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Theorem binom 13855
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 13854. This is Metamath 100 proof #44. (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 6304 . . . . . 6  |-  ( x  =  0  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^
0 ) )
2 oveq2 6304 . . . . . . 7  |-  ( x  =  0  ->  (
0 ... x )  =  ( 0 ... 0
) )
3 oveq1 6303 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  _C  k )  =  ( 0  _C  k ) )
4 oveq1 6303 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
x  -  k )  =  ( 0  -  k ) )
54oveq2d 6312 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A ^ ( x  -  k ) )  =  ( A ^ (
0  -  k ) ) )
65oveq1d 6311 . . . . . . . . 9  |-  ( x  =  0  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( 0  -  k ) )  x.  ( B ^ k
) ) )
73, 6oveq12d 6314 . . . . . . . 8  |-  ( x  =  0  ->  (
( x  _C  k
)  x.  ( ( A ^ ( x  -  k ) )  x.  ( B ^
k ) ) )  =  ( ( 0  _C  k )  x.  ( ( A ^
( 0  -  k
) )  x.  ( B ^ k ) ) ) )
87adantr 466 . . . . . . 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 13739 . . . . . 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 2442 . . . . 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 317 . . . 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 6304 . . . . . 6  |-  ( x  =  n  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^
n ) )
13 oveq2 6304 . . . . . . 7  |-  ( x  =  n  ->  (
0 ... x )  =  ( 0 ... n
) )
14 oveq1 6303 . . . . . . . . 9  |-  ( x  =  n  ->  (
x  _C  k )  =  ( n  _C  k ) )
15 oveq1 6303 . . . . . . . . . . 11  |-  ( x  =  n  ->  (
x  -  k )  =  ( n  -  k ) )
1615oveq2d 6312 . . . . . . . . . 10  |-  ( x  =  n  ->  ( A ^ ( x  -  k ) )  =  ( A ^ (
n  -  k ) ) )
1716oveq1d 6311 . . . . . . . . 9  |-  ( x  =  n  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( n  -  k ) )  x.  ( B ^ k
) ) )
1814, 17oveq12d 6314 . . . . . . . 8  |-  ( x  =  n  ->  (
( x  _C  k
)  x.  ( ( A ^ ( x  -  k ) )  x.  ( B ^
k ) ) )  =  ( ( n  _C  k )  x.  ( ( A ^
( n  -  k
) )  x.  ( B ^ k ) ) ) )
1918adantr 466 . . . . . . 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 13739 . . . . . 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 2442 . . . . 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 317 . . . 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 6304 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^
( n  +  1 ) ) )
24 oveq2 6304 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (
0 ... x )  =  ( 0 ... (
n  +  1 ) ) )
25 oveq1 6303 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
x  _C  k )  =  ( ( n  +  1 )  _C  k ) )
26 oveq1 6303 . . . . . . . . . . 11  |-  ( x  =  ( n  + 
1 )  ->  (
x  -  k )  =  ( ( n  +  1 )  -  k ) )
2726oveq2d 6312 . . . . . . . . . 10  |-  ( x  =  ( n  + 
1 )  ->  ( A ^ ( x  -  k ) )  =  ( A ^ (
( n  +  1 )  -  k ) ) )
2827oveq1d 6311 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( ( n  +  1 )  -  k ) )  x.  ( B ^ k
) ) )
2925, 28oveq12d 6314 . . . . . . . 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 466 . . . . . . 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 13739 . . . . . 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 2442 . . . . 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 317 . . . 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 6304 . . . . . 6  |-  ( x  =  N  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^ N ) )
35 oveq2 6304 . . . . . . 7  |-  ( x  =  N  ->  (
0 ... x )  =  ( 0 ... N
) )
36 oveq1 6303 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  _C  k )  =  ( N  _C  k ) )
37 oveq1 6303 . . . . . . . . . . 11  |-  ( x  =  N  ->  (
x  -  k )  =  ( N  -  k ) )
3837oveq2d 6312 . . . . . . . . . 10  |-  ( x  =  N  ->  ( A ^ ( x  -  k ) )  =  ( A ^ ( N  -  k )
) )
3938oveq1d 6311 . . . . . . . . 9  |-  ( x  =  N  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( N  -  k ) )  x.  ( B ^ k
) ) )
4036, 39oveq12d 6314 . . . . . . . 8  |-  ( x  =  N  ->  (
( x  _C  k
)  x.  ( ( A ^ ( x  -  k ) )  x.  ( B ^
k ) ) )  =  ( ( N  _C  k )  x.  ( ( A ^
( N  -  k
) )  x.  ( B ^ k ) ) ) )
