MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bpoly2 Structured version   Unicode version

Theorem bpoly2 14088
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 10886 . . 3  |-  2  e.  NN0
2 bpolyval 14080 . . 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 674 . 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 2m1e1 10724 . . . . . . 7  |-  ( 2  -  1 )  =  1
5 0p1e1 10721 . . . . . . 7  |-  ( 0  +  1 )  =  1
64, 5eqtr4i 2461 . . . . . 6  |-  ( 2  -  1 )  =  ( 0  +  1 )
76oveq2i 6316 . . . . 5  |-  ( 0 ... ( 2  -  1 ) )  =  ( 0 ... (
0  +  1 ) )
87sumeq1i 13742 . . . 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 ) ) )
9 0nn0 10884 . . . . . . . . 9  |-  0  e.  NN0
10 nn0uz 11193 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
119, 10eleqtri 2515 . . . . . . . 8  |-  0  e.  ( ZZ>= `  0 )
1211a1i 11 . . . . . . 7  |-  ( X  e.  CC  ->  0  e.  ( ZZ>= `  0 )
)
13 0z 10948 . . . . . . . . . . 11  |-  0  e.  ZZ
14 fzpr 11849 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  (
0 ... ( 0  +  1 ) )  =  { 0 ,  ( 0  +  1 ) } )
1513, 14ax-mp 5 . . . . . . . . . 10  |-  ( 0 ... ( 0  +  1 ) )  =  { 0 ,  ( 0  +  1 ) }
1615eleq2i 2507 . . . . . . . . 9  |-  ( k  e.  ( 0 ... ( 0  +  1 ) )  <->  k  e.  { 0 ,  ( 0  +  1 ) } )
17 vex 3090 . . . . . . . . . 10  |-  k  e. 
_V
1817elpr 4020 . . . . . . . . 9  |-  ( k  e.  { 0 ,  ( 0  +  1 ) }  <->  ( k  =  0  \/  k  =  ( 0  +  1 ) ) )
1916, 18bitri 252 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( 0  +  1 ) )  <->  ( k  =  0  \/  k  =  ( 0  +  1 ) ) )
20 oveq2 6313 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
2  _C  k )  =  ( 2  _C  0 ) )
21 bcn0 12492 . . . . . . . . . . . . . 14  |-  ( 2  e.  NN0  ->  ( 2  _C  0 )  =  1 )
221, 21ax-mp 5 . . . . . . . . . . . . 13  |-  ( 2  _C  0 )  =  1
2320, 22syl6eq 2486 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
2  _C  k )  =  1 )
24 oveq1 6312 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
k BernPoly  X )  =  ( 0 BernPoly  X ) )
25 oveq2 6313 . . . . . . . . . . . . . . 15  |-  ( k  =  0  ->  (
2  -  k )  =  ( 2  -  0 ) )
2625oveq1d 6320 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
( 2  -  k
)  +  1 )  =  ( ( 2  -  0 )  +  1 ) )
27 2cn 10680 . . . . . . . . . . . . . . . . 17  |-  2  e.  CC
2827subid1i 9945 . . . . . . . . . . . . . . . 16  |-  ( 2  -  0 )  =  2
2928oveq1i 6315 . . . . . . . . . . . . . . 15  |-  ( ( 2  -  0 )  +  1 )  =  ( 2  +  1 )
30 df-3 10669 . . . . . . . . . . . . . . 15  |-  3  =  ( 2  +  1 )
3129, 30eqtr4i 2461 . . . . . . . . . . . . . 14  |-  ( ( 2  -  0 )  +  1 )  =  3
3226, 31syl6eq 2486 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
( 2  -  k
)  +  1 )  =  3 )
3324, 32oveq12d 6323 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
( k BernPoly  X )  /  ( ( 2  -  k )  +  1 ) )  =  ( ( 0 BernPoly  X
)  /  3 ) )
3423, 33oveq12d 6323 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  3
) ) )
35 bpoly0 14081 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
0 BernPoly  X )  =  1 )
3635oveq1d 6320 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  (
( 0 BernPoly  X )  /  3 )  =  ( 1  /  3
) )
3736oveq2d 6321 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  =  ( 1  x.  ( 1  /  3 ) ) )
38 3cn 10684 . . . . . . . . . . . . . 14  |-  3  e.  CC
39 3ne0 10704 . . . . . . . . . . . . . 14  |-  3  =/=  0
4038, 39reccli 10336 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  CC
4140mulid2i 9645 . . . . . . . . . . . 12  |-  ( 1  x.  ( 1  / 
3 ) )  =  ( 1  /  3
)
4237, 41syl6eq 2486 . . . . . . . . . . 11  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  =  ( 1  /  3 ) )
4334, 42sylan9eqr 2492 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  0 )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( 1  /  3 ) )
