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Theorem binom 12165
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 12164. (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
StepHypRef Expression
1 oveq2 5718 . . . . . 6  |-  ( x  =  0  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^
0 ) )
2 oveq2 5718 . . . . . . 7  |-  ( x  =  0  ->  (
0 ... x )  =  ( 0 ... 0
) )
3 oveq1 5717 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  _C  k )  =  ( 0  _C  k ) )
4 oveq1 5717 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
x  -  k )  =  ( 0  -  k ) )
54oveq2d 5726 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A ^ ( x  -  k ) )  =  ( A ^ (
0  -  k ) ) )
65oveq1d 5725 . . . . . . . . 9  |-  ( x  =  0  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( 0  -  k ) )  x.  ( B ^ k
) ) )
73, 6oveq12d 5728 . . . . . . . 8  |-  ( x  =  0  ->  (
( x  _C  k
)  x.  ( ( A ^ ( x  -  k ) )  x.  ( B ^
k ) ) )  =  ( ( 0  _C  k )  x.  ( ( A ^
( 0  -  k
) )  x.  ( B ^ k ) ) ) )
87adantr 453 . . . . . . 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 12056 . . . . . 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 2267 . . . . 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 309 . . . 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 5718 . . . . . 6  |-  ( x  =  n  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^
n ) )
13 oveq2 5718 . . . . . . 7  |-  ( x  =  n  ->  (
0 ... x )  =  ( 0 ... n
) )
14 oveq1 5717 . . . . . . . . 9  |-  ( x  =  n  ->  (
x  _C  k )  =  ( n  _C  k ) )
15 oveq1 5717 . . . . . . . . . . 11  |-  ( x  =  n  ->  (
x  -  k )  =  ( n  -  k ) )
1615oveq2d 5726 . . . . . . . . . 10  |-  ( x  =  n  ->  ( A ^ ( x  -  k ) )  =  ( A ^ (
n  -  k ) ) )
1716oveq1d 5725 . . . . . . . . 9  |-  ( x  =  n  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( n  -  k ) )  x.  ( B ^ k
) ) )
1814, 17oveq12d 5728 . . . . . . . 8  |-  ( x  =  n  ->  (
( x  _C  k
)  x.  ( ( A ^ ( x  -  k ) )  x.  ( B ^
k ) ) )  =  ( ( n  _C  k )  x.  ( ( A ^
( n  -  k
) )  x.  ( B ^ k ) ) ) )
1918adantr 453 . . . . . . 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 12056 . . . . . 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 2267 . . . . 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 309 . . . 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 5718 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^
( n  +  1 ) ) )
24 oveq2 5718 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (
0 ... x )  =  ( 0 ... (
n  +  1 ) ) )
25 oveq1 5717 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
x  _C  k )  =  ( ( n  +  1 )  _C  k ) )
26 oveq1 5717 . . . . . . . . . . 11  |-  ( x  =  ( n  + 
1 )  ->  (
x  -  k )  =  ( ( n  +  1 )  -  k ) )
2726oveq2d 5726 . . . . . . . . . 10  |-  ( x  =  ( n  + 
1 )  ->  ( A ^ ( x  -  k ) )  =  ( A ^ (
( n  +  1 )  -  k ) ) )
2827oveq1d 5725 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( ( n  +  1 )  -  k ) )  x.  ( B ^ k
) ) )
2925, 28oveq12d 5728 . . . . . . . 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 453 . . . . . . 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 12056 . . . . . 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 2267 . . . . 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 309 . . . 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 5718 . . . . . 6  |-  ( x  =  N  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^ N ) )
35 oveq2 5718 . . . . . . 7  |-  ( x  =  N  ->  (
0 ... x )  =  ( 0 ... N
) )
36 oveq1 5717 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  _C  k )  =  ( N  _C  k ) )
37 oveq1 5717 . . . . . . . . . . 11  |-  ( x  =  N  ->  (
x  -  k )  =  ( N  -  k ) )
3837oveq2d 5726 . . . . . . . . . 10  |-  ( x  =  N  ->  ( A ^ ( x  -  k ) )  =  ( A ^ ( N  -  k )
) )
3938oveq1d 5725 . . . . . . . . 9  |-  ( x  =  N  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( N  -  k ) )  x.  ( B ^ k
) ) )
4036, 39oveq12d 5728 . . . . . . . 8  |-  ( x  =  N  ->  (
