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Theorem bpoly2 14103
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 10883 . . 3  |-  2  e.  NN0
2 bpolyval 14095 . . 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 675 . 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 10721 . . . . . . 7  |-  ( 2  -  1 )  =  1
5 0p1e1 10718 . . . . . . 7  |-  ( 0  +  1 )  =  1
64, 5eqtr4i 2475 . . . . . 6  |-  ( 2  -  1 )  =  ( 0  +  1 )
76oveq2i 6299 . . . . 5  |-  ( 0 ... ( 2  -  1 ) )  =  ( 0 ... (
0  +  1 ) )
87sumeq1i 13757 . . . 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 10881 . . . . . . . . 9  |-  0  e.  NN0
10 nn0uz 11190 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
119, 10eleqtri 2526 . . . . . . . 8  |-  0  e.  ( ZZ>= `  0 )
1211a1i 11 . . . . . . 7  |-  ( X  e.  CC  ->  0  e.  ( ZZ>= `  0 )
)
13 0z 10945 . . . . . . . . . . 11  |-  0  e.  ZZ
14 fzpr 11848 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  (
0 ... ( 0  +  1 ) )  =  { 0 ,  ( 0  +  1 ) } )
1513, 14ax-mp 5 . . . . . . . . . 10  |-  ( 0 ... ( 0  +  1 ) )  =  { 0 ,  ( 0  +  1 ) }
1615eleq2i 2520 . . . . . . . . 9  |-  ( k  e.  ( 0 ... ( 0  +  1 ) )  <->  k  e.  { 0 ,  ( 0  +  1 ) } )
17 vex 3047 . . . . . . . . . 10  |-  k  e. 
_V
1817elpr 3985 . . . . . . . . 9  |-  ( k  e.  { 0 ,  ( 0  +  1 ) }  <->  ( k  =  0  \/  k  =  ( 0  +  1 ) ) )
1916, 18bitri 253 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( 0  +  1 ) )  <->  ( k  =  0  \/  k  =  ( 0  +  1 ) ) )
20 oveq2 6296 . . . . . . . . . . . . 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 2500 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
2  _C  k )  =  1 )
24 oveq1 6295 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
k BernPoly  X )  =  ( 0 BernPoly  X ) )
25 oveq2 6296 . . . . . . . . . . . . . . 15  |-  ( k  =  0  ->  (
2  -  k )  =  ( 2  -  0 ) )
2625oveq1d 6303 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
( 2  -  k
)  +  1 )  =  ( ( 2  -  0 )  +  1 ) )
27 2cn 10677 . . . . . . . . . . . . . . . . 17  |-  2  e.  CC
2827subid1i 9943 . . . . . . . . . . . . . . . 16  |-  ( 2  -  0 )  =  2
2928oveq1i 6298 . . . . . . . . . . . . . . 15  |-  ( ( 2  -  0 )  +  1 )  =  ( 2  +  1 )
30 df-3 10666 . . . . . . . . . . . . . . 15  |-  3  =  ( 2  +  1 )
3129, 30eqtr4i 2475 . . . . . . . . . . . . . 14  |-  ( ( 2  -  0 )  +  1 )  =  3
3226, 31syl6eq 2500 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
( 2  -  k
)  +  1 )  =  3 )
3324, 32oveq12d 6306 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
( k BernPoly  X )  /  ( ( 2  -  k )  +  1 ) )  =  ( ( 0 BernPoly  X
)  /  3 ) )
3423, 33oveq12d 6306 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  3
) ) )
35 bpoly0 14096 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
0 BernPoly  X )  =  1 )
3635oveq1d 6303 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  (
( 0 BernPoly  X )  /  3 )  =  ( 1  /  3
) )
3736oveq2d 6304 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  =  ( 1  x.  ( 1  /  3 ) ) )
38 3cn 10681 . . . . . . . . . . . . . 14  |-  3  e.  CC
39 3ne0 10701 . . . . . . . . . . . . . 14  |-  3  =/=  0
4038, 39reccli 10334 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  CC
4140mulid2i 9643 . . . . . . . . . . . 12  |-  ( 1  x.  ( 1  / 
3 ) )  =  ( 1  /  3
)
4237, 41syl6eq 2500 . . . . . . . . . . 11  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  =  ( 1  /  3 ) )
