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Theorem bccolsum 30446
Description: A column-sum rule for binomial coefficents. (Contributed by Scott Fenton, 24-Jun-2020.)
Assertion
Ref Expression
bccolsum  |-  ( ( N  e.  NN0  /\  C  e.  NN0 )  ->  sum_ k  e.  ( 0 ... N ) ( k  _C  C )  =  ( ( N  +  1 )  _C  ( C  +  1 ) ) )
Distinct variable groups:    k, N    C, k

Proof of Theorem bccolsum
Dummy variables  n  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6316 . . . . . 6  |-  ( m  =  0  ->  (
0 ... m )  =  ( 0 ... 0
) )
21sumeq1d 13844 . . . . 5  |-  ( m  =  0  ->  sum_ k  e.  ( 0 ... m
) ( k  _C  C )  =  sum_ k  e.  ( 0 ... 0 ) ( k  _C  C ) )
3 oveq1 6315 . . . . . . 7  |-  ( m  =  0  ->  (
m  +  1 )  =  ( 0  +  1 ) )
4 0p1e1 10743 . . . . . . 7  |-  ( 0  +  1 )  =  1
53, 4syl6eq 2521 . . . . . 6  |-  ( m  =  0  ->  (
m  +  1 )  =  1 )
65oveq1d 6323 . . . . 5  |-  ( m  =  0  ->  (
( m  +  1 )  _C  ( C  +  1 ) )  =  ( 1  _C  ( C  +  1 ) ) )
72, 6eqeq12d 2486 . . . 4  |-  ( m  =  0  ->  ( sum_ k  e.  ( 0 ... m ) ( k  _C  C )  =  ( ( m  +  1 )  _C  ( C  +  1 ) )  <->  sum_ k  e.  ( 0 ... 0
) ( k  _C  C )  =  ( 1  _C  ( C  +  1 ) ) ) )
87imbi2d 323 . . 3  |-  ( m  =  0  ->  (
( C  e.  NN0  -> 
sum_ k  e.  ( 0 ... m ) ( k  _C  C
)  =  ( ( m  +  1 )  _C  ( C  + 
1 ) ) )  <-> 
( C  e.  NN0  -> 
sum_ k  e.  ( 0 ... 0 ) ( k  _C  C
)  =  ( 1  _C  ( C  + 
1 ) ) ) ) )
9 oveq2 6316 . . . . . 6  |-  ( m  =  n  ->  (
0 ... m )  =  ( 0 ... n
) )
109sumeq1d 13844 . . . . 5  |-  ( m  =  n  ->  sum_ k  e.  ( 0 ... m
) ( k  _C  C )  =  sum_ k  e.  ( 0 ... n ) ( k  _C  C ) )
11 oveq1 6315 . . . . . 6  |-  ( m  =  n  ->  (
m  +  1 )  =  ( n  + 
1 ) )
1211oveq1d 6323 . . . . 5  |-  ( m  =  n  ->  (
( m  +  1 )  _C  ( C  +  1 ) )  =  ( ( n  +  1 )  _C  ( C  +  1 ) ) )
1310, 12eqeq12d 2486 . . . 4  |-  ( m  =  n  ->  ( sum_ k  e.  ( 0 ... m ) ( k  _C  C )  =  ( ( m  +  1 )  _C  ( C  +  1 ) )  <->  sum_ k  e.  ( 0 ... n
) ( k  _C  C )  =  ( ( n  +  1 )  _C  ( C  +  1 ) ) ) )
1413imbi2d 323 . . 3  |-  ( m  =  n  ->  (
( C  e.  NN0  -> 
sum_ k  e.  ( 0 ... m ) ( k  _C  C
)  =  ( ( m  +  1 )  _C  ( C  + 
1 ) ) )  <-> 
( C  e.  NN0  -> 
sum_ k  e.  ( 0 ... n ) ( k  _C  C
)  =  ( ( n  +  1 )  _C  ( C  + 
