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Theorem binom 12288
Description: The binomial theorem:  ( A  +  B ) ^ N is the sum from  k  =  0 to  N of  ( N  _C  k )  x.  ( ( A ^
k )  x.  ( B ^ ( N  -  k ) ). Theorem 15-2.8 of [Gleason] p. 296. This part of the proof sets up the induction and does the base case, with the bulk of the work (the induction step) in binomlem 12287. (Contributed by NM, 7-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
Assertion
Ref Expression
binom  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  (
( A  +  B
) ^ N )  =  sum_ k  e.  ( 0 ... N ) ( ( N  _C  k )  x.  (
( A ^ ( N  -  k )
)  x.  ( B ^ k ) ) ) )
Distinct variable groups:    A, k    B, k    k, N

Proof of Theorem binom
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5866 . . . . . 6  |-  ( x  =  0  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^
0 ) )
2 oveq2 5866 . . . . . . 7  |-  ( x  =  0  ->  (
0 ... x )  =  ( 0 ... 0
) )
3 oveq1 5865 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  _C  k )  =  ( 0  _C  k ) )
4 oveq1 5865 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
x  -  k )  =  ( 0  -  k ) )
54oveq2d 5874 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A ^ ( x  -  k ) )  =  ( A ^ (
0  -  k ) ) )
65oveq1d 5873 . . . . . . . . 9  |-  ( x  =  0  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( 0  -  k ) )  x.  ( B ^ k
) ) )
73, 6oveq12d 5876 . . . . . . . 8  |-  ( x  =  0  ->  (
( x  _C  k
)  x.  ( ( A ^ ( x  -  k ) )  x.  ( B ^
k ) ) )  =  ( ( 0  _C  k )  x.  ( ( A ^
( 0  -  k
) )  x.  ( B ^ k ) ) ) )
87adantr 451 . . . . . . 7  |-  ( ( x  =  0  /\  k  e.  ( 0 ... x ) )  ->  ( ( x  _C  k )  x.  ( ( A ^
( x  -  k
) )  x.  ( B ^ k ) ) )  =  ( ( 0  _C  k )  x.  ( ( A ^ ( 0  -  k ) )  x.  ( B ^ k
) ) ) )
92, 8sumeq12dv 12179 . . . . . 6  |-  ( x  =  0  ->  sum_ k  e.  ( 0 ... x
) ( ( x  _C  k )  x.  ( ( A ^
( x  -  k
) )  x.  ( B ^ k ) ) )  =  sum_ k  e.  ( 0 ... 0
) ( ( 0  _C  k )  x.  ( ( A ^
( 0  -  k
) )  x.  ( B ^ k ) ) ) )
101, 9eqeq12d 2297 . . . . 5  |-  ( x  =  0  ->  (
( ( A  +  B ) ^ x
)  =  sum_ k  e.  ( 0 ... x
) ( ( x  _C  k )  x.  ( ( A ^
( x  -  k
) )  x.  ( B ^ k ) ) )  <->  ( ( A  +  B ) ^
0 )  =  sum_ k  e.  ( 0 ... 0 ) ( ( 0  _C  k
)  x.  ( ( A ^ ( 0  -  k ) )  x.  ( B ^
k ) ) ) ) )
1110imbi2d 307 . . . 4  |-  ( x  =  0  ->  (
( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
) ^ x )  =  sum_ k  e.  ( 0 ... x ) ( ( x  _C  k )  x.  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) ) ) )  <->  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
) ^ 0 )  =  sum_ k  e.  ( 0 ... 0 ) ( ( 0  _C  k )  x.  (
( A ^ (
0  -  k ) )  x.  ( B ^ k ) ) ) ) ) )
12 oveq2 5866 . . . . . 6  |-  ( x  =  n  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^
n ) )
