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Theorem binom 12385
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 12384. (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 5953 . . . . . 6  |-  ( x  =  0  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^
0 ) )
2 oveq2 5953 . . . . . . 7  |-  ( x  =  0  ->  (
0 ... x )  =  ( 0 ... 0
) )
3 oveq1 5952 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  _C  k )  =  ( 0  _C  k ) )
4 oveq1 5952 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
x  -  k )  =  ( 0  -  k ) )
54oveq2d 5961 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A ^ ( x  -  k ) )  =  ( A ^ (
0  -  k ) ) )
65oveq1d 5960 . . . . . . . . 9  |-  ( x  =  0  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( 0  -  k ) )  x.  ( B ^ k
) ) )
73, 6oveq12d 5963 . . . . . . . 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 12276 . . . . . 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 2372 . . . . 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 5953 . . . . . 6  |-  ( x  =  n  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^
n ) )
13 oveq2 5953 . . . . . . 7  |-  ( x  =  n  ->  (
0 ... x )  =  ( 0 ... n
) )
14 oveq1 5952 . . . . . . . . 9  |-  ( x  =  n  ->  (
x  _C  k )  =  ( n  _C  k ) )
15 oveq1 5952 . . . . . . . . . . 11  |-  ( x  =  n  ->  (
x  -  k )  =  ( n  -  k ) )
1615oveq2d 5961 . . . . . . . . . 10  |-  ( x  =  n  ->  ( A ^ ( x  -  k ) )  =  ( A ^ (
n  -  k ) ) )
1716oveq1d 5960 . . . . . . . . 9  |-  ( x  =  n  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( n  -  k ) )  x.  ( B ^ k
) ) )
1814, 17oveq12d 5963 . . . . . . . 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 12276 . . . . . 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 2372 . . . . 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 5953 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^
( n  +  1 ) ) )
24 oveq2 5953 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (
0 ... x )  =  ( 0 ... (
n  +  1 ) ) )
25 oveq1 5952 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
x  _C  k )  =  ( ( n  +  1 )  _C  k ) )
26 oveq1 5952 . . . . . . . . . . 11  |-  ( x  =  ( n  + 
1 )  ->  (
x  -  k )  =  ( ( n  +  1 )  -  k ) )
2726oveq2d 5961 . . . . . . . . . 10  |-  ( x  =  ( n  + 
1 )  ->  ( A ^ ( x  -  k ) )  =  ( A ^ (
( n  +  1 )  -  k ) ) )
2827oveq1d 5960 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( ( n  +  1 )  -  k ) )  x.  ( B ^ k
) ) )
2925, 28oveq12d 5963 . . . . . . . 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 12276 . . . . . 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 2372 . . . . 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 5953 . . . . . 6  |-  ( x  =  N  ->  (
( A  +  B
) ^ x )  =  ( ( A  +  B ) ^ N ) )
35 oveq2 5953 . . . . . . 7  |-  ( x  =  N  ->  (
0 ... x )  =  ( 0 ... N
) )
36 oveq1 5952 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  _C  k )  =  ( N  _C  k ) )
37 oveq1 5952 . . . . . . . . . . 11  |-  ( x  =  N  ->  (
x  -  k )  =  ( N  -  k ) )
3837oveq2d 5961 . . . . . . . . . 10  |-  ( x  =  N  ->  ( A ^ ( x  -  k ) )  =  ( A ^ ( N  -  k )
) )
3938oveq1d 5960 . . . . . . . . 9  |-  ( x  =  N  ->  (
( A ^ (
x  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ ( N  -  k ) )  x.  ( B ^ k
) ) )
4036, 39oveq12d 5963 . . . . . . . 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 12276 . . . . . 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 2372 . . . . 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 11201 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
46 exp0 11201 . . . . . . . . 9  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
4745, 46oveqan12d 5964 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  ( 1  x.  1 ) )
48 1t1e1 9962 . . . . . . . 8  |-  ( 1  x.  1 )  =  1
4947, 48syl6eq 2406 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  1 )
5049oveq2d 5961 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  =  ( 1  x.  1 ) )
5150, 48syl6eq 2406 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  =  1 )
52 0z 10127 . . . . . 6  |-  0  e.  ZZ
53 ax-1cn 8885 . . . . . . 7  |-  1  e.  CC
5451, 53syl6eqel 2446 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  (
( A ^ 0 )  x.  ( B ^ 0 ) ) )  e.  CC )
55 oveq2 5953 . . . . . . . . 9  |-  ( k  =  0  ->  (
0  _C  k )  =  ( 0  _C  0 ) )
56 0nn0 10072 . . . . . . . . . 10  |-  0  e.  NN0
57 bcn0 11416 . . . . . . . . . 10  |-  ( 0  e.  NN0  ->  ( 0  _C  0 )  =  1 )
5856, 57ax-mp 8 . . . . . . . . 9  |-  ( 0  _C  0 )  =  1
5955, 58syl6eq 2406 . . . . . . . 8  |-  ( k  =  0  ->  (
0  _C  k )  =  1 )
60 oveq2 5953 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
0  -  k )  =  ( 0  -  0 ) )
61 0cn 8921 . . . . . . . . . . . 12  |-  0  e.  CC
6261subidi 9207 . . . . . . . . . . 11  |-  ( 0  -  0 )  =  0
6360, 62syl6eq 2406 . . . . . . . . . 10  |-  ( k  =  0  ->  (
0  -  k )  =  0 )
6463oveq2d 5961 . . . . . . . . 9  |-  ( k  =  0  ->  ( A ^ ( 0  -  k ) )  =  ( A ^ 0 ) )
65 oveq2 5953 . . . . . . . . 9  |-  ( k  =  0  ->  ( B ^ k )  =  ( B ^ 0 ) )
6664, 65oveq12d 5963 . . . . . . . 8  |-  ( k  =  0  ->  (
( A ^ (
0  -  k ) )  x.  ( B ^ k ) )  =  ( ( A ^ 0 )  x.  ( B ^ 0 ) ) )
6759, 66oveq12d 5963 . . . . . . 7  |-  ( k  =  0  ->  (
( 0  _C  k
)  x.  ( ( A ^ ( 0  -  k ) )  x.  ( B ^
k ) ) )  =  ( 1  x.  ( ( A ^
0 )  x.  ( B ^ 0 ) ) ) )
6867fsum1 12311 . . . . . 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 8909 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
7170exp0d 11332 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^ 0 )  =  1 )
7251, 69, 713eqtr4rd 2401 . . . 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 12384 . . . . . 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 10200 . . 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 1642    e. wcel 1710  (class class class)co 5945   CCcc 8825   0cc0 8827   1c1 8828    + caddc 8830    x. cmul 8832    - cmin 9127   NN0cn0 10057   ZZcz 10116   ...cfz 10874   ^cexp 11197    _C cbc 11408   sum_csu 12255
This theorem is referenced by:  binom1p  12386  efaddlem  12471  basellem3  20432  jm2.22  26411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-fz 10875  df-fzo 10963  df-seq 11139  df-exp 11198  df-fac 11382  df-bc 11409  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-sum 12256
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