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Theorem coemulhi 22413
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 22393 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
31, 2syl5eqel 2559 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
4 coemulhi.4 . . . . 5  |-  N  =  (deg `  G )
5 dgrcl 22393 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
64, 5syl5eqel 2559 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
7 nn0addcl 10831 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  +  N
)  e.  NN0 )
83, 6, 7syl2an 477 . . 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 22411 . . 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 1325 . 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 466 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  NN0 )
1413nn0ge0d 10855 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  0  <_  N )
153adantr 465 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  NN0 )
1615nn0red 10853 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  RR )
1713nn0red 10853 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  RR )
1816, 17addge01d 10140 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( 0  <_  N  <->  M  <_  ( M  +  N ) ) )
1914, 18mpbid 210 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  <_  ( M  +  N ) )
20 nn0uz 11116 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2115, 20syl6eleq 2565 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  ( ZZ>= `  0 )
)
228nn0zd 10964 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  +  N )  e.  ZZ )
23 elfz5 11680 . . . . . 6  |-  ( ( M  e.  ( ZZ>= ` 
0 )  /\  ( M  +  N )  e.  ZZ )  ->  ( M  e.  ( 0 ... ( M  +  N ) )  <->  M  <_  ( M  +  N ) ) )
2421, 22, 23syl2anc 661 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  e.  ( 0 ... ( M  +  N )
)  <->  M  <_  ( M  +  N ) ) )
2519, 24mpbird 232 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  ( 0 ... ( M  +  N )
) )
2625snssd 4172 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  { M }  C_  ( 0 ... ( M  +  N
) ) )
27 elsni 4052 . . . . . 6  |-  ( k  e.  { M }  ->  k  =  M )
2827adantl 466 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
k  =  M )
29 fveq2 5866 . . . . . 6  |-  ( k  =  M  ->  ( A `  k )  =  ( A `  M ) )
30 oveq2 6292 . . . . . . 7  |-  ( k  =  M  ->  (
( M  +  N
)  -  k )  =  ( ( M  +  N )  -  M ) )
3130fveq2d 5870 . . . . . 6  |-  ( k  =  M  ->  ( B `  ( ( M  +  N )  -  k ) )  =  ( B `  ( ( M  +  N )  -  M
) ) )
3229, 31oveq12d 6302 . . . . 5  |-  ( k  =  M  ->  (
( A `  k
)  x.  ( B `
 ( ( M  +  N )  -  k ) ) )  =  ( ( A `
 M )  x.  ( B `  (
( M  +  N
)  -  M ) ) ) )
3328, 32syl 16 . . . 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 9622 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  CC )
3517recnd 9622 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  CC )
3634, 35pncan2d 9932 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( M  +  N )  -  M )  =  N )
3736fveq2d 5870 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B `  ( ( M  +  N )  -  M
) )  =  ( B `  N ) )
3837oveq2d 6300 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  (
( M  +  N
)  -  M ) ) )  =  ( ( A `  M
)  x.  ( B `
 N ) ) )
399coef3 22392 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
4039adantr 465 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
4140, 15ffvelrnd 6022 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A `  M )  e.  CC )
4210coef3 22392 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
4342adantl 466 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
4443, 13ffvelrnd 6022 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B `  N )  e.  CC )
4541, 44mulcld 9616 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  N
) )  e.  CC )
4638, 45eqeltrd 2555 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  (
( M  +  N
)  -  M ) ) )  e.  CC )
4746adantr 465 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
( ( A `  M )  x.  ( B `  ( ( M  +  N )  -  M ) ) )  e.  CC )
4833, 47eqeltrd 2555 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
( ( A `  k )  x.  ( B `  ( ( M  +  N )  -  k ) ) )  e.  CC )
49 simpl 457 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
50 eldifi 3626 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  e.  ( 0 ... ( M  +  N ) ) )
51 elfznn0 11770 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... ( M  +  N
) )  ->  k  e.  NN0 )
5250, 51syl 16 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  e.  NN0 )
539, 1dgrub 22394 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  M )
54533expia 1198 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  M )
)
5549, 52, 54syl2an 477 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( ( A `  k )  =/=  0  ->  k  <_  M )
)
5655necon1bd 2685 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( -.  k  <_  M  ->  ( A `  k )  =  0 ) )
5756imp 429 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  ( A `  k )  =  0 )
5857oveq1d 6299 . . . . 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 725 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  B : NN0 --> CC )