13 oveq2 5866 . . . . . . 7  |-  ( x  =  n  ->  (
0 ... x )  =  ( 0 ... n
) )
14 oveq1 5865 . . . . . . . . 9  |-  ( x  =  n  ->  (
x  _C  k )  =  ( n  _C  k ) )
15 oveq1 5865 . . . . . . . . . . 11  |-  ( x  =  n  ->  (
x  -  k )  =  ( n  -  k ) )
1615oveq2d 5874 . . . . . . . . . 10  |-  ( x  =  n  ->  ( A ^ ( x  -  k ) )  =  ( A ^ (
n  -  k ) ) )
1716oveq1d 5873 . . . . . . . . 9  |-  ( x  =  n  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( n  -  k ) )  x.  ( B ^ k
) ) )
1814, 17oveq12d 5876 . . . . . . . 8  |-  ( x  =  n  ->  (
( x  _C  k
)  x.  ( ( A ^ ( x  -  k ) )  x.  ( B ^
k ) ) )  =  ( ( n  _C  k )  x.  ( ( A ^
( n  -  k
) )  x.  ( B ^ k ) ) ) )
1918adantr 451 . . . . . . 7  |-  ( ( x  =  n  /\  k  e.  ( 0 ... x ) )  ->  ( ( x  _C  k )  x.  ( ( A ^
( x  -  k
) )  x.  ( B ^ k ) ) )  =  ( ( n  _C  k )  x.  ( ( A ^ ( n  -  k ) )  x.  ( B ^ k
) ) ) )
2013, 19sumeq12dv 12179 . . . . . 6  |-  ( x  =  n  ->  sum_ k  e.  ( 0 ... x
) ( ( x  _C  k )  x.  ( ( A ^
( x  -  k
) )  x.  ( B ^ k ) ) )  =  sum_ k  e.  ( 0 ... n
) ( ( n  _C  k )  x.  ( ( A ^
( n  -  k
) )  x.  ( B ^ k ) ) ) )
2112, 20eqeq12d 2297 . . . . 5  |-  ( x  =  n  ->  (
( ( A  +  B ) ^ x
)  =  sum_ k  e.  ( 0 ... x
) ( ( x  _C  k )  x.  ( ( A ^
( x  -  k
) )  x.  ( B ^ k ) ) )  <->  ( ( A  +  B ) ^
n )  =  sum_ k  e.  ( 0 ... n ) ( ( n  _C  k
)  x.  ( ( A ^ ( n  -  k ) )  x.  ( B ^
k ) ) ) ) )
2221imbi2d 307 . . . 4  |-  ( x  =  n  ->  (
( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
) ^ x )  =  sum_ k  e.  ( 0 ... x ) ( ( x  _C  k )  x.  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) ) ) )  <->  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
) ^ n )  =  sum_ k  e.  ( 0 ... n ) ( ( n  _C  k )  x.  (
( A ^ (
n  -  k ) )  x.  ( B ^ k ) ) ) ) ) )
23 oveq2 5866 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^
( n  +  1 ) ) )
24 oveq2 5866 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (
0 ... x )  =  ( 0 ... (
n  +  1 ) ) )
25 oveq1 5865 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
x  _C  k )  =  ( ( n  +  1 )  _C  k ) )
26 oveq1 5865 . . . . . . . . . . 11  |-  ( x  =  ( n  + 
1 )  ->  (
x  -  k )  =  ( ( n  +  1 )  -  k ) )
2726oveq2d 5874 . . . . . . . . . 10  |-  ( x  =  ( n  + 
1 )  ->  ( A ^ ( x  -  k ) )  =  ( A ^ (
( n  +  1 )  -  k ) ) )
2827oveq1d 5873 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( ( n  +  1 )  -  k ) )  x.  ( B ^ k
) ) )
2925, 28oveq12d 5876 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( x  _C  k
)  x.  ( ( A ^ ( x  -  k ) )  x.  ( B ^
k ) ) )  =  ( ( ( n  +  1 )  _C  k )  x.  ( ( A ^
( ( n  + 
1 )  -  k
) )  x.  ( B ^ k ) ) ) )
3029adantr 451 . . . . . . 7  |-  ( ( x  =  ( n  +  1 )  /\  k  e.  ( 0 ... x ) )  ->  ( ( x  _C  k )  x.  ( ( A ^
( x  -  k