1 ) ) ) ) )
15 oveq2 6316 . . . . . 6  |-  ( m  =  ( n  + 
1 )  ->  (
0 ... m )  =  ( 0 ... (
n  +  1 ) ) )
1615sumeq1d 13844 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  sum_ k  e.  ( 0 ... m
) ( k  _C  C )  =  sum_ k  e.  ( 0 ... ( n  + 
1 ) ) ( k  _C  C ) )
17 oveq1 6315 . . . . . 6  |-  ( m  =  ( n  + 
1 )  ->  (
m  +  1 )  =  ( ( n  +  1 )  +  1 ) )
1817oveq1d 6323 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (
( m  +  1 )  _C  ( C  +  1 ) )  =  ( ( ( n  +  1 )  +  1 )  _C  ( C  +  1 ) ) )
1916, 18eqeq12d 2486 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  ( sum_ k  e.  ( 0 ... m ) ( k  _C  C )  =  ( ( m  +  1 )  _C  ( C  +  1 ) )  <->  sum_ k  e.  ( 0 ... (
n  +  1 ) ) ( k  _C  C )  =  ( ( ( n  + 
1 )  +  1 )  _C  ( C  +  1 ) ) ) )
2019imbi2d 323 . . 3  |-  ( m  =  ( n  + 
1 )  ->  (
( C  e.  NN0  -> 
sum_ k  e.  ( 0 ... m ) ( k  _C  C
)  =  ( ( m  +  1 )  _C  ( C  + 
1 ) ) )  <-> 
( C  e.  NN0  -> 
sum_ k  e.  ( 0 ... ( n  +  1 ) ) ( k  _C  C
)  =  ( ( ( n  +  1 )  +  1 )  _C  ( C  + 
1 ) ) ) ) )
21 oveq2 6316 . . . . . 6  |-  ( m  =  N  ->  (
0 ... m )  =  ( 0 ... N
) )
2221sumeq1d 13844 . . . . 5  |-  ( m  =  N  ->  sum_ k  e.  ( 0 ... m
) ( k  _C  C )  =  sum_ k  e.  ( 0 ... N ) ( k  _C  C ) )
23 oveq1 6315 . . . . . 6  |-  ( m  =  N  ->  (
m  +  1 )  =  ( N  + 
1 ) )
2423oveq1d 6323 . . . . 5  |-  ( m  =  N  ->  (
( m  +  1 )  _C  ( C  +  1 ) )  =  ( ( N  +  1 )  _C  ( C  +  1 ) ) )
2522, 24eqeq12d 2486 . . . 4  |-  ( m  =  N  ->  ( sum_ k  e.  ( 0 ... m ) ( k  _C  C )  =  ( ( m  +  1 )  _C  ( C  +  1 ) )  <->  sum_ k  e.  ( 0 ... N
) ( k  _C  C )  =  ( ( N  +  1 )  _C  ( C  +  1 ) ) ) )
2625imbi2d 323 . . 3  |-  ( m  =  N  ->  (
( C  e.  NN0  -> 
sum_ k  e.  ( 0 ... m ) ( k  _C  C
)  =  ( ( m  +  1 )  _C  ( C  + 
1 ) ) )  <-> 
( C  e.  NN0  -> 
sum_ k  e.  ( 0 ... N ) ( k  _C  C
)  =  ( ( N  +  1 )  _C  ( C  + 
1 ) ) ) ) )
27 0z 10972 . . . . 5  |-  0  e.  ZZ
28 0nn0 10908 . . . . . . 7  |-  0  e.  NN0
29 nn0z 10984 . . . . . . 7  |-  ( C  e.  NN0  ->  C  e.  ZZ )
30 bccl 12545 . . . . . . 7  |-  ( ( 0  e.  NN0  /\  C  e.  ZZ )  ->  ( 0  _C  C
)  e.  NN0 )
3128, 29, 30sylancr 676 . . . . . 6  |-  ( C  e.  NN0  ->  ( 0  _C  C )  e. 
