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Theorem coemulhi 23287
Description: The leading coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
coefv0.1  |-  A  =  (coeff `  F )
coeadd.2  |-  B  =  (coeff `  G )
coemulhi.3  |-  M  =  (deg `  F )
coemulhi.4  |-  N  =  (deg `  G )
Assertion
Ref Expression
coemulhi  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  oF  x.  G ) ) `
 ( M  +  N ) )  =  ( ( A `  M )  x.  ( B `  N )
) )

Proof of Theorem coemulhi
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 coemulhi.3 . . . . 5  |-  M  =  (deg `  F )
2 dgrcl 23266 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
31, 2syl5eqel 2553 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
4 coemulhi.4 . . . . 5  |-  N  =  (deg `  G )
5 dgrcl 23266 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
64, 5syl5eqel 2553 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
7 nn0addcl 10929 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  +  N
)  e.  NN0 )
83, 6, 7syl2an 485 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  +  N )  e.  NN0 )
9 coefv0.1 . . . 4  |-  A  =  (coeff `  F )
10 coeadd.2 . . . 4  |-  B  =  (coeff `  G )
119, 10coemul 23285 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  ( M  +  N
)  e.  NN0 )  ->  ( (coeff `  ( F  oF  x.  G
) ) `  ( M  +  N )
)  =  sum_ k  e.  ( 0 ... ( M  +  N )
) ( ( A `
 k )  x.  ( B `  (
( M  +  N
)  -  k ) ) ) )
128, 11mpd3an3 1391 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  oF  x.  G ) ) `
 ( M  +  N ) )  = 
sum_ k  e.  ( 0 ... ( M  +  N ) ) ( ( A `  k )  x.  ( B `  ( ( M  +  N )  -  k ) ) ) )
136adantl 473 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  NN0 )
1413nn0ge0d 10952 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  0  <_  N )
153adantr 472 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  NN0 )
1615nn0red 10950 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  RR )
1713nn0red 10950 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  RR )
1816, 17addge01d 10222 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( 0  <_  N  <->  M  <_  ( M  +  N ) ) )
1914, 18mpbid 215 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  <_  ( M  +  N ) )
20 nn0uz 11217 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2115, 20syl6eleq 2559 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  ( ZZ>= `  0 )
)
228nn0zd 11061 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  +  N )  e.  ZZ )
23 elfz5 11818 . . . . . 6  |-  ( ( M  e.  ( ZZ>= ` 
0 )  /\  ( M  +  N )  e.  ZZ )  ->  ( M  e.  ( 0 ... ( M  +  N ) )  <->  M  <_  ( M  +  N ) ) )
2421, 22, 23syl2anc 673 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  e.  ( 0 ... ( M  +  N )
)  <->  M  <_  ( M  +  N ) ) )
2519, 24mpbird 240 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  ( 0 ... ( M  +  N )
) )
2625snssd 4108 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  { M }  C_  ( 0 ... ( M  +  N
) ) )
27 elsni 3985 . . . . . 6  |-  ( k  e.  { M }  ->  k  =  M )
2827adantl 473 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
k  =  M )
29 fveq2 5879 . . . . . 6  |-  ( k  =  M  ->  ( A `  k )  =  ( A `  M ) )
30 oveq2 6316 . . . . . . 7  |-  ( k  =  M  ->  (
( M  +  N
)  -  k )  =  ( ( M  +  N )  -  M ) )
3130fveq2d 5883 . . . . . 6  |-  ( k  =  M  ->  ( B `  ( ( M  +  N )  -  k ) )  =  ( B `  ( ( M  +  N )  -  M
) ) )
3229, 31oveq12d 6326 . . . . 5  |-  ( k  =  M  ->  (
( A `  k
)  x.  ( B `
 ( ( M  +  N )  -  k ) ) )  =  ( ( A `
 M )  x.  ( B `  (
( M  +  N
)  -  M ) ) ) )
3328, 32syl 17 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
( ( A `  k )  x.  ( B `  ( ( M  +  N )  -  k ) ) )  =  ( ( A `  M )  x.  ( B `  ( ( M  +  N )  -  M
) ) ) )
3416recnd 9687 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  CC )
3517recnd 9687 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  CC )
3634, 35pncan2d 10007 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( M  +  N )  -  M )  =  N )
3736fveq2d 5883 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B `  ( ( M  +  N )  -  M
) )  =  ( B `  N ) )
3837oveq2d 6324 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  (
( M  +  N
)  -  M ) ) )  =  ( ( A `  M
)  x.  ( B `
 N ) ) )