) )  x.  ( B ^ k ) ) )  =  ( ( ( n  +  1 )  _C  k )  x.  ( ( A ^ ( ( n  +  1 )  -  k ) )  x.  ( B ^ k
) ) ) )
3124, 30sumeq12dv 12179 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  sum_ k  e.  ( 0 ... x
) ( ( x  _C  k )  x.  ( ( A ^
( x  -  k
) )  x.  ( B ^ k ) ) )  =  sum_ k  e.  ( 0 ... (
n  +  1 ) ) ( ( ( n  +  1 )  _C  k )  x.  ( ( A ^
( ( n  + 
1 )  -  k
) )  x.  ( B ^ k ) ) ) )
3223, 31eqeq12d 2297 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
( ( A  +  B ) ^ x
)  =  sum_ k  e.  ( 0 ... x
) ( ( x  _C  k )  x.  ( ( A ^
( x  -  k
) )  x.  ( B ^ k ) ) )  <->  ( ( A  +  B ) ^
( n  +  1 ) )  =  sum_ k  e.  ( 0 ... ( n  + 
1 ) ) ( ( ( n  + 
1 )  _C  k
)  x.  ( ( A ^ ( ( n  +  1 )  -  k ) )  x.  ( B ^
k ) ) ) ) )
3332imbi2d 307 . . . 4  |-  ( x  =  ( n  + 
1 )  ->  (
( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
) ^ x )  =  sum_ k  e.  ( 0 ... x ) ( ( x  _C  k )  x.  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) ) ) )  <->  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
) ^ ( n  +  1 ) )  =  sum_ k  e.  ( 0 ... ( n  +  1 ) ) ( ( ( n  +  1 )  _C  k )  x.  (
( A ^ (
( n  +  1 )  -  k ) )  x.  ( B ^ k ) ) ) ) ) )
34 oveq2 5866 . . . . . 6  |-  ( x  =  N  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^ N ) )
35 oveq2 5866 . . . . . . 7  |-  ( x  =  N  ->  (
0 ... x )  =  ( 0 ... N
) )
36 oveq1 5865 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  _C  k )  =  ( N  _C  k ) )
37 oveq1 5865 . . . . . . . . . . 11  |-  ( x  =  N  ->  (
x  -  k )  =  ( N  -  k ) )
3837oveq2d 5874 . . . . . . . . . 10  |-  ( x  =  N  ->  ( A ^ ( x  -  k ) )  =  ( A ^ ( N  -  k )
) )
3938oveq1d 5873 . . . . . . . . 9  |-  ( x  =  N  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( N  -  k ) )  x.  ( B ^ k
) ) )
4036, 39oveq12d 5876 . . . . . . . 8  |-  ( x  =  N  ->  (
( x  _C  k
)  x.  ( ( A ^ ( x  -  k ) )  x.  ( B ^
k ) ) )  =  ( ( N  _C  k )  x.  ( ( A ^
( N  -  k
) )  x.  ( B ^ k ) ) ) )
4140adantr 451 . . . . . . 7  |-  ( ( x  =  N  /\  k  e.  ( 0 ... x ) )  ->  ( ( x  _C  k )  x.  ( ( A ^
( x  -  k
) )  x.  ( B ^ k ) ) )  =  ( ( N  _C  k )  x.  ( ( A ^ ( N  -  k ) )  x.  ( B ^ k
) ) ) )
4235, 41sumeq12dv 12179 . . . . . 6  |-  ( x  =  N  ->  sum_ k  e.  ( 0 ... x
) ( ( x  _C  k )  x.  ( ( A ^
( x  -  k
) )  x.  ( B ^ k ) ) )  =  sum_ k  e.  ( 0 ... N
) ( ( N  _C  k )  x.  ( ( A ^
( N  -  k
) )  x.  ( B ^ k ) ) ) )
4334, 42eqeq12d 2297 . . . . 5  |-  ( x  =  N  ->  (
( ( A  +  B ) ^ x
)  =  sum_ k  e.  ( 0 ... x
) ( ( x  _C  k )  x.  ( ( A ^
( x  -  k
) )  x.  ( B ^ k ) ) )  <->  ( ( A  +  B ) ^ N )  =  sum_ k  e.  ( 0 ... N ) ( ( N  _C  k
)  x.  ( ( A ^ ( N  -  k ) )  x.  ( B ^
k ) ) ) ) )
4443imbi2d 307 . . . 4  |-  ( x  =  N  ->  (
( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
