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Theorem coemulhi 21680
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 21660 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
31, 2syl5eqel 2525 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
4 coemulhi.4 . . . . 5  |-  N  =  (deg `  G )
5 dgrcl 21660 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
64, 5syl5eqel 2525 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
7 nn0addcl 10611 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  +  N
)  e.  NN0 )
83, 6, 7syl2an 474 . . 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 21678 . . 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 1310 . 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 463 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  NN0 )
1413nn0ge0d 10635 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  0  <_  N )
153adantr 462 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  NN0 )
1615nn0red 10633 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  RR )
1713nn0red 10633 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  RR )
1816, 17addge01d 9923 . . . . . 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 10891 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2115, 20syl6eleq 2531 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  ( ZZ>= `  0 )
)
228nn0zd 10741 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  +  N )  e.  ZZ )
23 elfz5 11441 . . . . . 6  |-  ( ( M  e.  ( ZZ>= ` 
0 )  /\  ( M  +  N )  e.  ZZ )  ->  ( M  e.  ( 0 ... ( M  +  N ) )  <->  M  <_  ( M  +  N ) ) )
2421, 22, 23syl2anc 656 . . . . 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 4015 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  { M }  C_  ( 0 ... ( M  +  N
) ) )
27 elsni 3899 . . . . . 6  |-  ( k  e.  { M }  ->  k  =  M )
2827adantl 463 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
k  =  M )
29 fveq2 5688 . . . . . 6  |-  ( k  =  M  ->  ( A `  k )  =  ( A `  M ) )
30 oveq2 6098 . . . . . . 7  |-  ( k  =  M  ->  (
( M  +  N
)  -  k )  =  ( ( M  +  N )  -  M ) )
3130fveq2d 5692 . . . . . 6  |-  ( k  =  M  ->  ( B `  ( ( M  +  N )  -  k ) )  =  ( B `  ( ( M  +  N )  -  M
) ) )
3229, 31oveq12d 6108 . . . . 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 9408 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  CC )
3517recnd 9408 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  CC )
3634, 35pncan2d 9717 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( M  +  N )  -  M )  =  N )
3736fveq2d 5692 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B `  ( ( M  +  N )  -  M
) )  =  ( B `  N ) )
3837oveq2d 6106 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  (
( M  +  N
)  -  M ) ) )  =  ( ( A `  M
)  x.  ( B `
 N ) ) )
399coef3 21659 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
4039adantr 462 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
4140, 15ffvelrnd 5841 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A `  M )  e.  CC )
4210coef3 21659 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
4342adantl 463 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
4443, 13ffvelrnd 5841 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B `  N )  e.  CC )
4541, 44mulcld 9402 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  N
) )  e.  CC )
4638, 45eqeltrd 2515 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  (
( M  +  N
)  -  M ) ) )  e.  CC )
4746adantr 462 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
( ( A `  M )  x.  ( B `  ( ( M  +  N )  -  M ) ) )  e.  CC )
4833, 47eqeltrd 2515 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
( ( A `  k )  x.  ( B `  ( ( M  +  N )  -  k ) ) )  e.  CC )
49 simpl 454 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
50 eldifi 3475 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  e.  ( 0 ... ( M  +  N ) ) )
51 elfznn0 11477 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... ( M  +  N
) )  ->  k  e.  NN0 )
5250, 51syl 16 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  e.  NN0 )
539, 1dgrub 21661 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  M )
54533expia 1184 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  M )
)
5549, 52, 54syl2an 474 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( ( A `  k )  =/=  0  ->  k  <_  M )
)
5655necon1bd 2677 . . . . . . 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 6105 . . . . 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 720 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  B : NN0 --> CC )
6050ad2antlr 721 . . . . . . . 8  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  k  e.  ( 0 ... ( M  +  N )
) )
61 fznn0sub 11483 . . . . . . . 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 5841 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  ( B `  ( ( M  +  N )  -  k ) )  e.  CC )
6463mul02d 9563 . . . . 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 2473 . . . 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 462 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  ->  M  e.  RR )
6750adantl 463 . . . . . . . . . . . . 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 10633 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  e.  RR )
7017adantr 462 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  ->  N  e.  RR )
7166, 69, 70leadd1d 9929 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  <_  k  <->  ( M  +  N )  <_  ( k  +  N ) ) )
728adantr 462 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  +  N
)  e.  NN0 )
7372nn0red 10633 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  +  N
)  e.  RR )
7473, 69, 70lesubadd2d 9934 . . . . . . . . . 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 481 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  -.  ( ( M  +  N )  -  k
)  <_  N )
78 simpr 458 . . . . . . . . . 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 21661 . . . . . . . . . . 11  |-  ( ( G  e.  (Poly `  S )  /\  (
( M  +  N
)  -  k )  e.  NN0  /\  ( B `  ( ( M  +  N )  -  k ) )  =/=  0 )  -> 
( ( M  +  N )  -  k
)  <_  N )
81803expia 1184 . . . . . . . . . 10  |-  ( ( G  e.  (Poly `  S )  /\  (
( M  +  N
)  -  k )  e.  NN0 )  -> 
( ( B `  ( ( M  +  N )  -  k
) )  =/=  0  ->  ( ( M  +  N )  -  k
)  <_  N )
)
8278, 79, 81syl2an 474 . . . . . . . . 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 2677 . . . . . . . 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 467 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  ( B `  ( ( M  +  N )  -  k ) )  =  0 )
8685oveq2d 6106 . . . . 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 720 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  A : NN0 --> CC )
8852ad2antlr 721 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  k  e.  NN0 )
8987, 88ffvelrnd 5841 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  ( A `  k )  e.  CC )
9089mul01d 9564 . . . . 5  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  (
( A `  k
)  x.  0 )  =  0 )
9186, 90eqtrd 2473 . . . 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 3998 . . . . . . 7  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  =/=  M )
9392adantl 463 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  =/=  M )
9469, 66letri3d 9512 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( k  =  M  <-> 
( k  <_  M  /\  M  <_  k ) ) )
9594necon3abid 2639 . . . . . 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 485 . . . . 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 779 . . 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 11791 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( 0 ... ( M  +  N ) )  e. 
Fin )
10126, 48, 99, 100fsumss 13198 . 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 13213 . . . 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 656 . . 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 2473 . 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 2479 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 1364    e. wcel 1761    =/= wne 2604    \ cdif 3322   {csn 3874   class class class wbr 4289   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   CCcc 9276   RRcr 9277   0cc0 9278    + caddc 9281    x. cmul 9283    <_ cle 9415    - cmin 9591   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   sum_csu 13159  Polycply 21611  coeffccoe 21613  degcdgr 21614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-0p 21107  df-ply 21615  df-coe 21617  df-dgr 21618
This theorem is referenced by:  dgrmul  21696  plymul0or  21706  plydivlem4  21721  vieta1lem2  21736
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