6050ad2antlr 726 . . . . . . . 8  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  k  e.  ( 0 ... ( M  +  N )
) )
61 fznn0sub 11716 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( M  +  N
) )  ->  (
( M  +  N
)  -  k )  e.  NN0 )
6260, 61syl 16 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  (
( M  +  N
)  -  k )  e.  NN0 )
6359, 62ffvelrnd 6022 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  ( B `  ( ( M  +  N )  -  k ) )  e.  CC )
6463mul02d 9777 . . . . 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 2508 . . . 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 465 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  ->  M  e.  RR )
6750adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  e.  ( 0 ... ( M  +  N ) ) )
6867, 51syl 16 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  e.  NN0 )
6968nn0red 10853 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  e.  RR )
7017adantr 465 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  ->  N  e.  RR )
7166, 69, 70leadd1d 10146 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  <_  k  <->  ( M  +  N )  <_  ( k  +  N ) ) )
728adantr 465 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  +  N
)  e.  NN0 )
7372nn0red 10853 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  +  N
)  e.  RR )
7473, 69, 70lesubadd2d 10151 . . . . . . . . . 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 256 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  <_  k  <->  ( ( M  +  N
)  -  k )  <_  N ) )
7675notbid 294 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( -.  M  <_ 
k  <->  -.  ( ( M  +  N )  -  k )  <_  N ) )
7776biimpa 484 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  -.  ( ( M  +  N )  -  k
)  <_  N )
78 simpr 461 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
7950, 61syl 16 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
( ( M  +  N )  -  k
)  e.  NN0 )
8010, 4dgrub 22394 . . . . . . . . . . 11  |-  ( ( G  e.  (Poly `  S )  /\  (
( M  +  N
)  -  k )  e.  NN0  /\  ( B `  ( ( M  +  N )  -  k ) )  =/=  0 )  -> 
( ( M  +  N )  -  k
)  <_  N )
81803expia 1198 . . . . . . . . . 10  |-  ( ( G  e.  (Poly `  S )  /\  (
( M  +  N
)  -  k )  e.  NN0 )  -> 
( ( B `  ( ( M  +  N )  -  k
) )  =/=  0  ->  ( ( M  +  N )  -  k
)  <_  N )
)
8278, 79, 81syl2an 477 . . . . . . . . 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 2685 . . . . . . . 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 429 . . . . . . 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 470 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  ( B `  ( ( M  +  N )  -  k ) )  =  0 )
8685oveq2d 6300 . . . . 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 725 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  A : NN0 --> CC )
8852ad2antlr 726 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  k  e.  NN0 )
8987, 88ffvelrnd 6022 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  ( A `  k )  e.  CC )
9089mul01d 9778 . . . . 5  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  (
( A `  k
)  x.  0 )  =  0 )
9186, 90eqtrd 2508 . . . 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 4153 . . . . . . 7  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  =/=  M )
9392adantl 466 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  =/=  M )
9469, 66letri3d 9726 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( k  =  M  <-> 
( k  <_  M  /\  M  <_  k ) ) )
9594necon3abid 2713 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( k  =/=  M  <->  -.  ( k  <_  M  /\  M  <_  k ) ) )
9693, 95mpbid 210 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  ->  -.  ( k  <_  M  /\  M  <_  k ) )
97 ianor 488 . . . . 5  |-  ( -.  ( k  <_  M  /\  M  <_  k )  <-> 
( -.  k  <_  M  \/  -.  M  <_  k ) )
9896, 97sylib 196 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( -.  k  <_  M  \/  -.  M  <_  k ) )
9965, 91, 98mpjaodan 784 . . 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 12051 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( 0 ... ( M  +  N ) )  e. 
Fin )
10126, 48, 99, 100fsumss 13510 . 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 13526 . . . 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 661 . . 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 2508 . 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 2514 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 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473   {csn 4027   class class class wbr 4447   -->wf 5584   ` cfv 5588  (class class class)co 6284    oFcof 6522   CCcc 9490   RRcr 9491   0cc0 9492    + caddc 9495    x. cmul 9497    <_ cle 9629    - cmin 9805   NN0cn0 10795   ZZcz 10864   ZZ>=cuz 11082   ...cfz 11672   sum_csu 13471  Polycply 22344  coeffccoe 22346  degcdgr 22347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-0p 21840  df-ply 22348  df-coe 22350  df-dgr 22351
This theorem is referenced by:  dgrmul  22429  plymul0or  22439  plydivlem4  22454  vieta1lem2  22469
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