) ^ x )  =  sum_ k  e.  ( 0 ... x ) ( ( x  _C  k )  x.  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) ) ) )  <->  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
) ^ N )  =  sum_ k  e.  ( 0 ... N ) ( ( N  _C  k )  x.  (
( A ^ ( N  -  k )
)  x.  ( B ^ k ) ) ) ) ) )
45 exp0 11108 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
46 exp0 11108 . . . . . . . . 9  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
4745, 46oveqan12d 5877 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  ( 1  x.  1 ) )
48 1t1e1 9870 . . . . . . . 8  |-  ( 1  x.  1 )  =  1
4947, 48syl6eq 2331 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  1 )
5049oveq2d 5874 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  =  ( 1  x.  1 ) )
5150, 48syl6eq 2331 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  =  1 )
52 0z 10035 . . . . . 6  |-  0  e.  ZZ
53 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
5451, 53syl6eqel 2371 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  e.  CC )
55 oveq2 5866 . . . . . . . . 9  |-  ( k  =  0  ->  (
0  _C  k )  =  ( 0  _C  0 ) )
56 0nn0 9980 . . . . . . . . . 10  |-  0  e.  NN0
57 bcn0 11323 . . . . . . . . . 10  |-  ( 0  e.  NN0  ->  ( 0  _C  0 )  =  1 )
5856, 57ax-mp 8 . . . . . . . . 9  |-  ( 0  _C  0 )  =  1
5955, 58syl6eq 2331 . . . . . . . 8  |-  ( k  =  0  ->  (
0  _C  k )  =  1 )
60 oveq2 5866 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
0  -  k )  =  ( 0  -  0 ) )
61 0cn 8831 . . . . . . . . . . . 12  |-  0  e.  CC
6261subidi 9117 . . . . . . . . . . 11  |-  ( 0  -  0 )  =  0
6360, 62syl6eq 2331 . . . . . . . . . 10  |-  ( k  =  0  ->  (
0  -  k )  =  0 )
6463oveq2d 5874 . . . . . . . . 9  |-  ( k  =  0  ->  ( A ^ ( 0  -  k ) )  =  ( A ^ 0 ) )
65 oveq2 5866 . . . . . . . . 9  |-  ( k  =  0  ->  ( B ^ k )  =  ( B ^ 0 ) )
6664, 65oveq12d 5876 . . . . . . . 8  |-  ( k  =  0  ->  (
( A ^ (
0  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ 0 )  x.  ( B ^ 0 ) ) )
6759, 66oveq12d 5876 . . . . . . 7  |-  ( k  =  0  ->  (
( 0  _C  k
)  x.  ( ( A ^ ( 0  -  k ) )  x.  ( B ^
k ) ) )  =  ( 1  x.  ( ( A ^
0 )  x.  ( B ^ 0 ) ) ) )
6867fsum1 12214 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  e.  CC )  ->  sum_ k  e.  ( 0 ... 0 ) ( ( 0  _C  k )  x.  (
( A ^ (
0  -  k ) )  x.  ( B ^ k ) ) )  =  ( 1  x.  ( ( A ^ 0 )  x.  ( B ^ 0 ) ) ) )
6952, 54, 68sylancr 644 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
sum_ k  e.  ( 0 ... 0 ) ( ( 0  _C  k )  x.  (
( A ^ (
0  -  k ) )  x.  ( B ^ k ) ) )  =  ( 1  x.  ( ( A ^ 0 )  x.  ( B ^ 0 ) ) ) )
70 addcl 8819 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
7170exp0d 11239 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^ 0 )  =  1 )
7251, 69, 713eqtr4rd 2326 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^ 0 )  =  sum_ k  e.  ( 0 ... 0