4140adantr 466 . . . . . . 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 13739 . . . . . 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 2442 . . . . 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 317 . . . 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 12262 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
46 exp0 12262 . . . . . . . . 9  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
4745, 46oveqan12d 6315 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  ( 1  x.  1 ) )
48 1t1e1 10746 . . . . . . . 8  |-  ( 1  x.  1 )  =  1
4947, 48syl6eq 2477 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  1 )
5049oveq2d 6312 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  =  ( 1  x.  1 ) )
5150, 48syl6eq 2477 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  =  1 )
52 0z 10937 . . . . . 6  |-  0  e.  ZZ
53 ax-1cn 9586 . . . . . . 7  |-  1  e.  CC
5451, 53syl6eqel 2516 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  e.  CC )
55 oveq2 6304 . . . . . . . . 9  |-  ( k  =  0  ->  (
0  _C  k )  =  ( 0  _C  0 ) )
56 0nn0 10873 . . . . . . . . . 10  |-  0  e.  NN0
57 bcn0 12481 . . . . . . . . . 10  |-  ( 0  e.  NN0  ->  ( 0  _C  0 )  =  1 )
5856, 57ax-mp 5 . . . . . . . . 9  |-  ( 0  _C  0 )  =  1
5955, 58syl6eq 2477 . . . . . . . 8  |-  ( k  =  0  ->  (
0  _C  k )  =  1 )
60 oveq2 6304 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
0  -  k )  =  ( 0  -  0 ) )
61 0m0e0 10708 . . . . . . . . . . 11  |-  ( 0  -  0 )  =  0
6260, 61syl6eq 2477 . . . . . . . . . 10  |-  ( k  =  0  ->  (
0  -  k )  =  0 )
6362oveq2d 6312 . . . . . . . . 9  |-  ( k  =  0  ->  ( A ^ ( 0  -  k ) )  =  ( A ^ 0 ) )
64 oveq2 6304 . . . . . . . . 9  |-  ( k  =  0  ->  ( B ^ k )  =  ( B ^ 0 ) )
6563, 64oveq12d 6314 . . . . . . . 8  |-  ( k  =  0  ->  (
( A ^ (
0  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ 0 )  x.  ( B ^ 0 ) ) )
6659, 65oveq12d 6314 . . . . . . 7  |-  ( k  =  0  ->  (
( 0  _C  k
)  x.  ( ( A ^ ( 0  -  k ) )  x.  ( B ^
k ) ) )  =  ( 1  x.  ( ( A ^
0 )  x.  ( B ^ 0 ) ) ) )
6766fsum1 13775 . . . . . 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 ) ) ) )
6852, 54, 67sylancr 667 . . . . 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 ) ) ) )
69 addcl 9610 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
7069exp0d 12396 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^ 0 )  =  1 )
7151, 68, 703eqtr4rd 2472 . . . 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 ) ) ) )
72 simprl 762 . . . . . . 7  |-  ( ( n  e.  NN0  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  A  e.  CC )
73 simprr 764 . . . . . . 7  |-  ( ( n  e.  NN0  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  B  e.  CC )
74 simpl 458 . . . . . . 7  |-  ( ( n  e.  NN0  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  n  e.  NN0 )
75 id 23 . . . . . . 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 ) ) ) )
7672, 73, 74, 75binomlem 13854 . . . . . 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 ) ) ) )
7776exp31 607 . . . . 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 ) ) ) ) ) )
7877a2d 29 . . . 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 ) ) ) ) ) )
7911, 22, 33, 44, 71, 78nn0ind 11019 . . 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 ) ) ) ) )
8079impcom 431 . 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 ) ) ) )
81803impa 1200 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867  (class class class)co 6296   CCcc 9526   0cc0 9528   1c1 9529    + caddc 9531    x. cmul 9533    - cmin 9849   NN0cn0 10858   ZZcz 10926   ...cfz 11771   ^cexp 12258    _C cbc 12473   sum_csu 13719
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 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
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 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-fz 11772  df-fzo 11903  df-seq 12200  df-exp 12259  df-fac 12446  df-bc 12474  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519  df-sum 13720
This theorem is referenced by:  binom1p  13856  efaddlem  14114  basellem3  23911  jm2.22  35604  binomcxplemnn0  36383  altgsumbc  38950
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