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Theorem bpoly2 24792
Description: The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)
Assertion
Ref Expression
bpoly2  |-  ( X  e.  CC  ->  (
2 BernPoly  X )  =  ( ( ( X ^
2 )  -  X
)  +  ( 1  /  6 ) ) )

Proof of Theorem bpoly2
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 2nn0 9982 . . 3  |-  2  e.  NN0
2 bpolyval 24784 . . 3  |-  ( ( 2  e.  NN0  /\  X  e.  CC )  ->  ( 2 BernPoly  X )  =  ( ( X ^ 2 )  -  sum_ k  e.  ( 0 ... ( 2  -  1 ) ) ( ( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) ) ) )
31, 2mpan 651 . 2  |-  ( X  e.  CC  ->  (
2 BernPoly  X )  =  ( ( X ^ 2 )  -  sum_ k  e.  ( 0 ... (
2  -  1 ) ) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) ) ) )
4 2cn 9816 . . . . . . . 8  |-  2  e.  CC
5 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
6 1p1e2 9840 . . . . . . . 8  |-  ( 1  +  1 )  =  2
74, 5, 5, 6subaddrii 9135 . . . . . . 7  |-  ( 2  -  1 )  =  1
85addid2i 9000 . . . . . . 7  |-  ( 0  +  1 )  =  1
97, 8eqtr4i 2306 . . . . . 6  |-  ( 2  -  1 )  =  ( 0  +  1 )
109oveq2i 5869 . . . . 5  |-  ( 0 ... ( 2  -  1 ) )  =  ( 0 ... (
0  +  1 ) )
1110sumeq1i 12171 . . . 4  |-  sum_ k  e.  ( 0 ... (
2  -  1 ) ) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  sum_ k  e.  ( 0 ... (
0  +  1 ) ) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )
12 0nn0 9980 . . . . . . . . 9  |-  0  e.  NN0
13 nn0uz 10262 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
1412, 13eleqtri 2355 . . . . . . . 8  |-  0  e.  ( ZZ>= `  0 )
1514a1i 10 . . . . . . 7  |-  ( X  e.  CC  ->  0  e.  ( ZZ>= `  0 )
)
16 0z 10035 . . . . . . . . . . 11  |-  0  e.  ZZ
17 fzpr 10840 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  (
0 ... ( 0  +  1 ) )  =  { 0 ,  ( 0  +  1 ) } )
1816, 17ax-mp 8 . . . . . . . . . 10  |-  ( 0 ... ( 0  +  1 ) )  =  { 0 ,  ( 0  +  1 ) }
1918eleq2i 2347 . . . . . . . . 9  |-  ( k  e.  ( 0 ... ( 0  +  1 ) )  <->  k  e.  { 0 ,  ( 0  +  1 ) } )
20 vex 2791 . . . . . . . . . 10  |-  k  e. 
_V
2120elpr 3658 . . . . . . . . 9  |-  ( k  e.  { 0 ,  ( 0  +  1 ) }  <->  ( k  =  0  \/  k  =  ( 0  +  1 ) ) )
2219, 21bitri 240 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( 0  +  1 ) )  <->  ( k  =  0  \/  k  =  ( 0  +  1 ) ) )
23 oveq2 5866 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
2  _C  k )  =  ( 2  _C  0 ) )
24 bcn0 11323 . . . . . . . . . . . . . 14  |-  ( 2  e.  NN0  ->  ( 2  _C  0 )  =  1 )
251, 24ax-mp 8 . . . . . . . . . . . . 13  |-  ( 2  _C  0 )  =  1
2623, 25syl6eq 2331 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
2  _C  k )  =  1 )
27 oveq1 5865 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
k BernPoly  X )  =  ( 0 BernPoly  X ) )
28 oveq2 5866 . . . . . . . . . . . . . . 15  |-  ( k  =  0  ->  (
2  -  k )  =  ( 2  -  0 ) )
2928oveq1d 5873 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
( 2  -  k
)  +  1 )  =  ( ( 2  -  0 )  +  1 ) )
304subid1i 9118 . . . . . . . . . . . . . . . 16  |-  ( 2  -  0 )  =  2
3130oveq1i 5868 . . . . . . . . . . . . . . 15  |-  ( ( 2  -  0 )  +  1 )  =  ( 2  +  1 )
32 df-3 9805 . . . . . . . . . . . . . . 15  |-  3  =  ( 2  +  1 )
