NN0  \  { 0 } ) ) )
3834, 37eqtr3d 2510 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " ( _V  \  { 0 } ) )  =  ( `' A " ( NN0  \  { 0 } ) ) )
3938eleq1d 2536 . . . . . . 7  |-  ( A  e.  ( T  i^i  R )  ->  ( ( `' A " ( _V 
\  { 0 } ) )  e.  Fin  <->  ( `' A " ( NN0  \  { 0 } ) )  e.  Fin )
)
40 dfn2 10808 . . . . . . . . 9  |-  NN  =  ( NN0  \  { 0 } )
4140imaeq2i 5335 . . . . . . . 8  |-  ( `' A " NN )  =  ( `' A " ( NN0  \  {
0 } ) )
4241eleq1i 2544 . . . . . . 7  |-  ( ( `' A " NN )  e.  Fin  <->  ( `' A " ( NN0  \  {
0 } ) )  e.  Fin )
4339, 42syl6bbr 263 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( ( `' A " ( _V 
\  { 0 } ) )  e.  Fin  <->  ( `' A " NN )  e.  Fin ) )
4431, 43mpbird 232 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " ( _V  \  { 0 } ) )  e.  Fin )
45 resss 5297 . . . . . . 7  |-  ( A  |`  J )  C_  A
46 cnvss 5175 . . . . . . 7  |-  ( ( A  |`  J )  C_  A  ->  `' ( A  |`  J )  C_  `' A )
47 imass1 5371 . . . . . . 7  |-  ( `' ( A  |`  J ) 
C_  `' A  -> 
( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  C_  ( `' A " ( _V 
\  { 0 } ) ) )
4845, 46, 47mp2b 10 . . . . . 6  |-  ( `' ( A  |`  J )
" ( _V  \  { 0 } ) )  C_  ( `' A " ( _V  \  { 0 } ) )
49 ssfi 7740 . . . . . 6  |-  ( ( ( `' A "
( _V  \  {
0 } ) )  e.  Fin  /\  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  C_  ( `' A " ( _V 
\  { 0 } ) ) )  -> 
( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  e.  Fin )
5048, 49mpan2 671 . . . . 5  |-  ( ( `' A " ( _V 
\  { 0 } ) )  e.  Fin  ->  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  e.  Fin )
5144, 50syl 16 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( A  |`  J )
" ( _V  \  { 0 } ) )  e.  Fin )
52 cnvco 5188 . . . . . . 7  |-  `' (bits 
o.  ( A  |`  J ) )  =  ( `' ( A  |`  J )  o.  `'bits )
5352imaeq1i 5334 . . . . . 6  |-  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  =  ( ( `' ( A  |`  J )  o.  `'bits ) " ( _V 
\  { (/) } ) )
54 imaco 5512 . . . . . 6  |-  ( ( `' ( A  |`  J )  o.  `'bits ) " ( _V  \  { (/) } ) )  =  ( `' ( A  |`  J ) " ( `'bits " ( _V  \  { (/) } ) ) )
5553, 54eqtri 2496 . . . . 5  |-  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  =  ( `' ( A  |`  J ) " ( `'bits " ( _V  \  { (/) } ) ) )
56 ffun 5733 . . . . . . 7  |-  ( A : NN --> NN0  ->  Fun 
A )
57 funres 5627 . . . . . . 7  |-  ( Fun 
A  ->  Fun  ( A  |`  J ) )
5819, 56, 573syl 20 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  Fun  ( A  |`  J ) )
59 ssv 3524 . . . . . . . 8  |-  ( `'bits " _V )  C_  _V
60 ssdif 3639 . . . . . . . 8  |-  ( ( `'bits " _V )  C_  _V  ->  ( ( `'bits " _V )  \  ( `'bits " { (/) } ) )  C_  ( _V  \  ( `'bits " { (/)
} ) ) )
6159, 60ax-mp 5 . . . . . . 7  |-  ( ( `'bits " _V )  \ 