4443, 40syl6eqel 2525 . . . . . . . . 9  |-  ( ( X  e.  CC  /\  k  =  0 )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
455eqeq2i 2447 . . . . . . . . . . . 12  |-  ( k  =  ( 0  +  1 )  <->  k  = 
1 )
46 oveq2 6313 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
2  _C  k )  =  ( 2  _C  1 ) )
47 bcn1 12495 . . . . . . . . . . . . . . 15  |-  ( 2  e.  NN0  ->  ( 2  _C  1 )  =  2 )
481, 47ax-mp 5 . . . . . . . . . . . . . 14  |-  ( 2  _C  1 )  =  2
4946, 48syl6eq 2486 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  (
2  _C  k )  =  2 )
50 oveq1 6312 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
k BernPoly  X )  =  ( 1 BernPoly  X ) )
51 oveq2 6313 . . . . . . . . . . . . . . . 16  |-  ( k  =  1  ->  (
2  -  k )  =  ( 2  -  1 ) )
5251oveq1d 6320 . . . . . . . . . . . . . . 15  |-  ( k  =  1  ->  (
( 2  -  k
)  +  1 )  =  ( ( 2  -  1 )  +  1 ) )
53 ax-1cn 9596 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
54 npcan 9883 . . . . . . . . . . . . . . . 16  |-  ( ( 2  e.  CC  /\  1  e.  CC )  ->  ( ( 2  -  1 )  +  1 )  =  2 )
5527, 53, 54mp2an 676 . . . . . . . . . . . . . . 15  |-  ( ( 2  -  1 )  +  1 )  =  2
5652, 55syl6eq 2486 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
( 2  -  k
)  +  1 )  =  2 )
5750, 56oveq12d 6323 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  (
( k BernPoly  X )  /  ( ( 2  -  k )  +  1 ) )  =  ( ( 1 BernPoly  X
)  /  2 ) )
5849, 57oveq12d 6323 . . . . . . . . . . . 12  |-  ( k  =  1  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 2  x.  ( ( 1 BernPoly  X )  /  2
) ) )
5945, 58sylbi 198 . . . . . . . . . . 11  |-  ( k  =  ( 0  +  1 )  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 2  x.  ( ( 1 BernPoly  X )  /  2
) ) )
60 bpoly1 14082 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
1 BernPoly  X )  =  ( X  -  ( 1  /  2 ) ) )
6160oveq1d 6320 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  (
( 1 BernPoly  X )  /  2 )  =  ( ( X  -  ( 1  /  2
) )  /  2
) )
6261oveq2d 6321 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
2  x.  ( ( 1 BernPoly  X )  /  2
) )  =  ( 2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) ) )
63 halfcn 10829 . . . . . . . . . . . . . 14  |-  ( 1  /  2 )  e.  CC
64 subcl 9873 . . . . . . . . . . . . . 14  |-  ( ( X  e.  CC  /\  ( 1  /  2
)  e.  CC )  ->  ( X  -  ( 1  /  2
) )  e.  CC )
6563, 64mpan2 675 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  ( X  -  ( 1  /  2 ) )  e.  CC )
66 2ne0 10702 . . . . . . . . . . . . . 14  |-  2  =/=  0
67 divcan2 10277 . . . . . . . . . . . . . 14  |-  ( ( ( X  -  (
1  /  2 ) )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) )  =  ( X  -  ( 1  /  2
) ) )
6827, 66, 67mp3an23 1352 . . . . . . . . . . . . 13  |-  ( ( X  -  ( 1  /  2 ) )  e.  CC  ->  (
2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) )  =  ( X  -  ( 1  /  2
) ) )
6965, 68syl 17 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) )  =  ( X  -  ( 1  /  2
) ) )
7062, 69eqtrd 2470 . . . . . . . . . . 11  |-  ( X  e.  CC  ->  (
2  x.  ( ( 1 BernPoly  X )  /  2
) )  =  ( X  -  ( 1  /  2 ) ) )
7159, 70sylan9eqr 2492 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( X  -  ( 1  / 
2 ) ) )
7265adantr 466 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( X  -  ( 1  /  2
) )  e.  CC )
7371, 72eqeltrd 2517 . . . . . . . . 9  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
7444, 73jaodan 792 . . . . . . . 8  |-  ( ( X  e.  CC  /\  ( k  =  0  \/  k  =  ( 0  +  1 ) ) )  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  e.  CC )
7519, 74sylan2b 477 . . . . . . 7  |-  ( ( X  e.  CC  /\  k  e.  ( 0 ... ( 0  +  1 ) ) )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
7612, 75, 59fsump1 13795 . . . . . 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