3331, 32eqtr4i 2306 . . . . . . . . . . . . . 14  |-  ( ( 2  -  0 )  +  1 )  =  3
3429, 33syl6eq 2331 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
( 2  -  k
)  +  1 )  =  3 )
3527, 34oveq12d 5876 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
( k BernPoly  X )  /  ( ( 2  -  k )  +  1 ) )  =  ( ( 0 BernPoly  X
)  /  3 ) )
3626, 35oveq12d 5876 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  3
) ) )
37 bpoly0 24785 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
0 BernPoly  X )  =  1 )
3837oveq1d 5873 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  (
( 0 BernPoly  X )  /  3 )  =  ( 1  /  3
) )
3938oveq2d 5874 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  =  ( 1  x.  ( 1  /  3 ) ) )
40 3cn 9818 . . . . . . . . . . . . . 14  |-  3  e.  CC
41 3ne0 9831 . . . . . . . . . . . . . 14  |-  3  =/=  0
4240, 41reccli 9490 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  CC
4342mulid2i 8840 . . . . . . . . . . . 12  |-  ( 1  x.  ( 1  / 
3 ) )  =  ( 1  /  3
)
4439, 43syl6eq 2331 . . . . . . . . . . 11  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  =  ( 1  /  3 ) )
4536, 44sylan9eqr 2337 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  0 )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( 1  /  3 ) )
4645, 42syl6eqel 2371 . . . . . . . . 9  |-  ( ( X  e.  CC  /\  k  =  0 )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
478eqeq2i 2293 . . . . . . . . . . . 12  |-  ( k  =  ( 0  +  1 )  <->  k  = 
1 )
48 oveq2 5866 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
2  _C  k )  =  ( 2  _C  1 ) )
49 bcn1 11325 . . . . . . . . . . . . . . 15  |-  ( 2  e.  NN0  ->  ( 2  _C  1 )  =  2 )
501, 49ax-mp 8 . . . . . . . . . . . . . 14  |-  ( 2  _C  1 )  =  2
5148, 50syl6eq 2331 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  (
2  _C  k )  =  2 )
52 oveq1 5865 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
k BernPoly  X )  =  ( 1 BernPoly  X ) )
53 oveq2 5866 . . . . . . . . . . . . . . . 16  |-  ( k  =  1  ->  (
2  -  k )  =  ( 2  -  1 ) )
5453oveq1d 5873 . . . . . . . . . . . . . . 15  |-  ( k  =  1  ->  (
( 2  -  k
)  +  1 )  =  ( ( 2  -  1 )  +  1 ) )
55 npcan 9060 . . . . . . . . . . . . . . . 16  |-  ( ( 2  e.  CC  /\  1  e.  CC )  ->  ( ( 2  -  1 )  +  1 )  =  2 )
564, 5, 55mp2an 653 . . . . . . . . . . . . . . 15  |-  ( ( 2  -  1 )  +  1 )  =  2
5754, 56syl6eq 2331 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
( 2  -  k
)  +  1 )  =  2 )
5852, 57oveq12d 5876 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  (
( k BernPoly  X )  /  ( ( 2  -  k )  +  1 ) )  =  ( ( 1 BernPoly  X
)  /  2 ) )
5951, 58oveq12d 5876 . . . . . . . . . . . 12  |-  ( k  =  1  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 2  x.  ( ( 1 BernPoly  X )  /  2
) ) )
6047, 59sylbi 187 . . . . . . . . . . 11  |-  ( k  =  ( 0  +  1 )  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 2  x.  ( ( 1 BernPoly  X )  /  2
) ) )
61 bpoly1 24786 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
1 BernPoly  X )  =  ( X  -  ( 1  /  2 ) ) )
6261oveq1d 5873 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  (
( 1 BernPoly  X )  /  2 )  =  ( ( X  -  ( 1  /  2
