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Theorem coemulhi 19635
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  o F  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 19615 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
31, 2syl5eqel 2367 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
4 coemulhi.4 . . . . 5  |-  N  =  (deg `  G )
5 dgrcl 19615 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
64, 5syl5eqel 2367 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
7 nn0addcl 9999 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  +  N
)  e.  NN0 )
83, 6, 7syl2an 463 . . 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 19633 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  ( M  +  N
)  e.  NN0 )  ->  ( (coeff `  ( F  o F  x.  G
) ) `  ( M  +  N )
)  =  sum_ k  e.  ( 0 ... ( M  +  N )
) ( ( A `
 k )  x.  ( B `  (
( M  +  N
)  -  k ) ) ) )
128, 11mpd3an3 1278 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  o F  x.  G ) ) `
 ( M  +  N ) )  = 
sum_ k  e.  ( 0 ... ( M  +  N ) ) ( ( A `  k )  x.  ( B `  ( ( M  +  N )  -  k ) ) ) )
136adantl 452 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  NN0 )
1413nn0ge0d 10021 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  0  <_  N )
153adantr 451 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  NN0 )
1615nn0red 10019 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  RR )
1713nn0red 10019 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  RR )
1816, 17addge01d 9360 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( 0  <_  N  <->  M  <_  ( M  +  N ) ) )
1914, 18mpbid 201 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  <_  ( M  +  N ) )
20 nn0uz 10262 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2115, 20syl6eleq 2373 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  ( ZZ>= `  0 )
)
228nn0zd 10115 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  +  N )  e.  ZZ )
23 elfz5 10790 . . . . . 6  |-  ( ( M  e.  ( ZZ>= ` 
0 )  /\  ( M  +  N )  e.  ZZ )  ->  ( M  e.  ( 0 ... ( M  +  N ) )  <->  M  <_  ( M  +  N ) ) )
2421, 22, 23syl2anc 642 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  e.  ( 0 ... ( M  +  N )
)  <->  M  <_  ( M  +  N ) ) )
2519, 24mpbird 223 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  ( 0 ... ( M  +  N )
) )
2625snssd 3760 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  { M }  C_  ( 0 ... ( M  +  N
) ) )
27 elsni 3664 . . . . . 6  |-  ( k  e.  { M }  ->  k  =  M )
2827adantl 452 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
k  =  M )
29 fveq2 5525 . . . . . 6  |-  ( k  =  M  ->  ( A `  k )  =  ( A `  M ) )
30 oveq2 5866 . . . . . . 7  |-  ( k  =  M  ->  (
( M  +  N
)  -  k )  =  ( ( M  +  N )  -  M ) )
3130fveq2d 5529 . . . . . 6  |-  ( k  =  M  ->  ( B `  ( ( M  +  N )  -  k ) )  =  ( B `  ( ( M  +  N )  -  M
) ) )
3229, 31oveq12d 5876 . . . . 5  |-  ( k  =  M  ->  (
( A `  k
)  x.  ( B `
 ( ( M  +  N )  -  k ) ) )  =  ( ( A `
 M )  x.  ( B `  (
( M  +  N
)  -  M ) ) ) )
3328, 32syl 15 . . . 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 8861 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  CC )
3517recnd 8861 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  CC )
3634, 35pncan2d 9159 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( M  +  N )  -  M )  =  N )
3736fveq2d 5529 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B `  ( ( M  +  N )  -  M
) )  =  ( B `  N ) )
3837oveq2d 5874 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  (
( M  +  N
)  -  M ) ) )  =  ( ( A `  M
)  x.  ( B `
 N ) ) )
399coef3 19614 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
4039adantr 451 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
41 ffvelrn 5663 . . . . . . . 8  |-  ( ( A : NN0 --> CC  /\  M  e.  NN0 )  -> 
( A `  M
)  e.  CC )
4240, 15, 41syl2anc 642 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A `  M )  e.  CC )
4310coef3 19614 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
4443adantl 452 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
45 ffvelrn 5663 . . . . . . . 8  |-  ( ( B : NN0 --> CC  /\  N  e.  NN0 )  -> 
( B `  N
)  e.  CC )
4644, 13, 45syl2anc 642 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B `  N )  e.  CC )
4742, 46mulcld 8855 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  N
) )  e.  CC )
4838, 47eqeltrd 2357 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  (
( M  +  N
)  -  M ) ) )  e.  CC )
4948adantr 451 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
( ( A `  M )  x.  ( B `  ( ( M  +  N )  -  M ) ) )  e.  CC )
5033, 49eqeltrd 2357 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
( ( A `  k )  x.  ( B `  ( ( M  +  N )  -  k ) ) )  e.  CC )
51 simpl 443 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
52 eldifi 3298 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  e.  ( 0 ... ( M  +  N ) ) )
53 elfznn0 10822 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... ( M  +  N
) )  ->  k  e.  NN0 )
5452, 53syl 15 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  e.  NN0 )
559, 1dgrub 19616 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  M )
56553expia 1153 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  M )
)
5751, 54, 56syl2an 463 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( ( A `  k )  =/=  0  ->  k  <_  M )
)
5857necon1bd 2514 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( -.  k  <_  M  ->  ( A `  k )  =  0 ) )
