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Theorem coeaddlem 22812
Description: Lemma for coeadd 22814 and dgradd 22830. (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 22783 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  +  G
)  e.  (Poly `  CC ) )
2 coeadd.4 . . . . . 6  |-  N  =  (deg `  G )
3 dgrcl 22796 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
42, 3syl5eqel 2546 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
54adantl 464 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  NN0 )
6 coeadd.3 . . . . . 6  |-  M  =  (deg `  F )
7 dgrcl 22796 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
86, 7syl5eqel 2546 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
98adantr 463 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  NN0 )
105, 9ifcld 3972 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
11 addcl 9563 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
1211adantl 464 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
13 coefv0.1 . . . . . 6  |-  A  =  (coeff `  F )
1413coef3 22795 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
1514adantr 463 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
16 coeadd.2 . . . . . 6  |-  B  =  (coeff `  G )
1716coef3 22795 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
1817adantl 464 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
19 nn0ex 10797 . . . . 5  |-  NN0  e.  _V
2019a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  NN0  e.  _V )
21 inidm 3693 . . . 4  |-  ( NN0 
i^i  NN0 )  =  NN0
2212, 15, 18, 20, 20, 21off 6527 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  oF  +  B
) : NN0 --> CC )
23 oveq12 6279 . . . . . . . . . 10  |-  ( ( ( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A `
 k )  +  ( B `  k
) )  =  ( 0  +  0 ) )
24 00id 9744 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
2523, 24syl6eq 2511 . . . . . . . . 9  |-  ( ( ( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A `
 k )  +  ( B `  k
) )  =  0 )
26 ffn 5713 . . . . . . . . . . . 12  |-  ( A : NN0 --> CC  ->  A  Fn  NN0 )
2715, 26syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A  Fn  NN0 )
28 ffn 5713 . . . . . . . . . . . 12  |-  ( B : NN0 --> CC  ->  B  Fn  NN0 )
2918, 28syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B  Fn  NN0 )
30 eqidd 2455 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( A `  k )  =  ( A `  k ) )
31 eqidd 2455 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( B `  k )  =  ( B `  k ) )
3227, 29, 20, 20, 21, 30, 31ofval 6522 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A  oF  +  B
) `  k )  =  ( ( A `
 k )  +  ( B `  k
) ) )
3332eqeq1d 2456 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  oF  +  B ) `  k )  =  0  <-> 
( ( A `  k )  +  ( B `  k ) )  =  0 ) )
3425, 33syl5ibr 221 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A  oF  +  B
) `  k )  =  0 ) )
3534necon3ad 2664 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  oF  +  B ) `  k )  =/=  0  ->  -.  ( ( A `
 k )  =  0  /\  ( B `
 k )  =  0 ) ) )
36 neorian 2781 . . . . . . 7  |-  ( ( ( A `  k
)  =/=  0  \/  ( B `  k
)  =/=  0 )  <->  -.  ( ( A `  k )  =  0  /\  ( B `  k )  =  0 ) )
3735, 36syl6ibr 227 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  oF  +  B ) `  k )  =/=  0  ->  ( ( A `  k )  =/=  0  \/  ( B `  k
)  =/=  0 ) ) )
3813, 6dgrub2 22798 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  ( A " ( ZZ>= `  ( M  +  1 ) ) )  =  { 0 } )
3938adantr 463 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A " ( ZZ>= `  ( M  +  1 ) ) )  =  { 0 } )
40 plyco0 22755 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  M ) ) )
419, 15, 40syl2anc 659 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A " ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  M ) ) )
4239, 41mpbid 210 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  M ) )
4342r19.21bi 2823 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A `  k )  =/=  0  ->  k  <_  M ) )
449adantr 463 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  e.  NN0 )
4544nn0red 10849 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  e.  RR )
465adantr 463 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  e.  NN0 )
4746nn0red 10849 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  e.  RR )
48 max1 11389 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
4945, 47, 48syl2anc 659 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
50 nn0re 10800 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  k  e.  RR )
5150adantl 464 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  k  e.  RR )
