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Theorem coeaddlem 21691
Description: Lemma for coeadd 21693 and dgradd 21709. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
coefv0.1  |-  A  =  (coeff `  F )
coeadd.2  |-  B  =  (coeff `  G )
coeadd.3  |-  M  =  (deg `  F )
coeadd.4  |-  N  =  (deg `  G )
Assertion
Ref Expression
coeaddlem  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  oF  +  G ) )  =  ( A  oF  +  B )  /\  (deg `  ( F  oF  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
) )

Proof of Theorem coeaddlem
Dummy variables  k  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyaddcl 21663 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  +  G
)  e.  (Poly `  CC ) )
2 coeadd.4 . . . . . 6  |-  N  =  (deg `  G )
3 dgrcl 21676 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
42, 3syl5eqel 2522 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
54adantl 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  NN0 )
6 coeadd.3 . . . . . 6  |-  M  =  (deg `  F )
7 dgrcl 21676 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
86, 7syl5eqel 2522 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
98adantr 465 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  NN0 )
10 ifcl 3826 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
115, 9, 10syl2anc 661 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
12 addcl 9356 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
1312adantl 466 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
14 coefv0.1 . . . . . 6  |-  A  =  (coeff `  F )
1514coef3 21675 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
1615adantr 465 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
17 coeadd.2 . . . . . 6  |-  B  =  (coeff `  G )
1817coef3 21675 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
1918adantl 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
20 nn0ex 10577 . . . . 5  |-  NN0  e.  _V
2120a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  NN0  e.  _V )
22 inidm 3554 . . . 4  |-  ( NN0 
i^i  NN0 )  =  NN0
2313, 16, 19, 21, 21, 22off 6329 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  oF  +  B
) : NN0 --> CC )
24 oveq12 6095 . . . . . . . . . 10  |-  ( ( ( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A `
 k )  +  ( B `  k
) )  =  ( 0  +  0 ) )
25 00id 9536 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
2624, 25syl6eq 2486 . . . . . . . . 9  |-  ( ( ( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A `
 k )  +  ( B `  k
) )  =  0 )
27 ffn 5554 . . . . . . . . . . . 12  |-  ( A : NN0 --> CC  ->  A  Fn  NN0 )
2816, 27syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A  Fn  NN0 )
29 ffn 5554 . . . . . . . . . . . 12  |-  ( B : NN0 --> CC  ->  B  Fn  NN0 )
3019, 29syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B  Fn  NN0 )
31 eqidd 2439 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( A `  k )  =  ( A `  k ) )
32 eqidd 2439 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( B `  k )  =  ( B `  k ) )
3328, 30, 21, 21, 22, 31, 32ofval 6324 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A  oF  +  B
) `  k )  =  ( ( A `
 k )  +  ( B `  k
) ) )
3433eqeq1d 2446 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  oF  +  B ) `  k )  =  0  <-> 
( ( A `  k )  +  ( B `  k ) )  =  0 ) )
3526, 34syl5ibr 221 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A  oF  +  B
) `  k )  =  0 ) )
3635necon3ad 2639 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  oF  +  B ) `  k )  =/=  0  ->  -.  ( ( A `
 k )  =  0  /\  ( B `
 k )  =  0 ) ) )
37 neorian 2694 . . . . . . 7  |-  ( ( ( A `  k
)  =/=  0  \/  ( B `  k
)  =/=  0 )  <->  -.  ( ( A `  k )  =  0  /\  ( B `  k )  =  0 ) )
3836, 37syl6ibr 227 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  oF  +  B ) `  k )  =/=  0  ->  ( ( A `  k )  =/=  0  \/  ( B `  k
)  =/=  0 ) ) )
3914, 6dgrub2 21678 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  ( A " ( ZZ>= `  ( M  +  1 ) ) )  =  { 0 } )
4039adantr 465 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A " ( ZZ>= `  ( M  +  1 ) ) )  =  { 0 } )
41 plyco0 21635 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  M ) ) )
429, 16, 41syl2anc 661 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A " ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  M ) ) )
4340, 42mpbid 210 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  M ) )
4443r19.21bi 2809 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A `  k )  =/=  0  ->  k  <_  M ) )
459adantr 465 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  e.  NN0 )
4645nn0red 10629 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  e.  RR )
475adantr 465 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  e.  NN0 )
4847nn0red 10629 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  e.  RR )
49 max1 11149 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
5046, 48, 49syl2anc 661 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
51 nn0re 10580 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  k  e.  RR )
5251adantl 466 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  k  e.  RR )
5311adantr 465 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
5453nn0red 10629 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
55 letr 9460 . . . . . . . . . 10  |-  ( ( k  e.  RR  /\  M  e.  RR  /\  if ( M  <_  N ,  N ,  M )  e.  RR )  ->  (
