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Theorem eulerpartlemmf 26763
Description: Lemma for eulerpart 26770 (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 13646 . . . . . 6  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
2 f1of 5646 . . . . . 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 26757 . . . . . . . . 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 1000 . . . . . . 7  |-  ( A  e.  ( T  i^i  R )  ->  A  e.  ( NN0  ^m  NN ) )
16 nn0ex 10590 . . . . . . . 8  |-  NN0  e.  _V
17 nnex 10333 . . . . . . . 8  |-  NN  e.  _V
1816, 17elmap 7246 . . . . . . 7  |-  ( A  e.  ( NN0  ^m  NN )  <->  A : NN --> NN0 )
1915, 18sylib 196 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  A : NN
--> NN0 )
20 ssrab2 3442 . . . . . . 7  |-  { z  e.  NN  |  -.  2  ||  z }  C_  NN
217, 20eqsstri 3391 . . . . . 6  |-  J  C_  NN
22 fssres 5583 . . . . . 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 5574 . . . . 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 4480 . . . . . 6  |-  ~P NN0  e.  _V
2726inex1 4438 . . . . 5  |-  ( ~P
NN0  i^i  Fin )  e.  _V
2817, 21ssexi 4442 . . . . 5  |-  J  e. 
_V
2927, 28elmap 7246 . . . 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 1001 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " NN )  e. 
Fin )
32 0nn0 10599 . . . . . . . . . 10  |-  0  e.  NN0
33 suppimacnv 6706 . . . . . . . . . 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 6707 . . . . . . . . . . 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 2477 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " ( _V  \  { 0 } ) )  =  ( `' A " ( NN0  \  { 0 } ) ) )
3938eleq1d 2509 . . . . . . 7  |-  ( A  e.  ( T  i^i  R )  ->  ( ( `' A " ( _V 
\  { 0 } ) )  e.  Fin  <->  ( `' A " ( NN0  \  { 0 } ) )  e.  Fin )
)
40 dfn2 10597 . . . . . . . . 9  |-  NN  =  ( NN0  \  { 0 } )
4140imaeq2i 5172 . . . . . . . 8  |-  ( `' A " NN )  =  ( `' A " ( NN0  \  {
0 } ) )
4241eleq1i 2506 . . . . . . 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 5139 . . . . . . 7  |-  ( A  |`  J )  C_  A
46 cnvss 5017 . . . . . . 7  |-  ( ( A  |`  J )  C_  A  ->  `' ( A  |`  J )  C_  `' A )
47 imass1 5208 . . . . . . 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 7538 . . . . . 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 5030 . . . . . . 7  |-  `' (bits 
o.  ( A  |`  J ) )  =  ( `' ( A  |`  J )  o.  `'bits )
5352imaeq1i 5171 . . . . . 6  |-  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  =  ( ( `' ( A  |`  J )  o.  `'bits ) " ( _V 
\  { (/) } ) )
54 imaco 5348 . . . . . 6  |-  ( ( `' ( A  |`  J )  o.  `'bits ) " ( _V  \  { (/) } ) )  =  ( `' ( A  |`  J ) " ( `'bits " ( _V  \  { (/) } ) ) )
5553, 54eqtri 2463 . . . . 5  |-  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  =  ( `' ( A  |`  J ) " ( `'bits " ( _V  \  { (/) } ) ) )
56 ffun 5566 . . . . . . 7  |-  ( A : NN --> NN0  ->  Fun 
A )
57 funres 5462 . . . . . . 7  |-  ( Fun 
A  ->  Fun  ( A  |`  J ) )
5819, 56, 573syl 20 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  Fun  ( A  |`  J ) )
59 ssv 3381 . . . . . . . 8  |-  ( `'bits " _V )  C_  _V
60 ssdif 3496 . . . . . . . 8  |-  ( ( `'bits " _V )  C_  _V  ->  ( ( `'bits " _V )  \  ( `'bits " { (/) } ) )  C_  ( _V  \  ( `'bits " { (/)
} ) ) )
6159, 60ax-mp 5 . . . . . . 7  |-  ( ( `'bits " _V )  \ 
( `'bits " { (/) } ) )  C_  ( _V  \  ( `'bits " { (/)
} ) )
62 bitsf 13628 . . . . . . . . 9  |- bits : ZZ --> ~P NN0
63 ffun 5566 . . . . . . . . 9  |-  (bits : ZZ
--> ~P NN0  ->  Fun bits )
6462, 63ax-mp 5 . . . . . . . 8  |-  Fun bits