4334, 42sylan9eqr 2506 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  0 )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( 1  /  3 ) )
4443, 40syl6eqel 2536 . . . . . . . . 9  |-  ( ( X  e.  CC  /\  k  =  0 )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
455eqeq2i 2462 . . . . . . . . . . . 12  |-  ( k  =  ( 0  +  1 )  <->  k  = 
1 )
46 oveq2 6296 . . . . . . . . . . . . . 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 2500 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  (
2  _C  k )  =  2 )
50 oveq1 6295 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
k BernPoly  X )  =  ( 1 BernPoly  X ) )
51 oveq2 6296 . . . . . . . . . . . . . . . 16  |-  ( k  =  1  ->  (
2  -  k )  =  ( 2  -  1 ) )
5251oveq1d 6303 . . . . . . . . . . . . . . 15  |-  ( k  =  1  ->  (
( 2  -  k
)  +  1 )  =  ( ( 2  -  1 )  +  1 ) )
53 ax-1cn 9594 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
54 npcan 9881 . . . . . . . . . . . . . . . 16  |-  ( ( 2  e.  CC  /\  1  e.  CC )  ->  ( ( 2  -  1 )  +  1 )  =  2 )
5527, 53, 54mp2an 677 . . . . . . . . . . . . . . 15  |-  ( ( 2  -  1 )  +  1 )  =  2
5652, 55syl6eq 2500 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
( 2  -  k
)  +  1 )  =  2 )
5750, 56oveq12d 6306 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  (
( k BernPoly  X )  /  ( ( 2  -  k )  +  1 ) )  =  ( ( 1 BernPoly  X
)  /  2 ) )
5849, 57oveq12d 6306 . . . . . . . . . . . 12  |-  ( k  =  1  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 2  x.  ( ( 1 BernPoly  X )  /  2
) ) )
5945, 58sylbi 199 . . . . . . . . . . 11  |-  ( k  =  ( 0  +  1 )  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 2  x.  ( ( 1 BernPoly  X )  /  2
) ) )
60 bpoly1 14097 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
1 BernPoly  X )  =  ( X  -  ( 1  /  2 ) ) )
6160oveq1d 6303 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  (
( 1 BernPoly  X )  /  2 )  =  ( ( X  -  ( 1  /  2
) )  /  2
) )
6261oveq2d 6304 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
2  x.  ( ( 1 BernPoly  X )  /  2
) )  =  ( 2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) ) )
63 halfcn 10826 . . . . . . . . . . . . . 14  |-  ( 1  /  2 )  e.  CC
64 subcl 9871 . . . . . . . . . . . . . 14  |-  ( ( X  e.  CC  /\  ( 1  /  2
)  e.  CC )  ->  ( X  -  ( 1  /  2
) )  e.  CC )
6563, 64mpan2 676 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  ( X  -  ( 1  /  2 ) )  e.  CC )
66 2ne0 10699 . . . . . . . . . . . . . 14  |-  2  =/=  0
67 divcan2 10275 . . . . . . . . . . . . . 14  |-  ( ( ( X  -  (
1  /  2 ) )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) )  =  ( X  -  ( 1  /  2
) ) )
6827, 66, 67mp3an23 1355 . . . . . . . . . . . . 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 2484 . . . . . . . . . . 11  |-  ( X  e.  CC  ->  (
2  x.  ( ( 1 BernPoly  X )  /  2
) )  =  ( X  -  ( 1  /  2 ) ) )
7159, 70sylan9eqr 2506 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( X  -  ( 1  / 
2 ) ) )
7265adantr 467 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( X  -  ( 1  /  2
) )  e.  CC )
7371, 72eqeltrd 2528 . . . . . . . . 9  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
7444, 73jaodan 793 . . . . . . . 8  |-  ( ( X  e.  CC  /\  ( k  =  0  \/  k  =  ( 0  +  1 ) ) )  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  e.  CC )
7519, 74sylan2b 478 . . . . . . 7  |-  ( ( X  e.  CC  /\  k  e.  ( 0 ... ( 0  +  1 ) ) )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
