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Theorem coeaddlem 20167
Description: Lemma for coeadd 20169 and dgradd 20185. (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 20139 . . 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 20152 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
42, 3syl5eqel 2520 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
54adantl 453 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  NN0 )
6 coeadd.3 . . . . . 6  |-  M  =  (deg `  F )
7 dgrcl 20152 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
86, 7syl5eqel 2520 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
98adantr 452 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  NN0 )
10 ifcl 3775 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
115, 9, 10syl2anc 643 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
12 addcl 9072 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
1312adantl 453 . . . 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 20151 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
1615adantr 452 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
17 coeadd.2 . . . . . 6  |-  B  =  (coeff `  G )
1817coef3 20151 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
1918adantl 453 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
20 nn0ex 10227 . . . . 5  |-  NN0  e.  _V
2120a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  NN0  e.  _V )
22 inidm 3550 . . . 4  |-  ( NN0 
i^i  NN0 )  =  NN0
2313, 16, 19, 21, 21, 22off 6320 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  o F  +  B
) : NN0 --> CC )
24 oveq12 6090 . . . . . . . . . 10  |-  ( ( ( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A `
 k )  +  ( B `  k
) )  =  ( 0  +  0 ) )
25 00id 9241 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
2624, 25syl6eq 2484 . . . . . . . . 9  |-  ( ( ( A `  k
)  =  0  /\  ( B `  k
)  =  0 )  ->  ( ( A `
 k )  +  ( B `  k
) )  =  0 )
27 ffn 5591 . . . . . . . . . . . 12  |-  ( A : NN0 --> CC  ->  A  Fn  NN0 )
2816, 27syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A  Fn  NN0 )
29 ffn 5591 . . . . . . . . . . . 12  |-  ( B : NN0 --> CC  ->  B  Fn  NN0 )
3019, 29syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B  Fn  NN0 )
31 eqidd 2437 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( A `  k )  =  ( A `  k ) )
32 eqidd 2437 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( B `  k )  =  ( B `  k ) )
3328, 30, 21, 21, 22, 31, 32ofval 6314 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A  o F  +  B
) `  k )  =  ( ( A `
 k )  +  ( B `  k
) ) )
3433eqeq1d 2444 . . . . . . . . 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 213 . . . . . . . 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 2637 . . . . . . 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 2691 . . . . . . 7  |-  ( ( ( A `  k
)  =/=  0  \/  ( B `  k
)  =/=  0 )  <->  -.  ( ( A `  k )  =  0  /\  ( B `  k )  =  0 ) )
3836, 37syl6ibr 219 . . . . . 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 20154 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  ( A " ( ZZ>= `  ( M  +  1 ) ) )  =  { 0 } )
4039adantr 452 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A " ( ZZ>= `  ( M  +  1 ) ) )  =  { 0 } )
41 plyco0 20111 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  M ) ) )
429, 16, 41syl2anc 643 . . . . . . . . . 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 202 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  M ) )
4443r19.21bi 2804 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A `  k )  =/=  0  ->  k  <_  M ) )
459adantr 452 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  e.  NN0 )
4645nn0red 10275 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  e.  RR )
475adantr 452 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  e.  NN0 )
4847nn0red 10275 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  e.  RR )
49 max1 10773 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
5046, 48, 49syl2anc 643 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
51 nn0re 10230 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  k  e.  RR )
5251adantl 453 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  k  e.  RR )
5311adantr 452 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
5453nn0red 10275 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
55 letr 9167 . . . . . . . . . 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 1184 . . . . . . . . 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 656 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( k  <_  M  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
5844, 57syld 42 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( A `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
5917, 2dgrub2 20154 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  ( B " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
6059adantl 453 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
61 plyco0 20111 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  B : NN0 --> CC )  ->  ( ( B
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( B `
 k )  =/=  0  ->  k  <_  N ) ) )
625, 19, 61syl2anc 643 . . . . . . . . . 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 202 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A. k  e.  NN0  ( ( B `
 k )  =/=  0  ->  k  <_  N ) )
6463r19.21bi 2804 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( B `  k )  =/=  0  ->  k  <_  N ) )
65 max2 10775 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
6646, 48, 65syl2anc 643 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
67 letr 9167 . . . . . . . . . 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 1184 . . . . . . . . 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 656 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( k  <_  N  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
7064, 69syld 42 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  NN0 )  ->  ( ( B `  k )  =/=  0  ->  k  <_  if ( M  <_  N ,  N ,  M ) ) )
7158, 70jaod 370 . . . . . 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 42 . . . . 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 2789 . . . 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 20111 . . . . 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 643 . . . 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 224 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A  o F  +  B
) " ( ZZ>= `  ( if ( M  <_  N ,  N ,  M )  +  1 ) ) )  =  { 0 } )
77 simpl 444 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
78 simpr 448 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
7914, 6coeid 20157 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
8079adantr 452 . . . 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 20157 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( B `  k
)  x.  ( z ^ k ) ) ) )
8281adantl 453 . . . 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 20132 . . 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 20146 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  o F  +  G
) )  =  ( A  o F  +  B ) )
85 elfznn0 11083 . . . 4  |-  ( k  e.  ( 0 ...
if ( M  <_  N ,  N ,  M ) )  -> 
k  e.  NN0 )
86 ffvelrn 5868 . . . 4  |-  ( ( ( A  o F  +  B ) : NN0 --> CC  /\  k  e.  NN0 )  ->  (
( A  o F  +  B ) `  k )  e.  CC )
8723, 85, 86syl2an 464 . . 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 20162 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  o F  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
8984, 88jca 519 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 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   _Vcvv 2956   ifcif 3739   {csn 3814   class class class wbr 4212    e. cmpt 4266   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    <_ cle 9121   NN0cn0 10221   ZZ>=cuz 10488   ...cfz 11043   ^cexp 11382   sum_csu 12479  Polycply 20103  coeffccoe 20105  degcdgr 20106
This theorem is referenced by:  coeadd  20169  dgradd  20185
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107  df-coe 20109  df-dgr 20110
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