65 difpreima 5836 . . . . . . . 8  |-  ( Fun bits  ->  ( `'bits " ( _V  \  { (/) } ) )  =  ( ( `'bits " _V )  \ 
( `'bits " { (/) } ) ) )
6664, 65ax-mp 5 . . . . . . 7  |-  ( `'bits " ( _V  \  { (/) } ) )  =  ( ( `'bits " _V )  \  ( `'bits " { (/) } ) )
67 bitsf1 13647 . . . . . . . . . 10  |- bits : ZZ -1-1-> ~P
NN0
68 0z 10662 . . . . . . . . . . 11  |-  0  e.  ZZ
69 snssi 4022 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  { 0 }  C_  ZZ )
7068, 69ax-mp 5 . . . . . . . . . 10  |-  { 0 }  C_  ZZ
71 f1imacnv 5662 . . . . . . . . . 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 5564 . . . . . . . . . . . . 13  |-  (bits : ZZ
--> ~P NN0  -> bits  Fn  ZZ )
7462, 73ax-mp 5 . . . . . . . . . . . 12  |- bits  Fn  ZZ
75 fnsnfv 5756 . . . . . . . . . . . 12  |-  ( (bits 
Fn  ZZ  /\  0  e.  ZZ )  ->  { (bits `  0 ) }  =  (bits " {
0 } ) )
7674, 68, 75mp2an 672 . . . . . . . . . . 11  |-  { (bits `  0 ) }  =  (bits " {
0 } )
77 0bits 13640 . . . . . . . . . . . 12  |-  (bits ` 
0 )  =  (/)
7877sneqi 3893 . . . . . . . . . . 11  |-  { (bits `  0 ) }  =  { (/) }
7976, 78eqtr3i 2465 . . . . . . . . . 10  |-  (bits " { 0 } )  =  { (/) }
8079imaeq2i 5172 . . . . . . . . 9  |-  ( `'bits " (bits " { 0 } ) )  =  ( `'bits " { (/) } )
8172, 80eqtr3i 2465 . . . . . . . 8  |-  { 0 }  =  ( `'bits " { (/) } )
8281difeq2i 3476 . . . . . . 7  |-  ( _V 
\  { 0 } )  =  ( _V 
\  ( `'bits " { (/)
} ) )
8361, 66, 823sstr4i 3400 . . . . . 6  |-  ( `'bits " ( _V  \  { (/) } ) ) 
C_  ( _V  \  { 0 } )
84 sspreima 25967 . . . . . 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 3405 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( `' (bits  o.  ( A  |`  J ) ) "
( _V  \  { (/)
} ) )  C_  ( `' ( A  |`  J ) " ( _V  \  { 0 } ) ) )
87 ssfi 7538 . . . 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 6103 . . . . 5  |-  ( r  =  (bits  o.  ( A  |`  J ) )  ->  ( r supp  (/) )  =  ( (bits  o.  ( A  |`  J ) ) supp  (/) ) )
9190eleq1d 2509 . . . 4  |-  ( r  =  (bits  o.  ( A  |`  J ) )  ->  ( ( r supp  (/) )  e.  Fin  <->  (
(bits  o.  ( A  |`  J ) ) supp  (/) )  e. 
Fin ) )
9291, 9elrab2 3124 . . 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 10660 . . . . . 6  |-  ZZ  e.  _V
94 fex 5955 . . . . . 6  |-  ( (bits
: ZZ --> ~P NN0  /\  ZZ  e.  _V )  -> bits  e.  _V )
9562, 93, 94mp2an 672 . . . . 5  |- bits  e.  _V
96 resexg 5154 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( A  |`  J )  e.  _V )
97 coexg 6533 . . . . 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 4427 . . . . . . 7  |-  (/)  e.  _V
100 suppimacnv 6706 . . . . . . 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 2509 . . . . 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 965    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   {crab 2724   _Vcvv 2977    \ cdif 3330    i^i cin 3332    C_ wss 3333   (/)c0 3642   ~Pcpw 3865   {csn 3882   class class class wbr 4297   {copab 4354    e. cmpt 4355   `'ccnv 4844    |` cres 4847   "cima 4848    o. ccom 4849   Fun wfun 5417    Fn wfn 5418   -->wf 5419   -1-1->wf1 5420   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   supp csupp 6695    ^m cmap 7219   Fincfn 7315   0cc0 9287   1c1 9288    x. cmul 9292    <_ cle 9424   NNcn 10327   2c2 10376   NN0cn0 10584   ZZcz 10651   ^cexp 11870   sum_csu 13168    || cdivides 13540  bitscbits 13620  𝟭cind 26472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-disj 4268  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-dvds 13541  df-bits 13623
This theorem is referenced by:  eulerpartlemgvv  26764  eulerpartlemgf  26767
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