7612, 75, 59fsump1 13810 . . . . . 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 2536 . . . . . . . . 9  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  e.  CC )
7834fsum1 13801 . . . . . . . . 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 668 . . . . . . . 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 2484 . . . . . . 7  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... 0
) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( 1  /  3 ) )
8180, 70oveq12d 6306 . . . . . 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 2484 . . . . 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 9885 . . . . . . 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 1354 . . . . . 6  |-  ( X  e.  CC  ->  (
( 1  /  3
)  +  ( X  -  ( 1  / 
2 ) ) )  =  ( X  +  ( ( 1  / 
3 )  -  (
1  /  2 ) ) ) )
8563, 40negsubdi2i 9958 . . . . . . . 8  |-  -u (
( 1  /  2
)  -  ( 1  /  3 ) )  =  ( ( 1  /  3 )  -  ( 1  /  2
) )
86 halfthird 11154 . . . . . . . . 9  |-  ( ( 1  /  2 )  -  ( 1  / 
3 ) )  =  ( 1  /  6
)
8786negeqi 9865 . . . . . . . 8  |-  -u (
( 1  /  2
)  -  ( 1  /  3 ) )  =  -u ( 1  / 
6 )
8885, 87eqtr3i 2474 . . . . . . 7  |-  ( ( 1  /  3 )  -  ( 1  / 
2 ) )  = 
-u ( 1  / 
6 )
8988oveq2i 6299 . . . . . 6  |-  ( X  +  ( ( 1  /  3 )  -  ( 1  /  2
) ) )  =  ( X  +  -u ( 1  /  6
) )
9084, 89syl6eq 2500 . . . . 5  |-  ( X  e.  CC  ->  (
( 1  /  3
)  +  ( X  -  ( 1  / 
2 ) ) )  =  ( X  +  -u ( 1  /  6
) ) )
91 6cn 10688 . . . . . . 7  |-  6  e.  CC
92 6re 10687 . . . . . . . 8  |-  6  e.  RR
93 6pos 10705 . . . . . . . 8  |-  0  <  6
9492, 93gt0ne0ii 10147 . . . . . . 7  |-  6  =/=  0
9591, 94reccli 10334 . . . . . 6  |-  ( 1  /  6 )  e.  CC
96 negsub 9919 . . . . . 6  |-  ( ( X  e.  CC  /\  ( 1  /  6
)  e.  CC )  ->  ( X  +  -u ( 1  /  6
) )  =  ( X  -  ( 1  /  6 ) ) )
9795, 96mpan2 676 . . . . 5  |-  ( X  e.  CC  ->  ( X  +  -u ( 1  /  6 ) )  =  ( X  -  ( 1  /  6
) ) )
9882, 90, 973eqtrd 2488 . . . 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 2496 . . 3  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... (
2  -  1 ) ) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( X  -  ( 1  / 
6 ) ) )
10099oveq2d 6304 . 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 9901 . . . 4  |-  ( ( ( X ^ 2 )  e.  CC  /\  X  e.  CC  /\  (
1  /  6 )  e.  CC )  -> 
( ( X ^
2 )  -  ( X  -  ( 1  /  6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  / 
6 ) ) )
10395, 102mp3an3 1352 . . 3  |-  ( ( ( X ^ 2 )  e.  CC  /\  X  e.  CC )  ->  ( ( X ^
2 )  -  ( X  -  ( 1  /  6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  / 
6 ) ) )
104101, 103mpancom 674 . 2  |-  ( X  e.  CC  ->  (
( X ^ 2 )  -  ( X  -  ( 1  / 
6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  /  6
) ) )
1053, 100, 1043eqtrd 2488 1  |-  ( X  e.  CC  ->  (
2 BernPoly  X )  =  ( ( ( X ^
2 )  -  X
)  +  ( 1  /  6 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621   {cpr 3969   ` cfv 5581  (class class class)co 6288   CCcc 9534   0cc0 9536   1c1 9537    + caddc 9539    x. cmul 9541    - cmin 9857   -ucneg 9858    / cdiv 10266   2c2 10656   3c3 10657   6c6 10660   NN0cn0 10866   ZZcz 10934   ZZ>=cuz 11156   ...cfz 11781   ^cexp 12269    _C cbc 12484   sum_csu 13745   BernPoly cbp 14092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-fz 11782  df-fzo 11913  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746  df-bpoly 14093
This theorem is referenced by:  bpoly3  14104  bpoly4  14105
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