NN0 )
3231nn0cnd 10951 . . . . 5  |-  ( C  e.  NN0  ->  ( 0  _C  C )  e.  CC )
33 oveq1 6315 . . . . . 6  |-  ( k  =  0  ->  (
k  _C  C )  =  ( 0  _C  C ) )
3433fsum1 13885 . . . . 5  |-  ( ( 0  e.  ZZ  /\  ( 0  _C  C
)  e.  CC )  ->  sum_ k  e.  ( 0 ... 0 ) ( k  _C  C
)  =  ( 0  _C  C ) )
3527, 32, 34sylancr 676 . . . 4  |-  ( C  e.  NN0  ->  sum_ k  e.  ( 0 ... 0
) ( k  _C  C )  =  ( 0  _C  C ) )
36 elnn0 10895 . . . . 5  |-  ( C  e.  NN0  <->  ( C  e.  NN  \/  C  =  0 ) )
37 1red 9676 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  1  e.  RR )
38 nnrp 11334 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  e.  RR+ )
3937, 38ltaddrp2d 11395 . . . . . . . . . 10  |-  ( C  e.  NN  ->  1  <  ( C  +  1 ) )
40 peano2nn 10643 . . . . . . . . . . . 12  |-  ( C  e.  NN  ->  ( C  +  1 )  e.  NN )
4140nnred 10646 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  ( C  +  1 )  e.  RR )
4237, 41ltnled 9799 . . . . . . . . . 10  |-  ( C  e.  NN  ->  (
1  <  ( C  +  1 )  <->  -.  ( C  +  1 )  <_  1 ) )
4339, 42mpbid 215 . . . . . . . . 9  |-  ( C  e.  NN  ->  -.  ( C  +  1
)  <_  1 )
44 elfzle2 11829 . . . . . . . . 9  |-  ( ( C  +  1 )  e.  ( 0 ... 1 )  ->  ( C  +  1 )  <_  1 )
4543, 44nsyl 125 . . . . . . . 8  |-  ( C  e.  NN  ->  -.  ( C  +  1
)  e.  ( 0 ... 1 ) )
4645iffalsed 3883 . . . . . . 7  |-  ( C  e.  NN  ->  if ( ( C  + 
1 )  e.  ( 0 ... 1 ) ,  ( ( ! `
 1 )  / 
( ( ! `  ( 1  -  ( C  +  1 ) ) )  x.  ( ! `  ( C  +  1 ) ) ) ) ,  0 )  =  0 )
47 1nn0 10909 . . . . . . . 8  |-  1  e.  NN0
4840nnzd 11062 . . . . . . . 8  |-  ( C  e.  NN  ->  ( C  +  1 )  e.  ZZ )
49 bcval 12527 . . . . . . . 8  |-  ( ( 1  e.  NN0  /\  ( C  +  1
)  e.  ZZ )  ->  ( 1  _C  ( C  +  1 ) )  =  if ( ( C  + 
1 )  e.  ( 0 ... 1 ) ,  ( ( ! `
 1 )  / 
( ( ! `  ( 1  -  ( C  +  1 ) ) )  x.  ( ! `  ( C  +  1 ) ) ) ) ,  0 ) )
5047, 48, 49sylancr 676 . . . . . . 7  |-  ( C  e.  NN  ->  (
1  _C  ( C  +  1 ) )  =  if ( ( C  +  1 )  e.  ( 0 ... 1 ) ,  ( ( ! `  1
)  /  ( ( ! `  ( 1  -  ( C  + 
1 ) ) )  x.  ( ! `  ( C  +  1
) ) ) ) ,  0 ) )
51 bc0k 12534 . . . . . . 7  |-  ( C  e.  NN  ->  (
0  _C  C )  =  0 )
5246, 50, 513eqtr4rd 2516 . . . . . 6  |-  ( C  e.  NN  ->  (
0  _C  C )  =  ( 1  _C  ( C  +  1 ) ) )
53 bcnn 12535 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( 0  _C  0 )  =  1 )
5428, 53ax-mp 5 . . . . . . . 8  |-  ( 0  _C  0 )  =  1
55 bcnn 12535 . . . . . . . . 9  |-  ( 1  e.  NN0  ->  ( 1  _C  1 )  =  1 )
5647, 55ax-mp 5 . . . . . . . 8  |-  ( 1  _C  1 )  =  1
5754, 56eqtr4i 2496 . . . . . . 7  |-  ( 0  _C  0 )  =  ( 1  _C  1
)
58 oveq2 6316 . . . . . . 7  |-  ( C  =  0  ->  (
0  _C  C )  =  ( 0  _C  0 ) )
59 oveq1 6315 . . . . . . . . 9  |-  ( C  =  0  ->  ( C  +  1 )  =  ( 0  +  1 ) )
6059, 4syl6eq 2521 . . . . . . . 8  |-  ( C  =  0  ->  ( C  +  1 )  =  1 )
6160oveq2d 6324 . . . . . . 7  |-  ( C  =  0  ->  (
1  _C  ( C  +  1 ) )  =  ( 1  _C  1 ) )
6257, 58, 613eqtr4a 2531 . . . . . 6  |-  ( C  =  0  ->  (
0  _C  C )  =  ( 1  _C  ( C  +  1 ) ) )
6352, 62jaoi 386 . . . . 5  |-  ( ( C  e.  NN  \/  C  =  0 )  ->  ( 0  _C  C )  =  ( 1  _C  ( C  +  1 ) ) )