399coef3 23265 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
4039adantr 472 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
4140, 15ffvelrnd 6038 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A `  M )  e.  CC )
4210coef3 23265 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
4342adantl 473 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
4443, 13ffvelrnd 6038 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B `  N )  e.  CC )
4541, 44mulcld 9681 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  N
) )  e.  CC )
4638, 45eqeltrd 2549 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  (
( M  +  N
)  -  M ) ) )  e.  CC )
4746adantr 472 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
( ( A `  M )  x.  ( B `  ( ( M  +  N )  -  M ) ) )  e.  CC )
4833, 47eqeltrd 2549 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
( ( A `  k )  x.  ( B `  ( ( M  +  N )  -  k ) ) )  e.  CC )
49 simpl 464 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
50 eldifi 3544 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  e.  ( 0 ... ( M  +  N ) ) )
51 elfznn0 11913 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... ( M  +  N
) )  ->  k  e.  NN0 )
5250, 51syl 17 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  e.  NN0 )
539, 1dgrub 23267 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  M )
54533expia 1233 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  M )
)
5549, 52, 54syl2an 485 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( ( A `  k )  =/=  0  ->  k  <_  M )
)
5655necon1bd 2661 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( -.  k  <_  M  ->  ( A `  k )  =  0 ) )
5756imp 436 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  ( A `  k )  =  0 )
5857oveq1d 6323 . . . . 5  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  (
( A `  k
)  x.  ( B `
 ( ( M  +  N )  -  k ) ) )  =  ( 0  x.  ( B `  (
( M  +  N
)  -  k ) ) ) )
5943ad2antrr 740 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  B : NN0 --> CC )
6050ad2antlr 741 . . . . . . . 8  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  k  e.  ( 0 ... ( M  +  N )
) )
61 fznn0sub 11857 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( M  +  N
) )  ->  (
( M  +  N
)  -  k )  e.  NN0 )
6260, 61syl 17 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  (
( M  +  N
)  -  k )  e.  NN0 )
6359, 62ffvelrnd 6038 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  ( B `  ( ( M  +  N )  -  k ) )  e.  CC )
6463mul02d 9849 . . . . 5  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  (
0  x.  ( B `
 ( ( M  +  N )  -  k ) ) )  =  0 )
6558, 64eqtrd 2505 . . . 4  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  (
( A `  k
)  x.  ( B `
 ( ( M  +  N )  -  k ) ) )  =  0 )
6616adantr 472 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  ->  M  e.  RR )
6750adantl 473 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  e.  ( 0 ... ( M  +  N ) ) )
6867, 51syl 17 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  e.  NN0 )
6968nn0red 10950 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  e.  RR )
7017adantr 472 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  ->  N  e.  RR )
7166, 69, 70leadd1d 10228 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  <_  k  <->  ( M  +  N )  <_  ( k  +  N ) ) )
728adantr 472 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  +  N
)  e.  NN0 )
7372nn0red 10950 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  +  N
)  e.  RR )
7473, 69, 70lesubadd2d 10233 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( ( ( M  +  N )  -  k )  <_  N  <->  ( M  +  N )  <_  ( k  +  N ) ) )
7571, 74bitr4d 264 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  <_  k  <->  ( ( M  +  N
)  -  k )  <_  N ) )
7675notbid 301 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( -.  M  <_ 
k  <->  -.  ( ( M  +  N )  -  k )  <_  N ) )
7776biimpa 492 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  -.  ( ( M  +  N )  -  k
)  <_  N )
78 simpr 468 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
7950, 61syl 17 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
( ( M  +  N )  -  k
)  e.  NN0 )
8010, 4dgrub 23267 . . . . . . . . . . 11  |-  ( ( G  e.  (Poly `  S )  /\  (