( `'bits " { (/) } ) )  C_  ( _V  \  ( `'bits " { (/)
} ) )
62 bitsf 13936 . . . . . . . . 9  |- bits : ZZ --> ~P NN0
63 ffun 5733 . . . . . . . . 9  |-  (bits : ZZ
--> ~P NN0  ->  Fun bits )
6462, 63ax-mp 5 . . . . . . . 8  |-  Fun bits
65 difpreima 6009 . . . . . . . 8  |-  ( Fun bits  ->  ( `'bits " ( _V  \  { (/) } ) )  =  ( ( `'bits " _V )  \ 
( `'bits " { (/) } ) ) )
6664, 65ax-mp 5 . . . . . . 7  |-  ( `'bits " ( _V  \  { (/) } ) )  =  ( ( `'bits " _V )  \  ( `'bits " { (/) } ) )
67 bitsf1 13955 . . . . . . . . . 10  |- bits : ZZ -1-1-> ~P
NN0
68 0z 10875 . . . . . . . . . . 11  |-  0  e.  ZZ
69 snssi 4171 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  { 0 }  C_  ZZ )
7068, 69ax-mp 5 . . . . . . . . . 10  |-  { 0 }  C_  ZZ
71 f1imacnv 5832 . . . . . . . . . 10  |-  ( (bits
: ZZ -1-1-> ~P NN0  /\ 
{ 0 }  C_  ZZ )  ->  ( `'bits " (bits " { 0 } ) )  =  {
0 } )
7267, 70, 71mp2an 672 . . . . . . . . 9  |-  ( `'bits " (bits " { 0 } ) )  =  {
0 }
73 ffn 5731 . . . . . . . . . . . . 13  |-  (bits : ZZ
--> ~P NN0  -> bits  Fn  ZZ )
7462, 73ax-mp 5 . . . . . . . . . . . 12  |- bits  Fn  ZZ
75 fnsnfv 5927 . . . . . . . . . . . 12  |-  ( (bits 
Fn  ZZ  /\  0  e.  ZZ )  ->  { (bits `  0 ) }  =  (bits " {
0 } ) )
7674, 68, 75mp2an 672 . . . . . . . . . . 11  |-  { (bits `  0 ) }  =  (bits " {
0 } )
77 0bits 13948 . . . . . . . . . . . 12  |-  (bits ` 
0 )  =  (/)
7877sneqi 4038 . . . . . . . . . . 11  |-  { (bits `  0 ) }  =  { (/) }
7976, 78eqtr3i 2498 . . . . . . . . . 10  |-  (bits " { 0 } )  =  { (/) }
8079imaeq2i 5335 . . . . . . . . 9  |-  ( `'bits " (bits " { 0 } ) )  =  ( `'bits " { (/) } )
8172, 80eqtr3i 2498 . . . . . . . 8  |-  { 0 }  =  ( `'bits " { (/) } )
8281difeq2i 3619 . . . . . . 7  |-  ( _V 
\  { 0 } )  =  ( _V 
\  ( `'bits " { (/)
} ) )
8361, 66, 823sstr4i 3543 . . . . . 6  |-  ( `'bits " ( _V  \  { (/) } ) ) 
C_  ( _V  \  { 0 } )
84 sspreima 27185 . . . . . 6  |-  ( ( Fun  ( A  |`  J )  /\  ( `'bits " ( _V  \  { (/) } ) ) 
C_  ( _V  \  { 0 } ) )  ->  ( `' ( A  |`  J )
" ( `'bits " ( _V  \  { (/) } ) ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
8558, 83, 84sylancl 662 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( A  |`  J )
" ( `'bits " ( _V  \  { (/) } ) ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
8655, 85syl5eqss 3548 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
87 ssfi 7740 . . . 4  |-  ( ( ( `' ( A  |`  J ) " ( _V  \  { 0 } ) )  e.  Fin  /\  ( `' (bits  o.  ( A  |`  J ) ) " ( _V 
\  { (/) } ) )  C_  ( `' ( A  |`  J )
" ( _V  \  { 0 } ) ) )  ->  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) )  e.  Fin )
8851, 86, 87syl2anc 661 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  e. 