) ) ) )
7742, 40syl6eqel 2525 . . . . . . . . 9  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  e.  CC )
7834fsum1 13786 . . . . . . . . 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
) ) )
7913, 77, 78sylancr 667 . . . . . . . 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
) ) )
8079, 42eqtrd 2470 . . . . . . 7  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... 0
) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( 1  /  3 ) )
8180, 70oveq12d 6323 . . . . . 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 ) ) ) )
8276, 81eqtrd 2470 . . . . 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
) ) ) )
83 addsub12 9887 . . . . . . 7  |-  ( ( ( 1  /  3
)  e.  CC  /\  X  e.  CC  /\  (
1  /  2 )  e.  CC )  -> 
( ( 1  / 
3 )  +  ( X  -  ( 1  /  2 ) ) )  =  ( X  +  ( ( 1  /  3 )  -  ( 1  /  2
) ) ) )
8440, 63, 83mp3an13 1351 . . . . . 6  |-  ( X  e.  CC  ->  (
( 1  /  3
)  +  ( X  -  ( 1  / 
2 ) ) )  =  ( X  +  ( ( 1  / 
3 )  -  (
1  /  2 ) ) ) )
8563, 40negsubdi2i 9960 . . . . . . . 8  |-  -u (
( 1  /  2
)  -  ( 1  /  3 ) )  =  ( ( 1  /  3 )  -  ( 1  /  2
) )
86 halfthird 11157 . . . . . . . . 9  |-  ( ( 1  /  2 )  -  ( 1  / 
3 ) )  =  ( 1  /  6
)
8786negeqi 9867 . . . . . . . 8  |-  -u (
( 1  /  2
)  -  ( 1  /  3 ) )  =  -u ( 1  / 
6 )
8885, 87eqtr3i 2460 . . . . . . 7  |-  ( ( 1  /  3 )  -  ( 1  / 
2 ) )  = 
-u ( 1  / 
6 )
8988oveq2i 6316 . . . . . 6  |-  ( X  +  ( ( 1  /  3 )  -  ( 1  /  2
) ) )  =  ( X  +  -u ( 1  /  6
) )
9084, 89syl6eq 2486 . . . . 5  |-  ( X  e.  CC  ->  (
( 1  /  3
)  +  ( X  -  ( 1  / 
2 ) ) )  =  ( X  +  -u ( 1  /  6
) ) )
91 6cn 10691 . . . . . . 7  |-  6  e.  CC
92 6re 10690 . . . . . . . 8  |-  6  e.  RR
93 6pos 10708 . . . . . . . 8  |-  0  <  6
9492, 93gt0ne0ii 10149 . . . . . . 7  |-  6  =/=  0
9591, 94reccli 10336 . . . . . 6  |-  ( 1  /  6 )  e.  CC
96 negsub 9921 . . . . . 6  |-  ( ( X  e.  CC  /\  ( 1  /  6
)  e.  CC )  ->  ( X  +  -u ( 1  /  6
) )  =  ( X  -  ( 1  /  6 ) ) )
9795, 96mpan2 675 . . . . 5  |-  ( X  e.  CC  ->  ( X  +  -u ( 1  /  6 ) )  =  ( X  -  ( 1  /  6
) ) )
9882, 90, 973eqtrd 2474 . . . 4  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... (
0  +  1 ) ) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( X  -  ( 1  / 
6 ) ) )
998, 98syl5eq 2482 . . 3  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... (
2  -  1 ) ) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( X  -  ( 1  / 
6 ) ) )
10099oveq2d 6321 . 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 ) ) ) )
101 sqcl 12334 . . 3  |-  ( X  e.  CC  ->  ( X ^ 2 )  e.  CC )
102 subsub 9903 . . . 4  |-  ( ( ( X ^ 2 )  e.  CC  /\  X  e.  CC  /\  (
1  /  6 )  e.  CC )  -> 
( ( X ^
2 )  -  ( X  -  ( 1  /  6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  / 
6 ) ) )
10395, 102mp3an3 1349 . . 3  |-  ( ( ( X ^ 2 )  e.  CC  /\  X  e.  CC )  ->  ( ( X ^
2 )  -  ( X  -  ( 1  /  6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  / 
6 ) ) )
104101, 103mpancom 673 . 2  |-  ( X  e.  CC  ->  (
( X ^ 2 )  -  ( X  -  ( 1  / 
6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  /  6
) ) )
1053, 100, 1043eqtrd 2474 1  |-  ( X  e.  CC  ->  (
2 BernPoly  X )  =  ( ( ( X ^
2 )  -  X
)  +  ( 1  /  6 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   {cpr 4004   ` cfv 5601  (class class class)co 6305   CCcc 9536   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543    - cmin 9859   -ucneg 9860    / cdiv 10268   2c2 10659   3c3 10660   6c6 10663   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11782   ^cexp 12269    _C cbc 12484   sum_csu 13730   BernPoly cbp 14077
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-bpoly 14078
This theorem is referenced by:  bpoly3  14089  bpoly4  14090
  Copyright terms: Public domain W3C validator