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

Theorem bpoly2 28151
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 10588 . . 3  |-  2  e.  NN0
2 bpolyval 28143 . . 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 10428 . . . . . . 7  |-  ( 2  -  1 )  =  1
5 0p1e1 10425 . . . . . . 7  |-  ( 0  +  1 )  =  1
64, 5eqtr4i 2461 . . . . . 6  |-  ( 2  -  1 )  =  ( 0  +  1 )
76oveq2i 6097 . . . . 5  |-  ( 0 ... ( 2  -  1 ) )  =  ( 0 ... (
0  +  1 ) )
87sumeq1i 13167 . . . 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 10586 . . . . . . . . 9  |-  0  e.  NN0
10 nn0uz 10887 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
119, 10eleqtri 2510 . . . . . . . 8  |-  0  e.  ( ZZ>= `  0 )
1211a1i 11 . . . . . . 7  |-  ( X  e.  CC  ->  0  e.  ( ZZ>= `  0 )
)
13 0z 10649 . . . . . . . . . . 11  |-  0  e.  ZZ
14 fzpr 11503 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  (
0 ... ( 0  +  1 ) )  =  { 0 ,  ( 0  +  1 ) } )
1513, 14ax-mp 5 . . . . . . . . . 10  |-  ( 0 ... ( 0  +  1 ) )  =  { 0 ,  ( 0  +  1 ) }
1615eleq2i 2502 . . . . . . . . 9  |-  ( k  e.  ( 0 ... ( 0  +  1 ) )  <->  k  e.  { 0 ,  ( 0  +  1 ) } )
17 vex 2970 . . . . . . . . . 10  |-  k  e. 
_V
1817elpr 3890 . . . . . . . . 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 6094 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
2  _C  k )  =  ( 2  _C  0 ) )
21 bcn0 12078 . . . . . . . . . . . . . 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 6093 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
k BernPoly  X )  =  ( 0 BernPoly  X ) )
25 oveq2 6094 . . . . . . . . . . . . . . 15  |-  ( k  =  0  ->  (
2  -  k )  =  ( 2  -  0 ) )
2625oveq1d 6101 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
( 2  -  k
)  +  1 )  =  ( ( 2  -  0 )  +  1 ) )
27 2cn 10384 . . . . . . . . . . . . . . . . 17  |-  2  e.  CC
2827subid1i 9672 . . . . . . . . . . . . . . . 16  |-  ( 2  -  0 )  =  2
2928oveq1i 6096 . . . . . . . . . . . . . . 15  |-  ( ( 2  -  0 )  +  1 )  =  ( 2  +  1 )
30 df-3 10373 . . . . . . . . . . . . . . 15  |-  3  =  ( 2  +  1 )
3129, 30eqtr4i 2461 . . . . . . . . . . . . . 14  |-  ( ( 2  -  0 )  +  1 )  =  3
3226, 31syl6eq 2486 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
( 2  -  k
)  +  1 )  =  3 )
3324, 32oveq12d 6104 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
( k BernPoly  X )  /  ( ( 2  -  k )  +  1 ) )  =  ( ( 0 BernPoly  X
)  /  3 ) )
3423, 33oveq12d 6104 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( 2  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 2  -  k
)  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  3
) ) )
35 bpoly0 28144 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
0 BernPoly  X )  =  1 )
3635oveq1d 6101 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  (
( 0 BernPoly  X )  /  3 )  =  ( 1  /  3
) )
3736oveq2d 6102 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  =  ( 1  x.  ( 1  /  3 ) ) )
38 3cn 10388 . . . . . . . . . . . . . 14  |-  3  e.  CC
39 3ne0 10408 . . . . . . . . . . . . . 14  |-  3  =/=  0
4038, 39reccli 10053 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  CC
4140mulid2i 9381 . . . . . . . . . . . 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 2526 . . . . . . . . 9  |-  ( ( X  e.  CC  /\  k  =  0 )  ->  ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  e.  CC )
455eqeq2i 2448 . . . . . . . . . . . 12  |-  ( k  =  ( 0  +  1 )  <->  k  = 
1 )
46 oveq2 6094 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
2  _C  k )  =  ( 2  _C  1 ) )
47 bcn1 12081 . . . . . . . . . . . . . . 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 6093 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
k BernPoly  X )  =  ( 1 BernPoly  X ) )
51 oveq2 6094 . . . . . . . . . . . . . . . 16  |-  ( k  =  1  ->  (
2  -  k )  =  ( 2  -  1 ) )
5251oveq1d 6101 . . . . . . . . . . . . . . 15  |-  ( k  =  1  ->  (
( 2  -  k
)  +  1 )  =  ( ( 2  -  1 )  +  1 ) )
53 ax-1cn 9332 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
54 npcan 9611 . . . . . . . . . . . . . . . 16  |-  ( ( 2  e.  CC  /\  1  e.  CC )  ->  ( ( 2  -  1 )  +  1 )  =  2 )