Fin )
8930, 88jca 532 . 2  |-  ( A  e.  ( T  i^i  R )  ->  ( (bits  o.  ( A  |`  J ) )  e.  ( ( ~P NN0  i^i  Fin )  ^m  J )  /\  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) )  e.  Fin ) )
90 oveq1 6291 . . . . 5  |-  ( r  =  (bits  o.  ( A  |`  J ) )  ->  ( r supp  (/) )  =  ( (bits  o.  ( A  |`  J ) ) supp  (/) ) )
9190eleq1d 2536 . . . 4  |-  ( r  =  (bits  o.  ( A  |`  J ) )  ->  ( ( r supp  (/) )  e.  Fin  <->  (
(bits  o.  ( A  |`  J ) ) supp  (/) )  e. 
Fin ) )
9291, 9elrab2 3263 . . 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 ) )
93 zex 10873 . . . . . 6  |-  ZZ  e.  _V
94 fex 6133 . . . . . 6  |-  ( (bits
: ZZ --> ~P NN0  /\  ZZ  e.  _V )  -> bits  e.  _V )
9562, 93, 94mp2an 672 . . . . 5  |- bits  e.  _V
96 resexg 5316 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( A  |`  J )  e.  _V )
97 coexg 6735 . . . . 5  |-  ( (bits 
e.  _V  /\  ( A  |`  J )  e. 
_V )  ->  (bits  o.  ( A  |`  J ) )  e.  _V )
9895, 96, 97sylancr 663 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  _V )
99 0ex 4577 . . . . . . 7  |-  (/)  e.  _V
100 suppimacnv 6912 . . . . . . 7  |-  ( ( (bits  o.  ( A  |`  J ) )  e. 
_V  /\  (/)  e.  _V )  ->  ( (bits  o.  ( A  |`  J ) ) supp  (/) )  =  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) ) )
10199, 100mpan2 671 . . . . . 6  |-  ( (bits 
o.  ( A  |`  J ) )  e. 
_V  ->  ( (bits  o.  ( A  |`  J ) ) supp  (/) )  =  ( `' (bits  o.  ( A  |`  J ) )
" ( _V  \  { (/) } ) ) )
102101eleq1d 2536 . . . . 5  |-  ( (bits 
o.  ( A  |`  J ) )  e. 
_V  ->  ( ( (bits 
o.  ( A  |`  J ) ) supp  (/) )  e. 
Fin 
<->  ( `' (bits  o.  ( A  |`  J ) ) " ( _V 
\  { (/) } ) )  e.  Fin )
)
103102anbi2d 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 ) ) )
10498, 103syl 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 ) ) )
10592, 104syl5bb 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 ) ) )
10689, 105mpbird 232 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 1379    e. wcel 1767   {cab 2452   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   class class class wbr 4447   {copab 4504    |-> cmpt 4505   `'ccnv 4998    |` cres 5001   "cima 5002    o. ccom 5003   Fun wfun 5582    Fn wfn 5583   -->wf 5584   -1-1->wf1 5585   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   supp csupp 6901    ^m cmap 7420   Fincfn 7516   0cc0 9492   1c1 9493    x. cmul 9497    <_ cle 9629   NNcn 10536   2c2 10585   NN0cn0 10795   ZZcz 10864   ^cexp 12134   sum_csu 13471    || cdivides 13847  bitscbits 13928  𝟭cind 27692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-dvds 13848  df-bits 13931
This theorem is referenced by:  eulerpartlemgvv  27983  eulerpartlemgf  27986
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