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Theorem coeaddlem 19630
Description: Lemma for coeadd 19632 and dgradd 19648. (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  o F  +  G ) )  =  ( A  o F  +  B )  /\  (deg `  ( F  o F  +  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 19602 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  +  G
)  e.  (Poly `  CC ) )
2 coeadd.4 . . . . . 6  |-  N  =  (deg `  G )
3 dgrcl 19615 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
42, 3syl5eqel 2367 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
54adantl 452 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  NN0 )
6 coeadd.3 . . . . . 6  |-  M  =  (deg `  F )
7 dgrcl 19615 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
86, 7syl5eqel 2367 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
98adantr 451 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  NN0 )
10 ifcl 3601 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
115, 9, 10syl2anc 642 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
12 addcl 8819 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
1312adantl 452 . . . 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 19614 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
1615adantr 451 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
17 coeadd.2 . . . . . 6  |-  B  =  (coeff `  G )
1817coef3 19614 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
1918adantl 452 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
20 nn0ex 9971 . . . . 5  |-  NN0  e.  _V
2120a1i 10 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  NN0  e.  _V )
22 inidm 3378 . . . 4  |-  ( NN0 
i^i  NN0 )  =  NN0
2313, 16, 19, 21, 21, 22off 6093 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  o F  +  B
) : NN0 --> CC )
24 oveq12 5867 . . . . . . . . . 10  |-  ( ( ( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A `
 k )  +  ( B `  k
) )  =  ( 0  +  0 ) )
25 00id 8987 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
2624, 25syl6eq 2331 . . . . . . . . 9  |-  ( ( ( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A `
 k )  +  ( B `  k
) )  =  0 )
27 ffn 5389 . . . . . . . . . . . 12  |-  ( A : NN0 --> CC  ->  A  Fn  NN0 )
2816, 27syl 15 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A  Fn  NN0 )
29 ffn 5389 . . . . . . . . . . . 12  |-  ( B : NN0 --> CC  ->  B  Fn  NN0 )
3019, 29syl 15 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B  Fn  NN0 )
31 eqidd 2284 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( A `  k )  =  ( A `  k ) )
32 eqidd 2284 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( B `  k )  =  ( B `  k ) )
3328, 30, 21, 21, 22, 31, 32ofval 6087 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A  o F  +  B
) `  k )  =  ( ( A `
 k )  +  ( B `  k
) ) )
3433eqeq1d 2291 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  o F  +  B ) `  k )  =  0  <-> 
( ( A `  k )  +  ( B `  k ) )  =  0 ) )
3526, 34syl5ibr 212 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A  o F  +  B
) `  k )  =  0 ) )
3635necon3ad 2482 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  o F  +  B ) `  k )  =/=  0  ->  -.  ( ( A `
 k )  =  0  /\  ( B `
 k )  =  0 ) ) )
37 neorian 2533 . . . . . . 7  |-  ( ( ( A `  k
)  =/=  0  \/  ( B `  k
)  =/=  0 )  <->  -.  ( ( A `  k )  =  0  /\  ( B `  k )  =  0 ) )
3836, 37syl6ibr 218 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  o F  +  B ) `  k )  =/=  0  ->  ( ( A `  k )  =/=  0  \/  ( B `  k
)  =/=  0 ) ) )
3914, 6dgrub2 19617 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  ( A " ( ZZ>= `  ( M  +  1 ) ) )  =  { 0 } )
4039adantr 451 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A " ( ZZ>= `  ( M  +  1 ) ) )  =  { 0 } )
41 plyco0 19574 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  M ) ) )
429, 16, 41syl2anc 642 . . . . . . . . . 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 201 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  M ) )
4443r19.21bi 2641 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A `  k )  =/=  0  ->  k  <_  M ) )
459adantr 451 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  e.  NN0 )
4645nn0red 10019 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  e.  RR )
475adantr 451 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  e.  NN0 )
4847nn0red 10019 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  e.  RR )
49 max1 10514 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
5046, 48, 49syl2anc 642 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
51 nn0re 9974 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  k  e.  RR )
5251adantl 452 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  k  e.  RR )