5527, 53, 54mp2an 672 . . . . . . . . . . . . . . 15  |-  ( ( 2  -  1 )  +  1 )  =  2
5652, 55syl6eq 2486 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  (
( 2  -  k
)  +  1 )  =  2 )
5750, 56oveq12d 6104 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  (
( k BernPoly  X )  /  ( ( 2  -  k )  +  1 ) )  =  ( ( 1 BernPoly  X
)  /  2 ) )
5849, 57oveq12d 6104 . . . . . . . . . . . 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 28145 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
1 BernPoly  X )  =  ( X  -  ( 1  /  2 ) ) )
6160oveq1d 6101 . . . . . . . . . . . . 13  |-  ( X  e.  CC  ->  (
( 1 BernPoly  X )  /  2 )  =  ( ( X  -  ( 1  /  2
) )  /  2
) )
6261oveq2d 6102 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
2  x.  ( ( 1 BernPoly  X )  /  2
) )  =  ( 2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) ) )
63 halfcn 10533 . . . . . . . . . . . . . 14  |-  ( 1  /  2 )  e.  CC
64 subcl 9601 . . . . . . . . . . . . . 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 10406 . . . . . . . . . . . . . 14  |-  2  =/=  0
67 divcan2 9994 . . . . . . . . . . . . . 14  |-  ( ( ( X  -  (
1  /  2 ) )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
2  x.  ( ( X  -  ( 1  /  2 ) )  /  2 ) )  =  ( X  -  ( 1  /  2
) ) )
6827, 66, 67mp3an23 1306 . . . . . . . . . . . . 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 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 465 . . . . . . . . . 10  |-  ( ( X  e.  CC  /\  k  =  ( 0  +  1 ) )  ->  ( X  -  ( 1  /  2
) )  e.  CC )
7371, 72eqeltrd 2512 . . . . . . . . 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 13215 . . . . . 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 2526 . . . . . . . . 9  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  3
) )  e.  CC )
7834fsum1 13210 . . . . . . . . 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 2470 . . . . . . 7  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... 0
) ( ( 2  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 2  -  k )  +  1 ) ) )  =  ( 1  /  3 ) )
8180, 70oveq12d 6104 . . . . . 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 9615 . . . . . . 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 1305 . . . . . 6  |-  ( X  e.  CC  ->  (
( 1  /  3
)  +  ( X  -  ( 1  / 
2 ) ) )  =  ( X  +  ( ( 1  / 
3 )  -  (
1  /  2 ) ) ) )
8563, 40negsubdi2i 9686 . . . . . . . 8  |-  -u (
( 1  /  2
)  -  ( 1  /  3 ) )  =  ( ( 1  /  3 )  -  ( 1  /  2
) )
86 halfthird 27343 . . . . . . . . 9  |-  ( ( 1  /  2 )  -  ( 1  / 
3 ) )  =  ( 1  /  6
)
8786negeqi 9595 . . . . . . . 8  |-  -u (
( 1  /  2
)  -  ( 1  /  3 ) )  =  -u ( 1  / 
6 )
8885, 87eqtr3i 2460 . . . . . . 7  |-  ( ( 1  /  3 )  -  ( 1  / 
2 ) )  = 
-u ( 1  / 
6 )
8988oveq2i 6097 . . . . . 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 10395 . . . . . . 7  |-  6  e.  CC
92 6re 10394 . . . . . . . 8  |-  6  e.  RR
93 6pos 10412 . . . . . . . 8  |-  0  <  6
9492, 93gt0ne0ii 9868 . . . . . . 7  |-  6  =/=  0
9591, 94reccli 10053 . . . . . 6  |-  ( 1  /  6 )  e.  CC
96 negsub 9649 . . . . . 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 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 6102 . 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 11920 . . 3  |-  ( X  e.  CC  ->  ( X ^ 2 )  e.  CC )
102 subsub 9631 . . . 4  |-  ( ( ( X ^ 2 )  e.  CC  /\  X  e.  CC  /\  (
1  /  6 )  e.  CC )  -> 
( ( X ^
2 )  -  ( X  -  ( 1  /  6 ) ) )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  / 
6 ) ) )
10395, 102mp3an3 1303 . . 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 2474 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 1369    e. wcel 1756    =/= wne 2601   {cpr 3874   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    - cmin 9587   -ucneg 9588    / cdiv 9985   2c2 10363   3c3 10364   6c6 10367   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429   ^cexp 11857    _C cbc 12070   sum_csu 13155   BernPoly cbp 28140
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-pred 27576  df-wrecs 27668  df-bpoly 28141
This theorem is referenced by:  bpoly3  28152  bpoly4  28153
  Copyright terms: Public domain W3C validator