) )  /  2
) )
6362oveq2d 5874 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
2  x.  ( ( 1 BernPoly  X )  /  2
) )  =  ( 2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) ) )
64 halfcl 9937 . . . . . . . . . . . . . . 15  |-  ( 1  e.  CC  ->  (
1  /  2 )  e.  CC )
655, 64ax-mp 8 . . . . . . . . . . . . . 14  |-  ( 1  /  2 )  e.  CC
66 subcl 9051 . . . . . . . . . . . . . 14  |-  ( ( X  e.  CC  /\  ( 1  /  2
)  e.  CC )  ->  ( X  -  ( 1  /  2
) )  e.  CC )
6765, 66mpan2 652 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  ( X  -  ( 1  /  2 ) )  e.  CC )
68 2ne0 9829 . . . . . . . . . . . . . 14  |-  2  =/=  0
69 divcan2 9432 . . . . . . . . . . . . . 14  |-  ( ( ( X  -  (
1  /  2 ) )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) )  =  ( X  -  ( 1  /  2
) ) )
704, 68, 69mp3an23 1269 . . . . . . . . . . . . 13  |-  ( ( X  -  ( 1  /  2 ) )  e.  CC  ->  (
2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) )  =  ( X  -  ( 1  /  2
) ) )
7167, 70syl 15 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) )  =  ( X  -  ( 1  /  2
) ) )
7263, 71eqtrd 2315 . . . . . . . . . . 11  |-  ( X  e.  CC  ->  (
2  x.  ( ( 1 BernPoly  X )  /  2
) )  =  ( X  -  ( 1  /  2 ) ) )
7360, 72sylan9eqr 2337 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( X  -  ( 1  / 
2 ) ) )
7467adantr 451 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( X  -  ( 1  /  2
) )  e.  CC )
7573, 74eqeltrd 2357 . . . . . . . . 9  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
7646, 75jaodan 760 . . . . . . . 8  |-  ( ( X  e.  CC  /\  ( k  =  0  \/  k  =  ( 0  +  1 ) ) )  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  e.  CC )
7722, 76sylan2b 461 . . . . . . 7  |-  ( ( X  e.  CC  /\  k  e.  ( 0 ... ( 0  +  1 ) ) )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
7815, 77, 60fsump1 12219 . . . . . 6  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... (
0  +  1 ) ) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( sum_ k  e.  ( 0 ... 0 ) ( ( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  +  ( 2  x.  ( ( 1 BernPoly  X )  /  2
) ) ) )
7944, 42syl6eqel 2371 . . . . . . . . 9  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  e.  CC )
8036fsum1 12214 . . . . . . . . 9  |-  ( ( 0  e.  ZZ  /\  ( 1  x.  (
( 0 BernPoly  X )  /  3 ) )  e.  CC )  ->  sum_ k  e.  ( 0 ... 0 ) ( ( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  3
) ) )
8116, 79, 80sylancr 644 . . . . . . . 8  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... 0
) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  3
) ) )
8281, 44eqtrd 2315 . . . . . . 7  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... 0
) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( 1  /  3 ) )
8382, 72oveq12d 5876 . . . . . 6  |-  ( X  e.  CC  ->  ( sum_ k  e.  ( 0 ... 0 ) ( ( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  +  ( 2  x.  ( ( 1 BernPoly  X )  /  2
) ) )  =  ( ( 1  / 
3 )  +  ( X  -  ( 1  /  2 ) ) ) )
8478, 83eqtrd 2315 . . . . 5  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... (
0  +  1 ) ) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( ( 1  /  3 )  +  ( X  -  ( 1  /  2
