Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bpoly2 Structured version   Unicode version

Theorem bpoly2 29737
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 10824 . . 3  |-  2  e.  NN0
2 bpolyval 29729 . . 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 670 . 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 10662 . . . . . . 7  |-  ( 2  -  1 )  =  1
5 0p1e1 10659 . . . . . . 7  |-  ( 0  +  1 )  =  1
64, 5eqtr4i 2499 . . . . . 6  |-  ( 2  -  1 )  =  ( 0  +  1 )
76oveq2i 6306 . . . . 5  |-  ( 0 ... ( 2  -  1 ) )  =  ( 0 ... (
0  +  1 ) )
87sumeq1i 13500 . . . 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 10822 . . . . . . . . 9  |-  0  e.  NN0
10 nn0uz 11128 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
119, 10eleqtri 2553 . . . . . . . 8  |-  0  e.  ( ZZ>= `  0 )
1211a1i 11 . . . . . . 7  |-  ( X  e.  CC  ->  0  e.  ( ZZ>= `  0 )
)
13 0z 10887 . . . . . . . . . . 11  |-  0  e.  ZZ
14 fzpr 11747 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  (
0 ... ( 0  +  1 ) )  =  { 0 ,  ( 0  +  1 ) } )
1513, 14ax-mp 5 . . . . . . . . . 10  |-  ( 0 ... ( 0  +  1 ) )  =  { 0 ,  ( 0  +  1 ) }
1615eleq2i 2545 . . . . . . . . 9  |-  ( k  e.  ( 0 ... ( 0  +  1 ) )  <->  k  e.  { 0 ,  ( 0  +  1 ) } )
17 vex 3121 . . . . . . . . . 10  |-  k  e. 
_V
1817elpr 4051 . . . . . . . . 9  |-  ( k  e.  { 0 ,  ( 0  +  1 ) }  <->  ( k  =  0  \/  k  =  ( 0  +  1 ) ) )
1916, 18bitri 249 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( 0  +  1 ) )  <->  ( k  =  0  \/  k  =  ( 0  +  1 ) ) )
20 oveq2 6303 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
2  _C  k )  =  ( 2  _C  0 ) )
21 bcn0 12368 . . . . . . . . . . . . . 14  |-  ( 2  e.  NN0  ->  ( 2  _C  0 )  =  1 )
221, 21ax-mp 5 . . . . . . . . . . . . 13  |-  ( 2  _C  0 )  =  1
2320, 22syl6eq 2524 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
2  _C  k )  =  1 )
24 oveq1 6302 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
k BernPoly  X )  =  ( 0 BernPoly  X ) )
25 oveq2 6303 . . . . . . . . . . . . . . 15  |-  ( k  =  0  ->  (
2  -  k )  =  ( 2  -  0 ) )
2625oveq1d 6310 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
( 2  -  k
)  +  1 )  =  ( ( 2  -  0 )  +  1 ) )
27 2cn 10618 . . . . . . . . . . . . . . . . 17  |-  2  e.  CC
2827subid1i 9903 . . . . . . . . . . . . . . . 16  |-  ( 2  -  0 )  =  2
2928oveq1i 6305 . . . . . . . . . . . . . . 15  |-  ( ( 2  -  0 )  +  1 )  =  ( 2  +  1 )
30 df-3 10607 . . . . . . . . . . . . . . 15  |-  3  =  ( 2  +  1 )
3129, 30eqtr4i 2499 . . . . . . . . . . . . . 14  |-  ( ( 2  -  0 )  +  1 )  =  3
3226, 31syl6eq 2524 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
( 2  -  k
)  +  1 )  =  3 )
3324, 32oveq12d 6313 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
( k BernPoly  X )  /  ( ( 2  -  k )  +  1 ) )  =  ( ( 0 BernPoly  X
)  /  3 ) )
3423, 33oveq12d 6313 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  3
) ) )
35 bpoly0 29730 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
0 BernPoly  X )  =  1 )
3635oveq1d 6310 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  (
( 0 BernPoly  X )  /  3 )  =  ( 1  /  3
) )
3736oveq2d 6311 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  =  ( 1  x.  ( 1  /  3 ) ) )
38 3cn 10622 . . . . . . . . . . . . . 14  |-  3  e.  CC
39 3ne0 10642 . . . . . . . . . . . . . 14  |-  3  =/=  0
4038, 39reccli 10286 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  CC
4140mulid2i 9611 . . . . . . . . . . . 12  |-  ( 1  x.  ( 1  / 
3 ) )  =  ( 1  /  3
)
4237, 41syl6eq 2524 . . . . . . . . . . 11  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  =  ( 1  /  3 ) )