5311adantr 451 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
5453nn0red 10019 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
55 letr 8914 . . . . . . . . . 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 1182 . . . . . . . . 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 655 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( k  <_  M  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
5844, 57syld 40 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
5917, 2dgrub2 19617 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  ( B " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
6059adantl 452 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
61 plyco0 19574 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  B : NN0 --> CC )  ->  ( ( B
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( B `
 k )  =/=  0  ->  k  <_  N ) ) )
625, 19, 61syl2anc 642 . . . . . . . . . 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 201 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( B `
 k )  =/=  0  ->  k  <_  N ) )
6463r19.21bi 2641 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( B `  k )  =/=  0  ->  k  <_  N ) )
65 max2 10516 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
6646, 48, 65syl2anc 642 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
67 letr 8914 . . . . . . . . . 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 1182 . . . . . . . . 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 655 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( k  <_  N  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
7064, 69syld 40 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( B `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
7158, 70jaod 369 . . . . . 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 40 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( (
( A  o F  +  B ) `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
7372ralrimiva 2626 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( ( A  o F  +  B ) `  k
)  =/=  0  -> 
k  <_  if ( M  <_  N ,  N ,  M ) ) )
74 plyco0 19574 . . . . 5  |-  ( ( if ( M  <_  N ,  N ,  M )  e.  NN0  /\  ( A  o F  +  B ) : NN0 --> CC )  -> 
( ( ( A  o F  +  B
) " ( ZZ>= `  ( if ( M  <_  N ,  N ,  M )  +  1 ) ) )  =  { 0 }  <->  A. k  e.  NN0  ( ( ( A  o F  +  B ) `  k
)  =/=  0  -> 
k  <_  if ( M  <_  N ,  N ,  M ) ) ) )
7511, 23, 74syl2anc 642 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
( A  o F  +  B ) "
( ZZ>= `  ( if ( M  <_  N ,  N ,  M )  +  1 ) ) )  =  { 0 }  <->  A. k  e.  NN0  ( ( ( A  o F  +  B
) `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) ) )
7673, 75mpbird 223 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A  o F  +  B
) " ( ZZ>= `  ( if ( M  <_  N ,  N ,  M )  +  1 ) ) )  =  { 0 } )
77 simpl 443 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
78 simpr 447 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
7914, 6coeid 19620 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
8079adantr 451 . . . 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 19620 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( B `  k
)  x.  ( z ^ k ) ) ) )
8281adantl 452 . . . 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 19595 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  +  G
)  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... if ( M  <_  N ,  N ,  M )
) ( ( ( A  o F  +  B ) `  k
)  x.  ( z ^ k ) ) ) )
841, 11, 23, 76, 83coeeq 19609 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  o F  +  G
) )  =  ( A  o F  +  B ) )
85 elfznn0 10822 . . . 4  |-  ( k  e.  ( 0 ...
if ( M  <_  N ,  N ,  M ) )  -> 
k  e.  NN0 )
86 ffvelrn 5663 . . . 4  |-  ( ( ( A  o F  +  B ) : NN0 --> CC  /\  k  e.  NN0 )  ->  (
( A  o F  +  B ) `  k )  e.  CC )
8723, 85, 86syl2an 463 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( 0 ... if ( M  <_  N ,  N ,  M )
) )  ->  (
( A  o F  +  B ) `  k )  e.  CC )
881, 11, 87, 83dgrle 19625 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  o F  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
8984, 88jca 518 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  o F  +  G ) )  =  ( A  o F  +  B )  /\  (deg `  ( F  o F  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788   ifcif 3565   {csn 3640   class class class wbr 4023    e. cmpt 4077   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    <_ cle 8868   NN0cn0 9965   ZZ>=cuz 10230   ...cfz 10782   ^cexp 11104   sum_csu 12158  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  coeadd  19632  dgradd  19648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573
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