( x  _C  k
)  x.  ( ( A ^ ( x  -  k ) )  x.  ( B ^
k ) ) )  =  ( ( N  _C  k )  x.  ( ( A ^
( N  -  k
) )  x.  ( B ^ k ) ) ) )
4140adantr 453 . . . . . . 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 12056 . . . . . 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 2267 . . . . 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 309 . . . 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 10986 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
46 exp0 10986 . . . . . . . . 9  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
4745, 46oveqan12d 5729 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  ( 1  x.  1 ) )
48 1t1e1 9749 . . . . . . . 8  |-  ( 1  x.  1 )  =  1
4947, 48syl6eq 2301 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  1 )
5049oveq2d 5726 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  =  ( 1  x.  1 ) )
5150, 48syl6eq 2301 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  =  1 )
52 0z 9914 . . . . . 6  |-  0  e.  ZZ
53 ax-1cn 8675 . . . . . . 7  |-  1  e.  CC
5451, 53syl6eqel 2341 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  e.  CC )
55 oveq2 5718 . . . . . . . . 9  |-  ( k  =  0  ->  (
0  _C  k )  =  ( 0  _C  0 ) )
56 0nn0 9859 . . . . . . . . . 10  |-  0  e.  NN0
57 bcn0 11201 . . . . . . . . . 10  |-  ( 0  e.  NN0  ->  ( 0  _C  0 )  =  1 )
5856, 57ax-mp 10 . . . . . . . . 9  |-  ( 0  _C  0 )  =  1
5955, 58syl6eq 2301 . . . . . . . 8  |-  ( k  =  0  ->  (
0  _C  k )  =  1 )
60 oveq2 5718 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
0  -  k )  =  ( 0  -  0 ) )
61 0cn 8711 . . . . . . . . . . . 12  |-  0  e.  CC
6261subidi 8997 . . . . . . . . . . 11  |-  ( 0  -  0 )  =  0
6360, 62syl6eq 2301 . . . . . . . . . 10  |-  ( k  =  0  ->  (
0  -  k )  =  0 )
6463oveq2d 5726 . . . . . . . . 9  |-  ( k  =  0  ->  ( A ^ ( 0  -  k ) )  =  ( A ^ 0 ) )
65 oveq2 5718 . . . . . . . . 9  |-  ( k  =  0  ->  ( B ^ k )  =  ( B ^ 0 ) )
6664, 65oveq12d 5728 . . . . . . . 8  |-  ( k  =  0  ->  (
( A ^ (
0  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ 0 )  x.  ( B ^ 0 ) ) )
6759, 66oveq12d 5728 . . . . . . 7  |-  ( k  =  0  ->  (
( 0  _C  k
)  x.  ( ( A ^ ( 0  -  k ) )  x.  ( B ^
k ) ) )  =  ( 1  x.  ( ( A ^
0 )  x.  ( B ^ 0 ) ) ) )
6867fsum1 12091 . . . . . 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 647 . . . . 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 8699 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
7170exp0d 11117 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^ 0 )  =  1 )
7251, 69, 713eqtr4rd 2296 . . . 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 735 . . . . . . 7  |-  ( ( n  e.  NN0  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  A  e.  CC )
74 simprr 736 . . . . . . 7  |-  ( ( n  e.  NN0  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  B  e.  CC )
75 simpl 445 . . . . . . 7  |-  ( ( n  e.  NN0  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  n  e.  NN0 )
76 id 21 . . . . . . 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 12164 . . . . . 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 590 . . . . 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 25 . . . 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 9987 . . 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 421 . 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 1151 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621  (class class class)co 5710   CCcc 8615   0cc0 8617   1c1 8618    + caddc 8620    x. cmul 8622    - cmin 8917   NN0cn0 9844   ZZcz 9903   ...cfz 10660   ^cexp 10982    _C cbc 11193   sum_csu 12035
This theorem is referenced by:  binom1p  12166  efaddlem  12248  basellem3  20152  jm2.22  26254
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-fz 10661  df-fzo 10749  df-seq 10925  df-exp 10983  df-fac 11167  df-bc 11194  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036
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