4334, 42sylan9eqr 2530 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  0 )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( 1  /  3 ) )
4443, 40syl6eqel 2563 . . . . . . . . 9  |-  ( ( X  e.  CC  /\  k  =  0 )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
455eqeq2i 2485 . . . . . . . . . . . 12  |-  ( k  =  ( 0  +  1 )  <->  k  = 
1 )
46 oveq2 6303 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
2  _C  k )  =  ( 2  _C  1 ) )
47 bcn1 12371 . . . . . . . . . . . . . . 15  |-  ( 2  e.  NN0  ->  ( 2  _C  1 )  =  2 )
481, 47ax-mp 5 . . . . . . . . . . . . . 14  |-  ( 2  _C  1 )  =  2
4946, 48syl6eq 2524 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  (
2  _C  k )  =  2 )
50 oveq1 6302 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
k BernPoly  X )  =  ( 1 BernPoly  X ) )
51 oveq2 6303 . . . . . . . . . . . . . . . 16  |-  ( k  =  1  ->  (
2  -  k )  =  ( 2  -  1 ) )
5251oveq1d 6310 . . . . . . . . . . . . . . 15  |-  ( k  =  1  ->  (
( 2  -  k
)  +  1 )  =  ( ( 2  -  1 )  +  1 ) )
53 ax-1cn 9562 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
54 npcan 9841 . . . . . . . . . . . . . . . 16  |-  ( ( 2  e.  CC  /\  1  e.  CC )  ->  ( ( 2  -  1 )  +  1 )  =  2 )
5527, 53, 54mp2an 672 . . . . . . . . . . . . . . 15  |-  ( ( 2  -  1 )  +  1 )  =  2
5652, 55syl6eq 2524 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
( 2  -  k
)  +  1 )  =  2 )
5750, 56oveq12d 6313 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  (
( k BernPoly  X )  /  ( ( 2  -  k )  +  1 ) )  =  ( ( 1 BernPoly  X
)  /  2 ) )
5849, 57oveq12d 6313 . . . . . . . . . . . 12  |-  ( k  =  1  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 2  x.  ( ( 1 BernPoly  X )  /  2
) ) )
5945, 58sylbi 195 . . . . . . . . . . 11  |-  ( k  =  ( 0  +  1 )  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 2  x.  ( ( 1 BernPoly  X )  /  2
) ) )
60 bpoly1 29731 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
1 BernPoly  X )  =  ( X  -  ( 1  /  2 ) ) )
6160oveq1d 6310 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  (
( 1 BernPoly  X )  /  2 )  =  ( ( X  -  ( 1  /  2
) )  /  2
) )
6261oveq2d 6311 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
2  x.  ( ( 1 BernPoly  X )  /  2
) )  =  ( 2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) ) )
63 halfcn 10767 . . . . . . . . . . . . . 14  |-  ( 1  /  2 )  e.  CC
64 subcl 9831 . . . . . . . . . . . . . 14  |-  ( ( X  e.  CC  /\  ( 1  /  2
)  e.  CC )  ->  ( X  -  ( 1  /  2
) )  e.  CC )
6563, 64mpan2 671 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  ( X  -  ( 1  /  2 ) )  e.  CC )
66 2ne0 10640 . . . . . . . . . . . . . 14  |-  2  =/=  0
67 divcan2 10227 . . . . . . . . . . . . . 14  |-  ( ( ( X  -  (
1  /  2 ) )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) )  =  ( X  -  ( 1  /  2
) ) )
6827, 66, 67mp3an23 1316 . . . . . . . . . . . . 13  |-  ( ( X  -  ( 1  /  2 ) )  e.  CC  ->  (
2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) )  =  ( X  -  ( 1  /  2
) ) )
6965, 68syl 16 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) )  =  ( X  -  ( 1  /  2
) ) )
7062, 69eqtrd 2508 . . . . . . . . . . 11  |-  ( X  e.  CC  ->  (
2  x.  ( ( 1 BernPoly  X )  /  2
) )  =  ( X  -  ( 1  /  2 ) ) )
7159, 70sylan9eqr 2530 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( X  -  ( 1  / 
2 ) ) )
7265adantr 465 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( X  -  ( 1  /  2
) )  e.  CC )
7371, 72eqeltrd 2555 . . . . . . . . 9  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
7444, 73jaodan 783 . . . . . . . 8  |-  ( ( X  e.  CC  /\  ( k  =  0  \/  k  =  ( 0  +  1 ) ) )  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  e.  CC )
