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Theorem coeaddlem 23140
Description: Lemma for coeadd 23142 and dgradd 23158. (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 23111 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  +  G
)  e.  (Poly `  CC ) )
2 coeadd.4 . . . . . 6  |-  N  =  (deg `  G )
3 dgrcl 23124 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
42, 3syl5eqel 2505 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
54adantl 467 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  NN0 )
6 coeadd.3 . . . . . 6  |-  M  =  (deg `  F )
7 dgrcl 23124 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
86, 7syl5eqel 2505 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
98adantr 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  NN0 )
105, 9ifcld 3892 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
11 addcl 9567 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
1211adantl 467 . . . 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 23123 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
1514adantr 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
16 coeadd.2 . . . . . 6  |-  B  =  (coeff `  G )
1716coef3 23123 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
1817adantl 467 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
19 nn0ex 10821 . . . . 5  |-  NN0  e.  _V
2019a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  NN0  e.  _V )
21 inidm 3609 . . . 4  |-  ( NN0 
i^i  NN0 )  =  NN0
2212, 15, 18, 20, 20, 21off 6499 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  oF  +  B
) : NN0 --> CC )
23 oveq12 6253 . . . . . . . . . 10  |-  ( ( ( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A `
 k )  +  ( B `  k
) )  =  ( 0  +  0 ) )
24 00id 9754 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
2523, 24syl6eq 2473 . . . . . . . . 9  |-  ( ( ( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A `
 k )  +  ( B `  k
) )  =  0 )
26 ffn 5684 . . . . . . . . . . . 12  |-  ( A : NN0 --> CC  ->  A  Fn  NN0 )
2715, 26syl 17 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A  Fn  NN0 )
28 ffn 5684 . . . . . . . . . . . 12  |-  ( B : NN0 --> CC  ->  B  Fn  NN0 )
2918, 28syl 17 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B  Fn  NN0 )
30 eqidd 2424 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( A `  k )  =  ( A `  k ) )
31 eqidd 2424 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( B `  k )  =  ( B `  k ) )
3227, 29, 20, 20, 21, 30, 31ofval 6493 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A  oF  +  B
) `  k )  =  ( ( A `
 k )  +  ( B `  k
) ) )
3332eqeq1d 2425 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  oF  +  B ) `  k )  =  0  <-> 
( ( A `  k )  +  ( B `  k ) )  =  0 ) )
3425, 33syl5ibr 224 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A  oF  +  B
) `  k )  =  0 ) )
3534necon3ad 2609 . . . . . . 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 2690 . . . . . . 7  |-  ( ( ( A `  k
)  =/=  0  \/  ( B `  k
)  =/=  0 )  <->  -.  ( ( A `  k )  =  0  /\  ( B `  k )  =  0 ) )
3735, 36syl6ibr 230 . . . . . 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 23126 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  ( A " ( ZZ>= `  ( M  +  1 ) ) )  =  { 0 } )
3938adantr 466 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A " ( ZZ>= `  ( M  +  1 ) ) )  =  { 0 } )
40 plyco0 23083 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  M ) ) )
419, 15, 40syl2anc 665 . . . . . . . . . 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 213 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  M ) )
4342r19.21bi 2729 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A `  k )  =/=  0  ->  k  <_  M ) )
449adantr 466 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  e.  NN0 )
4544nn0red 10872 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  e.  RR )
465adantr 466 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  e.  NN0 )
4746nn0red 10872 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  e.  RR )
48 max1 11426 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
4945, 47, 48syl2anc 665 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
50 nn0re 10824 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  k  e.  RR )
5150adantl 467 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  k  e.  RR )
5210adantr 466 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
5352nn0red 10872 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
54 letr 9673 . . . . . . . . . 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 1264 . . . . . . . . 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 678 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( k  <_  M  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
5743, 56syld 45 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
5816, 2dgrub2 23126 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  ( B " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
5958adantl 467 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
60 plyco0 23083 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  B : NN0 --> CC )  ->  ( ( B
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( B `
 k )  =/=  0  ->  k  <_  N ) ) )
615, 18, 60syl2anc 665 . . . . . . . . . 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 213 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( B `
 k )  =/=  0  ->  k  <_  N ) )
6362r19.21bi 2729 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( B `  k )  =/=  0  ->  k  <_  N ) )
64 max2 11428 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
6545, 47, 64syl2anc 665 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
66 letr 9673 . . . . . . . . . 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 1264 . . . . . . . . 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 678 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( k  <_  N  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
6963, 68syld 45 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( B `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
7057, 69jaod 381 . . . . . 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 45 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  oF  +  B ) `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
7271ralrimiva 2774 . . . 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 23083 . . . . 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 665 . . . 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 235 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A  oF  +  B
) " ( ZZ>= `  ( if ( M  <_  N ,  N ,  M )  +  1 ) ) )  =  { 0 } )
76 simpl 458 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
77 simpr 462 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
7813, 6coeid 23129 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
7978adantr 466 . . . 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 23129 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( B `  k
)  x.  ( z ^ k ) ) ) )
8180adantl 467 . . . 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 23104 . . 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 23118 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  oF  +  G
) )  =  ( A  oF  +  B ) )
84 elfznn0 11833 . . . 4  |-  ( k  e.  ( 0 ...
if ( M  <_  N ,  N ,  M ) )  -> 
k  e.  NN0 )
85 ffvelrn 5974 . . . 4  |-  ( ( ( A  oF  +  B ) : NN0 --> CC  /\  k  e.  NN0 )  ->  (
( A  oF  +  B ) `  k )  e.  CC )
8622, 84, 85syl2an 479 . . 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 23134 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  oF  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
8883, 87jca 534 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2594   A.wral 2709   _Vcvv 3017   ifcif 3849   {csn 3936   class class class wbr 4361    |-> cmpt 4420   "cima 4794    Fn wfn 5534   -->wf 5535   ` cfv 5539  (class class class)co 6244    oFcof 6482   CCcc 9483   RRcr 9484   0cc0 9485   1c1 9486    + caddc 9488    x. cmul 9490    <_ cle 9622   NN0cn0 10815   ZZ>=cuz 11105   ...cfz 11730   ^cexp 12217   sum_csu 13690  Polycply 23075  coeffccoe 23077  degcdgr 23078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-inf2 8094  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563  ax-addf 9564
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-se 4751  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-isom 5548  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-of 6484  df-om 6646  df-1st 6746  df-2nd 6747  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7520  df-dom 7521  df-sdom 7522  df-fin 7523  df-sup 7904  df-inf 7905  df-oi 7973  df-card 8320  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-n0 10816  df-z 10884  df-uz 11106  df-rp 11249  df-fz 11731  df-fzo 11862  df-fl 11973  df-seq 12159  df-exp 12218  df-hash 12461  df-cj 13101  df-re 13102  df-im 13103  df-sqrt 13237  df-abs 13238  df-clim 13490  df-rlim 13491  df-sum 13691  df-0p 22565  df-ply 23079  df-coe 23081  df-dgr 23082
This theorem is referenced by:  coeadd  23142  dgradd  23158
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