5210adantr 463 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
5352nn0red 10849 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
54 letr 9667 . . . . . . . . . 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 ) ) )
5551, 45, 53, 54syl3anc 1226 . . . . . . . . 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 ) ) )
5649, 55mpan2d 672 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( k  <_  M  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
5743, 56syld 44 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
5816, 2dgrub2 22798 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  ( B " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
5958adantl 464 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
60 plyco0 22755 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  B : NN0 --> CC )  ->  ( ( B
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( B `
 k )  =/=  0  ->  k  <_  N ) ) )
615, 18, 60syl2anc 659 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( B " ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( B `
 k )  =/=  0  ->  k  <_  N ) ) )
6259, 61mpbid 210 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( B `
 k )  =/=  0  ->  k  <_  N ) )
6362r19.21bi 2823 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( B `  k )  =/=  0  ->  k  <_  N ) )
64 max2 11391 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
6545, 47, 64syl2anc 659 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
66 letr 9667 . . . . . . . . . 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 ) ) )
6751, 47, 53, 66syl3anc 1226 . . . . . . . . 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 ) ) )
6865, 67mpan2d 672 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( k  <_  N  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
6963, 68syld 44 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( B `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
7057, 69jaod 378 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A `  k
)  =/=  0  \/  ( B `  k
)  =/=  0 )  ->  k  <_  if ( M  <_  N ,  N ,  M )
) )
7137, 70syld 44 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  oF  +  B ) `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
7271ralrimiva 2868 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( ( A  oF  +  B ) `  k
)  =/=  0  -> 
k  <_  if ( M  <_  N ,  N ,  M ) ) )
73 plyco0 22755 . . . . 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 ) ) ) )
7410, 22, 73syl2anc 659 . . . 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 ) ) ) )
7572, 74mpbird 232 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A  oF  +  B
) " ( ZZ>= `  ( if ( M  <_  N ,  N ,  M )  +  1 ) ) )  =  { 0 } )
76 simpl 455 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
77 simpr 459 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
7813, 6coeid 22801 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
7978adantr 463 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
8016, 2coeid 22801 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( B `  k
)  x.  ( z ^ k ) ) ) )
8180adantl 464 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( B `  k
)  x.  ( z ^ k ) ) ) )
8276, 77, 9, 5, 15, 18, 39, 59, 79, 81plyaddlem1 22776 . . 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 ) ) ) )
831, 10, 22, 75, 82coeeq 22790 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  oF  +  G
) )  =  ( A  oF  +  B ) )
84 elfznn0 11775 . . . 4  |-  ( k  e.  ( 0 ...
if ( M  <_  N ,  N ,  M ) )  -> 
k  e.  NN0 )
85 ffvelrn 6005 . . . 4  |-  ( ( ( A  oF  +  B ) : NN0 --> CC  /\  k  e.  NN0 )  ->  (
( A  oF  +  B ) `  k )  e.  CC )
8622, 84, 85syl2an 475 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( 0 ... if ( M  <_  N ,  N ,  M )
) )  ->  (
( A  oF  +  B ) `  k )  e.  CC )
871, 10, 86, 82dgrle 22806 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  oF  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
8883, 87jca 530 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 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106   ifcif 3929   {csn 4016   class class class wbr 4439    |-> cmpt 4497   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    oFcof 6511   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    <_ cle 9618   NN0cn0 10791   ZZ>=cuz 11082   ...cfz 11675   ^cexp 12148   sum_csu 13590  Polycply 22747  coeffccoe 22749  degcdgr 22750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-rlim 13394  df-sum 13591  df-0p 22243  df-ply 22751  df-coe 22753  df-dgr 22754
This theorem is referenced by:  coeadd  22814  dgradd  22830
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