7519, 74sylan2b 475 . . . . . . 7  |-  ( ( X  e.  CC  /\  k  e.  ( 0 ... ( 0  +  1 ) ) )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
7612, 75, 59fsump1 13551 . . . . . 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 2563 . . . . . . . . 9  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  e.  CC )
7834fsum1 13544 . . . . . . . . 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 663 . . . . . . . 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 2508 . . . . . . 7  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... 0
) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( 1  /  3 ) )
8180, 70oveq12d 6313 . . . . . 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 2508 . . . . 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 9845 . . . . . . 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 1315 . . . . . 6  |-  ( X  e.  CC  ->  (
( 1  /  3
)  +  ( X  -  ( 1  / 
2 ) ) )  =  ( X  +  ( ( 1  / 
3 )  -  (
1  /  2 ) ) ) )
8563, 40negsubdi2i 9917 . . . . . . . 8  |-  -u (
( 1  /  2
)  -  ( 1  /  3 ) )  =  ( ( 1  /  3 )  -  ( 1  /  2
) )
86 halfthird 28929 . . . . . . . . 9  |-  ( ( 1  /  2 )  -  ( 1  / 
3 ) )  =  ( 1  /  6
)
8786negeqi 9825 . . . . . . . 8  |-  -u (
( 1  /  2
)  -  ( 1  /  3 ) )  =  -u ( 1  / 
6 )
8885, 87eqtr3i 2498 . . . . . . 7  |-  ( ( 1  /  3 )  -  ( 1  / 
2 ) )  = 
-u ( 1  / 
6 )
8988oveq2i 6306 . . . . . 6  |-  ( X  +  ( ( 1  /  3 )  -  ( 1  /  2
) ) )  =  ( X  +  -u ( 1  /  6
) )
9084, 89syl6eq 2524 . . . . 5  |-  ( X  e.  CC  ->  (
( 1  /  3
)  +  ( X  -  ( 1  / 
2 ) ) )  =  ( X  +  -u ( 1  /  6
) ) )
91 6cn 10629 . . . . . . 7  |-  6  e.  CC
92 6re 10628 . . . . . . . 8  |-  6  e.  RR
93 6pos 10646 . . . . . . . 8  |-  0  <  6
9492, 93gt0ne0ii 10101 . . . . . . 7  |-  6  =/=  0
9591, 94reccli 10286 . . . . . 6  |-  ( 1  /  6 )  e.  CC
96 negsub 9879 . . . . . 6  |-  ( ( X  e.  CC  /\  ( 1  /  6
)  e.  CC )  ->  ( X  +  -u ( 1  /  6
) )  =  ( X  -  ( 1  /  6 ) ) )
9795, 96mpan2 671 . . . . 5  |-  ( X  e.  CC  ->  ( X  +  -u ( 1  /  6 ) )  =  ( X  -  ( 1  /  6
) ) )
9882, 90, 973eqtrd 2512 . . . 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 2520 . . 3  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... (
2  -  1 ) ) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( X  -  ( 1  / 
6 ) ) )
10099oveq2d 6311 . 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 12210 . . 3  |-  ( X  e.  CC  ->  ( X ^ 2 )  e.  CC )
102 subsub 9861 . . . 4  |-  ( ( ( X ^ 2 )  e.  CC  /\  X  e.  CC  /\  (
1  /  6 )  e.  CC )  -> 
( ( X ^
2 )  -  ( X  -  ( 1  /  6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  / 
6 ) ) )
10395, 102mp3an3 1313 . . 3  |-  ( ( ( X ^ 2 )  e.  CC  /\  X  e.  CC )  ->  ( ( X ^
2 )  -  ( X  -  ( 1  /  6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  / 
6 ) ) )
104101, 103mpancom 669 . 2  |-  ( X  e.  CC  ->  (
( X ^ 2 )  -  ( X  -  ( 1  / 
6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  /  6
) ) )
1053, 100, 1043eqtrd 2512 1  |-  ( X  e.  CC  ->  (
2 BernPoly  X )  =  ( ( ( X ^
2 )  -  X
)  +  ( 1  /  6 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {cpr 4035   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    - cmin 9817   -ucneg 9818    / cdiv 10218   2c2 10597   3c3 10598   6c6 10601   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094   ...cfz 11684   ^cexp 12146    _C cbc 12360   sum_csu 13488   BernPoly cbp 29726
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-pred 29162  df-wrecs 29254  df-bpoly 29727
This theorem is referenced by:  bpoly3  29738  bpoly4  29739
  Copyright terms: Public domain W3C validator