( k  <_  M  /\  M  <_  if ( M  <_  N ,  N ,  M )
)  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
5652, 46, 54, 55syl3anc 1218 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
k  <_  M  /\  M  <_  if ( M  <_  N ,  N ,  M ) )  -> 
k  <_  if ( M  <_  N ,  N ,  M ) ) )
5750, 56mpan2d 674 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( k  <_  M  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
5844, 57syld 44 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
5917, 2dgrub2 21678 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  ( B " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
6059adantl 466 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
61 plyco0 21635 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  B : NN0 --> CC )  ->  ( ( B
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( B `
 k )  =/=  0  ->  k  <_  N ) ) )
625, 19, 61syl2anc 661 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( B " ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( B `
 k )  =/=  0  ->  k  <_  N ) ) )
6360, 62mpbid 210 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( B `
 k )  =/=  0  ->  k  <_  N ) )
6463r19.21bi 2809 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( B `  k )  =/=  0  ->  k  <_  N ) )
65 max2 11151 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
6646, 48, 65syl2anc 661 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
67 letr 9460 . . . . . . . . . 10  |-  ( ( k  e.  RR  /\  N  e.  RR  /\  if ( M  <_  N ,  N ,  M )  e.  RR )  ->  (
( k  <_  N  /\  N  <_  if ( M  <_  N ,  N ,  M )
)  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
6852, 48, 54, 67syl3anc 1218 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
k  <_  N  /\  N  <_  if ( M  <_  N ,  N ,  M ) )  -> 
k  <_  if ( M  <_  N ,  N ,  M ) ) )
6966, 68mpan2d 674 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( k  <_  N  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
7064, 69syld 44 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( B `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
7158, 70jaod 380 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A `  k
)  =/=  0  \/  ( B `  k
)  =/=  0 )  ->  k  <_  if ( M  <_  N ,  N ,  M )
) )
7238, 71syld 44 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  oF  +  B ) `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
7372ralrimiva 2794 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( ( A  oF  +  B ) `  k
)  =/=  0  -> 
k  <_  if ( M  <_  N ,  N ,  M ) ) )
74 plyco0 21635 . . . . 5  |-  ( ( if ( M  <_  N ,  N ,  M )  e.  NN0  /\  ( A  oF  +  B ) : NN0 --> CC )  -> 
( ( ( A  oF  +  B
) " ( ZZ>= `  ( if ( M  <_  N ,  N ,  M )  +  1 ) ) )  =  { 0 }  <->  A. k  e.  NN0  ( ( ( A  oF  +  B ) `  k
)  =/=  0  -> 
k  <_  if ( M  <_  N ,  N ,  M ) ) ) )
7511, 23, 74syl2anc 661 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
( A  oF  +  B ) "
( ZZ>= `  ( if ( M  <_  N ,  N ,  M )  +  1 ) ) )  =  { 0 }  <->  A. k  e.  NN0  ( ( ( A  oF  +  B
) `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) ) )
7673, 75mpbird 232 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A  oF  +  B
) " ( ZZ>= `  ( if ( M  <_  N ,  N ,  M )  +  1 ) ) )  =  { 0 } )
77 simpl 457 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
78 simpr 461 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
7914, 6coeid 21681 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
8079adantr 465 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
8117, 2coeid 21681 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( B `  k
)  x.  ( z ^ k ) ) ) )
8281adantl 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( B `  k
)  x.  ( z ^ k ) ) ) )
8377, 78, 9, 5, 16, 19, 40, 60, 80, 82plyaddlem1 21656 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  +  G
)  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... if ( M  <_  N ,  N ,  M )
) ( ( ( A  oF  +  B ) `  k
)  x.  ( z ^ k ) ) ) )
841, 11, 23, 76, 83coeeq 21670 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  oF  +  G
) )  =  ( A  oF  +  B ) )
85 elfznn0 11473 . . . 4  |-  ( k  e.  ( 0 ...
if ( M  <_  N ,  N ,  M ) )  -> 
k  e.  NN0 )
86 ffvelrn 5836 . . . 4  |-  ( ( ( A  oF  +  B ) : NN0 --> CC  /\  k  e.  NN0 )  ->  (
( A  oF  +  B ) `  k )  e.  CC )
8723, 85, 86syl2an 477 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( 0 ... if ( M  <_  N ,  N ,  M )
) )  ->  (
( A  oF  +  B ) `  k )  e.  CC )
881, 11, 87, 83dgrle 21686 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  oF  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
8984, 88jca 532 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  oF  +  G ) )  =  ( A  oF  +  B )  /\  (deg `  ( F  oF  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   _Vcvv 2967   ifcif 3786   {csn 3872   class class class wbr 4287    e. cmpt 4345   "cima 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086    oFcof 6313   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    <_ cle 9411   NN0cn0 10571   ZZ>=cuz 10853   ...cfz 11429   ^cexp 11857   sum_csu 13155  Polycply 21627  coeffccoe 21629  degcdgr 21630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-0p 21123  df-ply 21631  df-coe 21633  df-dgr 21634
This theorem is referenced by:  coeadd  21693  dgradd  21709
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