6436, 63sylbi 200 . . . 4  |-  ( C  e.  NN0  ->  ( 0  _C  C )  =  ( 1  _C  ( C  +  1 ) ) )
6535, 64eqtrd 2505 . . 3  |-  ( C  e.  NN0  ->  sum_ k  e.  ( 0 ... 0
) ( k  _C  C )  =  ( 1  _C  ( C  +  1 ) ) )
66 elnn0uz 11220 . . . . . . . . . 10  |-  ( n  e.  NN0  <->  n  e.  ( ZZ>=
`  0 ) )
6766biimpi 199 . . . . . . . . 9  |-  ( n  e.  NN0  ->  n  e.  ( ZZ>= `  0 )
)
6867adantr 472 . . . . . . . 8  |-  ( ( n  e.  NN0  /\  C  e.  NN0 )  ->  n  e.  ( ZZ>= ` 
0 ) )
69 elfznn0 11913 . . . . . . . . . . 11  |-  ( k  e.  ( 0 ... ( n  +  1 ) )  ->  k  e.  NN0 )
7069adantl 473 . . . . . . . . . 10  |-  ( ( ( n  e.  NN0  /\  C  e.  NN0 )  /\  k  e.  (
0 ... ( n  + 
1 ) ) )  ->  k  e.  NN0 )
71 simplr 770 . . . . . . . . . . 11  |-  ( ( ( n  e.  NN0  /\  C  e.  NN0 )  /\  k  e.  (
0 ... ( n  + 
1 ) ) )  ->  C  e.  NN0 )
7271nn0zd 11061 . . . . . . . . . 10  |-  ( ( ( n  e.  NN0  /\  C  e.  NN0 )  /\  k  e.  (
0 ... ( n  + 
1 ) ) )  ->  C  e.  ZZ )
73 bccl 12545 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  C  e.  ZZ )  ->  ( k  _C  C
)  e.  NN0 )
7470, 72, 73syl2anc 673 . . . . . . . . 9  |-  ( ( ( n  e.  NN0  /\  C  e.  NN0 )  /\  k  e.  (
0 ... ( n  + 
1 ) ) )  ->  ( k  _C  C )  e.  NN0 )
7574nn0cnd 10951 . . . . . . . 8  |-  ( ( ( n  e.  NN0  /\  C  e.  NN0 )  /\  k  e.  (
0 ... ( n  + 
1 ) ) )  ->  ( k  _C  C )  e.  CC )
76 oveq1 6315 . . . . . . . 8  |-  ( k  =  ( n  + 
1 )  ->  (
k  _C  C )  =  ( ( n  +  1 )  _C  C ) )
7768, 75, 76fsump1 13894 . . . . . . 7  |-  ( ( n  e.  NN0  /\  C  e.  NN0 )  ->  sum_ k  e.  ( 0 ... ( n  + 
1 ) ) ( k  _C  C )  =  ( sum_ k  e.  ( 0 ... n
) ( k  _C  C )  +  ( ( n  +  1 )  _C  C ) ) )
7877adantr 472 . . . . . 6  |-  ( ( ( n  e.  NN0  /\  C  e.  NN0 )  /\  sum_ k  e.  ( 0 ... n ) ( k  _C  C
)  =  ( ( n  +  1 )  _C  ( C  + 
1 ) ) )  ->  sum_ k  e.  ( 0 ... ( n  +  1 ) ) ( k  _C  C
)  =  ( sum_ k  e.  ( 0 ... n ) ( k  _C  C )  +  ( ( n  +  1 )  _C  C ) ) )
79 id 22 . . . . . . 7  |-  ( sum_ k  e.  ( 0 ... n ) ( k  _C  C )  =  ( ( n  +  1 )  _C  ( C  +  1 ) )  ->  sum_ k  e.  ( 0 ... n
) ( k  _C  C )  =  ( ( n  +  1 )  _C  ( C  +  1 ) ) )
80 nn0cn 10903 . . . . . . . . . . 11  |-  ( C  e.  NN0  ->  C  e.  CC )
8180adantl 473 . . . . . . . . . 10  |-  ( ( n  e.  NN0  /\  C  e.  NN0 )  ->  C  e.  CC )
82 1cnd 9677 . . . . . . . . . 10  |-  ( ( n  e.  NN0  /\  C  e.  NN0 )  -> 
1  e.  CC )
8381, 82pncand 10006 . . . . . . . . 9  |-  ( ( n  e.  NN0  /\  C  e.  NN0 )  -> 
( ( C  + 
1 )  -  1 )  =  C )
8483oveq2d 6324 . . . . . . . 8  |-  ( ( n  e.  NN0  /\  C  e.  NN0 )  -> 
( ( n  + 
1 )  _C  (
( C  +  1 )  -  1 ) )  =  ( ( n  +  1 )  _C  C ) )
8584eqcomd 2477 . . . . . . 7  |-  ( ( n  e.  NN0  /\  C  e.  NN0 )  -> 
( ( n  + 
1 )  _C  C
)  =  ( ( n  +  1 )  _C  ( ( C  +  1 )  - 
1 ) ) )
8679, 85oveqan12rd 6328 . . . . . 6  |-  ( ( ( n  e.  NN0  /\  C  e.  NN0 )  /\  sum_ k  e.  ( 0 ... n ) ( k  _C  C
)  =  ( ( n  +  1 )  _C  ( C  + 
1 ) ) )  ->  ( sum_ k  e.  ( 0 ... n
) ( k  _C  C )  +  ( ( n  +  1 )  _C  C ) )  =  ( ( ( n  +  1 )  _C  ( C  +  1 ) )  +  ( ( n  +  1 )  _C  ( ( C  + 
1 )  -  1 ) ) ) )
87 peano2nn0 10934 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( n  +  1 )  e. 