( M  +  N
)  -  k )  e.  NN0  /\  ( B `  ( ( M  +  N )  -  k ) )  =/=  0 )  -> 
( ( M  +  N )  -  k
)  <_  N )
81803expia 1233 . . . . . . . . . 10  |-  ( ( G  e.  (Poly `  S )  /\  (
( M  +  N
)  -  k )  e.  NN0 )  -> 
( ( B `  ( ( M  +  N )  -  k
) )  =/=  0  ->  ( ( M  +  N )  -  k
)  <_  N )
)
8278, 79, 81syl2an 485 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( ( B `  ( ( M  +  N )  -  k
) )  =/=  0  ->  ( ( M  +  N )  -  k
)  <_  N )
)
8382necon1bd 2661 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( -.  ( ( M  +  N )  -  k )  <_  N  ->  ( B `  ( ( M  +  N )  -  k
) )  =  0 ) )
8483imp 436 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  ( ( M  +  N )  -  k )  <_  N )  ->  ( B `  ( ( M  +  N )  -  k ) )  =  0 )
8577, 84syldan 478 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  ( B `  ( ( M  +  N )  -  k ) )  =  0 )
8685oveq2d 6324 . . . . 5  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  (
( A `  k
)  x.  ( B `
 ( ( M  +  N )  -  k ) ) )  =  ( ( A `
 k )  x.  0 ) )
8740ad2antrr 740 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  A : NN0 --> CC )
8852ad2antlr 741 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  k  e.  NN0 )
8987, 88ffvelrnd 6038 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  ( A `  k )  e.  CC )
9089mul01d 9850 . . . . 5  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  (
( A `  k
)  x.  0 )  =  0 )
9186, 90eqtrd 2505 . . . 4  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  (
( A `  k
)  x.  ( B `
 ( ( M  +  N )  -  k ) ) )  =  0 )
92 eldifsni 4089 . . . . . . 7  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  =/=  M )
9392adantl 473 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  =/=  M )
9469, 66letri3d 9794 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( k  =  M  <-> 
( k  <_  M  /\  M  <_  k ) ) )
9594necon3abid 2679 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( k  =/=  M  <->  -.  ( k  <_  M  /\  M  <_  k ) ) )
9693, 95mpbid 215 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  ->  -.  ( k  <_  M  /\  M  <_  k ) )
97 ianor 496 . . . . 5  |-  ( -.  ( k  <_  M  /\  M  <_  k )  <-> 
( -.  k  <_  M  \/  -.  M  <_  k ) )
9896, 97sylib 201 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( -.  k  <_  M  \/  -.  M  <_  k ) )
9965, 91, 98mpjaodan 803 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( ( A `  k )  x.  ( B `  ( ( M  +  N )  -  k ) ) )  =  0 )
100 fzfid 12224 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( 0 ... ( M  +  N ) )  e. 
Fin )
10126, 48, 99, 100fsumss 13868 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  sum_ k  e. 
{ M }  (
( A `  k
)  x.  ( B `
 ( ( M  +  N )  -  k ) ) )  =  sum_ k  e.  ( 0 ... ( M  +  N ) ) ( ( A `  k )  x.  ( B `  ( ( M  +  N )  -  k ) ) ) )
10232sumsn 13884 . . . 4  |-  ( ( M  e.  NN0  /\  ( ( A `  M )  x.  ( B `  ( ( M  +  N )  -  M ) ) )  e.  CC )  ->  sum_ k  e.  { M }  ( ( A `
 k )  x.  ( B `  (
( M  +  N
)  -  k ) ) )  =  ( ( A `  M
)  x.  ( B `
 ( ( M  +  N )  -  M ) ) ) )
10315, 46, 102syl2anc 673 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  sum_ k  e. 
{ M }  (
( A `  k
)  x.  ( B `
 ( ( M  +  N )  -  k ) ) )  =  ( ( A `
 M )  x.  ( B `  (
( M  +  N
)  -  M ) ) ) )
104103, 38eqtrd 2505 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  sum_ k  e. 
{ M }  (
( A `  k
)  x.  ( B `
 ( ( M  +  N )  -  k ) ) )  =  ( ( A `
 M )  x.  ( B `  N
) ) )
10512, 101, 1043eqtr2d 2511 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  oF  x.  G ) ) `
 ( M  +  N ) )  =  ( ( A `  M )  x.  ( B `  N )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641    \ cdif 3387   {csn 3959   class class class wbr 4395   -->wf 5585   ` cfv 5589  (class class class)co 6308    oFcof 6548   CCcc 9555   RRcr 9556   0cc0 9557    + caddc 9560    x. cmul 9562    <_ cle 9694    - cmin 9880   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   sum_csu 13829  Polycply 23217  coeffccoe 23219  degcdgr 23220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-coe 23223  df-dgr 23224
This theorem is referenced by:  dgrmul  23303  plymul0or  23313  plydivlem4  23328  vieta1lem2  23343
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