) ) ) )
85 addsub12 9064 . . . . . . 7  |-  ( ( ( 1  /  3
)  e.  CC  /\  X  e.  CC  /\  (
1  /  2 )  e.  CC )  -> 
( ( 1  / 
3 )  +  ( X  -  ( 1  /  2 ) ) )  =  ( X  +  ( ( 1  /  3 )  -  ( 1  /  2
) ) ) )
8642, 65, 85mp3an13 1268 . . . . . 6  |-  ( X  e.  CC  ->  (
( 1  /  3
)  +  ( X  -  ( 1  / 
2 ) ) )  =  ( X  +  ( ( 1  / 
3 )  -  (
1  /  2 ) ) ) )
8765, 42negsubdi2i 9132 . . . . . . . 8  |-  -u (
( 1  /  2
)  -  ( 1  /  3 ) )  =  ( ( 1  /  3 )  -  ( 1  /  2
) )
88 halfthird 24100 . . . . . . . . 9  |-  ( ( 1  /  2 )  -  ( 1  / 
3 ) )  =  ( 1  /  6
)
8988negeqi 9045 . . . . . . . 8  |-  -u (
( 1  /  2
)  -  ( 1  /  3 ) )  =  -u ( 1  / 
6 )
9087, 89eqtr3i 2305 . . . . . . 7  |-  ( ( 1  /  3 )  -  ( 1  / 
2 ) )  = 
-u ( 1  / 
6 )
9190oveq2i 5869 . . . . . 6  |-  ( X  +  ( ( 1  /  3 )  -  ( 1  /  2
) ) )  =  ( X  +  -u ( 1  /  6
) )
9286, 91syl6eq 2331 . . . . 5  |-  ( X  e.  CC  ->  (
( 1  /  3
)  +  ( X  -  ( 1  / 
2 ) ) )  =  ( X  +  -u ( 1  /  6
) ) )
93 6re 9822 . . . . . . . 8  |-  6  e.  RR
9493recni 8849 . . . . . . 7  |-  6  e.  CC
95 6pos 9834 . . . . . . . 8  |-  0  <  6
9693, 95gt0ne0ii 9309 . . . . . . 7  |-  6  =/=  0
9794, 96reccli 9490 . . . . . 6  |-  ( 1  /  6 )  e.  CC
98 negsub 9095 . . . . . 6  |-  ( ( X  e.  CC  /\  ( 1  /  6
)  e.  CC )  ->  ( X  +  -u ( 1  /  6
) )  =  ( X  -  ( 1  /  6 ) ) )
9997, 98mpan2 652 . . . . 5  |-  ( X  e.  CC  ->  ( X  +  -u ( 1  /  6 ) )  =  ( X  -  ( 1  /  6
) ) )
10084, 92, 993eqtrd 2319 . . . 4  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... (
0  +  1 ) ) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( X  -  ( 1  / 
6 ) ) )
10111, 100syl5eq 2327 . . 3  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... (
2  -  1 ) ) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( X  -  ( 1  / 
6 ) ) )
102101oveq2d 5874 . 2  |-  ( X  e.  CC  ->  (
( X ^ 2 )  -  sum_ k  e.  ( 0 ... (
2  -  1 ) ) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) ) )  =  ( ( X ^ 2 )  -  ( X  -  ( 1  / 
6 ) ) ) )
103 sqcl 11166 . . 3  |-  ( X  e.  CC  ->  ( X ^ 2 )  e.  CC )
104 subsub 9077 . . . 4  |-  ( ( ( X ^ 2 )  e.  CC  /\  X  e.  CC  /\  (
1  /  6 )  e.  CC )  -> 
( ( X ^
2 )  -  ( X  -  ( 1  /  6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  / 
6 ) ) )
10597, 104mp3an3 1266 . . 3  |-  ( ( ( X ^ 2 )  e.  CC  /\  X  e.  CC )  ->  ( ( X ^
2 )  -  ( X  -  ( 1  /  6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  / 
6 ) ) )
106103, 105mpancom 650 . 2  |-  ( X  e.  CC  ->  (
( X ^ 2 )  -  ( X  -  ( 1  / 
6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  /  6
) ) )
1073, 102, 1063eqtrd 2319 1  |-  ( X  e.  CC  ->  (
2 BernPoly  X )  =  ( ( ( X ^
2 )  -  X
)  +  ( 1  /  6 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {cpr 3641   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   3c3 9796   6c6 9799   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   ^cexp 11104    _C cbc 11315   sum_csu 12158   BernPoly cbp 24781
This theorem is referenced by:  bpoly3  24793  bpoly4  24794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-pred 24168  df-bpoly 24782
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