) ( ( 0  _C  k )  x.  ( ( A ^
( 0  -  k
) )  x.  ( B ^ k ) ) ) )
73 simprl 732 . . . . . . 7  |-  ( ( n  e.  NN0  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  A  e.  CC )
74 simprr 733 . . . . . . 7  |-  ( ( n  e.  NN0  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  B  e.  CC )
75 simpl 443 . . . . . . 7  |-  ( ( n  e.  NN0  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  n  e.  NN0 )
76 id 19 . . . . . . 7  |-  ( ( ( A  +  B
) ^ n )  =  sum_ k  e.  ( 0 ... n ) ( ( n  _C  k )  x.  (
( A ^ (
n  -  k ) )  x.  ( B ^ k ) ) )  ->  ( ( A  +  B ) ^ n )  = 
sum_ k  e.  ( 0 ... n ) ( ( n  _C  k )  x.  (
( A ^ (
n  -  k ) )  x.  ( B ^ k ) ) ) )
7773, 74, 75, 76binomlem 12287 . . . . . 6  |-  ( ( ( n  e.  NN0  /\  ( A  e.  CC  /\  B  e.  CC ) )  /\  ( ( A  +  B ) ^ n )  = 
sum_ k  e.  ( 0 ... n ) ( ( n  _C  k )  x.  (
( A ^ (
n  -  k ) )  x.  ( B ^ k ) ) ) )  ->  (
( A  +  B
) ^ ( n  +  1 ) )  =  sum_ k  e.  ( 0 ... ( n  +  1 ) ) ( ( ( n  +  1 )  _C  k )  x.  (
( A ^ (
( n  +  1 )  -  k ) )  x.  ( B ^ k ) ) ) )
7877exp31 587 . . . . 5  |-  ( n  e.  NN0  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B ) ^
n )  =  sum_ k  e.  ( 0 ... n ) ( ( n  _C  k
)  x.  ( ( A ^ ( n  -  k ) )  x.  ( B ^
k ) ) )  ->  ( ( A  +  B ) ^
( n  +  1 ) )  =  sum_ k  e.  ( 0 ... ( n  + 
1 ) ) ( ( ( n  + 
1 )  _C  k
)  x.  ( ( A ^ ( ( n  +  1 )  -  k ) )  x.  ( B ^
k ) ) ) ) ) )
7978a2d 23 . . . 4  |-  ( n  e.  NN0  ->  ( ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^
n )  =  sum_ k  e.  ( 0 ... n ) ( ( n  _C  k
)  x.  ( ( A ^ ( n  -  k ) )  x.  ( B ^
k ) ) ) )  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  +  B
) ^ ( n  +  1 ) )  =  sum_ k  e.  ( 0 ... ( n  +  1 ) ) ( ( ( n  +  1 )  _C  k )  x.  (
( A ^ (
( n  +  1 )  -  k ) )  x.  ( B ^ k ) ) ) ) ) )
8011, 22, 33, 44, 72, 79nn0ind 10108 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^ N
)  =  sum_ k  e.  ( 0 ... N
) ( ( N  _C  k )  x.  ( ( A ^
( N  -  k
) )  x.  ( B ^ k ) ) ) ) )
8180impcom 419 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  N  e.  NN0 )  ->  ( ( A  +  B ) ^ N )  =  sum_ k  e.  ( 0 ... N ) ( ( N  _C  k
)  x.  ( ( A ^ ( N  -  k ) )  x.  ( B ^
k ) ) ) )
82813impa 1146 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  (
( A  +  B
) ^ N )  =  sum_ k  e.  ( 0 ... N ) ( ( N  _C  k )  x.  (
( A ^ ( N  -  k )
)  x.  ( B ^ k ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NN0cn0 9965   ZZcz 10024   ...cfz 10782   ^cexp 11104    _C cbc 11315   sum_csu 12158
This theorem is referenced by:  binom1p  12289  efaddlem  12374  basellem3  20320  jm2.22  27088
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-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
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