NN0 )
88 peano2nn0 10934 . . . . . . . . 9  |-  ( C  e.  NN0  ->  ( C  +  1 )  e. 
NN0 )
8988nn0zd 11061 . . . . . . . 8  |-  ( C  e.  NN0  ->  ( C  +  1 )  e.  ZZ )
90 bcpasc 12544 . . . . . . . 8  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( C  +  1
)  e.  ZZ )  ->  ( ( ( n  +  1 )  _C  ( C  + 
1 ) )  +  ( ( n  + 
1 )  _C  (
( C  +  1 )  -  1 ) ) )  =  ( ( ( n  + 
1 )  +  1 )  _C  ( C  +  1 ) ) )
9187, 89, 90syl2an 485 . . . . . . 7  |-  ( ( n  e.  NN0  /\  C  e.  NN0 )  -> 
( ( ( n  +  1 )  _C  ( C  +  1 ) )  +  ( ( n  +  1 )  _C  ( ( C  +  1 )  -  1 ) ) )  =  ( ( ( n  +  1 )  +  1 )  _C  ( C  + 
1 ) ) )
9291adantr 472 . . . . . 6  |-  ( ( ( n  e.  NN0  /\  C  e.  NN0 )  /\  sum_ k  e.  ( 0 ... n ) ( k  _C  C
)  =  ( ( n  +  1 )  _C  ( C  + 
1 ) ) )  ->  ( ( ( n  +  1 )  _C  ( C  + 
1 ) )  +  ( ( n  + 
1 )  _C  (
( C  +  1 )  -  1 ) ) )  =  ( ( ( n  + 
1 )  +  1 )  _C  ( C  +  1 ) ) )
9378, 86, 923eqtrd 2509 . . . . 5  |-  ( ( ( n  e.  NN0  /\  C  e.  NN0 )  /\  sum_ k  e.  ( 0 ... n ) ( k  _C  C
)  =  ( ( n  +  1 )  _C  ( C  + 
1 ) ) )  ->  sum_ k  e.  ( 0 ... ( n  +  1 ) ) ( k  _C  C
)  =  ( ( ( n  +  1 )  +  1 )  _C  ( C  + 
1 ) ) )
9493exp31 615 . . . 4  |-  ( n  e.  NN0  ->  ( C  e.  NN0  ->  ( sum_ k  e.  ( 0 ... n ) ( k  _C  C )  =  ( ( n  +  1 )  _C  ( C  +  1 ) )  ->  sum_ k  e.  ( 0 ... (
n  +  1 ) ) ( k  _C  C )  =  ( ( ( n  + 
1 )  +  1 )  _C  ( C  +  1 ) ) ) ) )
9594a2d 28 . . 3  |-  ( n  e.  NN0  ->  ( ( C  e.  NN0  ->  sum_ k  e.  ( 0 ... n ) ( k  _C  C )  =  ( ( n  +  1 )  _C  ( C  +  1 ) ) )  -> 
( C  e.  NN0  -> 
sum_ k  e.  ( 0 ... ( n  +  1 ) ) ( k  _C  C
)  =  ( ( ( n  +  1 )  +  1 )  _C  ( C  + 
1 ) ) ) ) )
968, 14, 20, 26, 65, 95nn0ind 11053 . 2  |-  ( N  e.  NN0  ->  ( C  e.  NN0  ->  sum_ k  e.  ( 0 ... N
) ( k  _C  C )  =  ( ( N  +  1 )  _C  ( C  +  1 ) ) ) )
9796imp 436 1  |-  ( ( N  e.  NN0  /\  C  e.  NN0 )  ->  sum_ k  e.  ( 0 ... N ) ( k  _C  C )  =  ( ( N  +  1 )  _C  ( C  +  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904   ifcif 3872   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   !cfa 12497    _C cbc 12525   sum_csu 13829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830
This theorem is referenced by: (None)
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