5958imp 418 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  ( A `  k )  =  0 )
6059oveq1d 5873 . . . . 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 ) ) ) )
6144ad2antrr 706 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  B : NN0 --> CC )
6252ad2antlr 707 . . . . . . . 8  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  k  e.  ( 0 ... ( M  +  N )
) )
63 fznn0sub 10824 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( M  +  N
) )  ->  (
( M  +  N
)  -  k )  e.  NN0 )
6462, 63syl 15 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  (
( M  +  N
)  -  k )  e.  NN0 )
65 ffvelrn 5663 . . . . . . 7  |-  ( ( B : NN0 --> CC  /\  ( ( M  +  N )  -  k
)  e.  NN0 )  ->  ( B `  (
( M  +  N
)  -  k ) )  e.  CC )
6661, 64, 65syl2anc 642 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  ( B `  ( ( M  +  N )  -  k ) )  e.  CC )
6766mul02d 9010 . . . . 5  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  k  <_  M )  ->  (
0  x.  ( B `
 ( ( M  +  N )  -  k ) ) )  =  0 )
6860, 67eqtrd 2315 . . . 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 )
6916adantr 451 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  ->  M  e.  RR )
7052adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  e.  ( 0 ... ( M  +  N ) ) )
7170, 53syl 15 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  e.  NN0 )
7271nn0red 10019 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  e.  RR )
7317adantr 451 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  ->  N  e.  RR )
7469, 72, 73leadd1d 9366 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  <_  k  <->  ( M  +  N )  <_  ( k  +  N ) ) )
758adantr 451 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  +  N
)  e.  NN0 )
7675nn0red 10019 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  +  N
)  e.  RR )
7776, 72, 73lesubadd2d 9371 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( ( ( M  +  N )  -  k )  <_  N  <->  ( M  +  N )  <_  ( k  +  N ) ) )
7874, 77bitr4d 247 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  <_  k  <->  ( ( M  +  N
)  -  k )  <_  N ) )
7978notbid 285 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( -.  M  <_ 
k  <->  -.  ( ( M  +  N )  -  k )  <_  N ) )
8079biimpa 470 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  -.  ( ( M  +  N )  -  k
)  <_  N )
81 simpr 447 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
8252, 63syl 15 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
( ( M  +  N )  -  k
)  e.  NN0 )
8310, 4dgrub 19616 . . . . . . . . . . 11  |-  ( ( G  e.  (Poly `  S )  /\  (
( M  +  N
)  -  k )  e.  NN0  /\  ( B `  ( ( M  +  N )  -  k ) )  =/=  0 )  -> 
( ( M  +  N )  -  k
)  <_  N )
84833expia 1153 . . . . . . . . . 10  |-  ( ( G  e.  (Poly `  S )  /\  (
( M  +  N
)  -  k )  e.  NN0 )  -> 
( ( B `  ( ( M  +  N )  -  k
) )  =/=  0  ->  ( ( M  +  N )  -  k
)  <_  N )
)
8581, 82, 84syl2an 463 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( ( B `  ( ( M  +  N )  -  k
) )  =/=  0  ->  ( ( M  +  N )  -  k
)  <_  N )
)
8685necon1bd 2514 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( -.  ( ( M  +  N )  -  k )  <_  N  ->  ( B `  ( ( M  +  N )  -  k
) )  =  0 ) )
8786imp 418 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  ( ( M  +  N )  -  k )  <_  N )  ->  ( B `  ( ( M  +  N )  -  k ) )  =  0 )
8880, 87syldan 456 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  ( B `  ( ( M  +  N )  -  k ) )  =  0 )
8988oveq2d 5874 . . . . 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 ) )
9040ad2antrr 706 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  A : NN0 --> CC )
9154ad2antlr 707 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  k  e.  NN0 )
92 ffvelrn 5663 . . . . . . 7  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
9390, 91, 92syl2anc 642 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  ( A `  k )  e.  CC )
9493mul01d 9011 . . . . 5  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  (
( A `  k
)  x.  0 )  =  0 )
9589, 94eqtrd 2315 . . . 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 )
96 eldifsni 3750 . . . . . . 7  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  =/=  M )
9796adantl 452 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
k  =/=  M )
9872, 69letri3d 8961 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( k  =  M  <-> 
( k  <_  M  /\  M  <_  k ) ) )
9998necon3abid 2479 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( k  =/=  M  <->  -.  ( k  <_  M  /\  M  <_  k ) ) )
10097, 99mpbid 201 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  ->  -.  ( k  <_  M  /\  M  <_  k ) )
101 ianor 474 . . . . 5  |-  ( -.  ( k  <_  M  /\  M  <_  k )  <-> 
( -.  k  <_  M  \/  -.  M  <_  k ) )
102100, 101sylib 188 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( -.  k  <_  M  \/  -.  M  <_  k ) )
10368, 95, 102mpjaodan 761 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( ( A `  k )  x.  ( B `  ( ( M  +  N )  -  k ) ) )  =  0 )
104 fzfid 11035 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( 0 ... ( M  +  N ) )  e. 
Fin )
10526, 50, 103, 104fsumss 12198 . 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 ) ) ) )
10632sumsn 12213 . . . 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 ) ) ) )
10715, 48, 106syl2anc 642 . . 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 ) ) ) )
108107, 38eqtrd 2315 . 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
) ) )
10912, 105, 1083eqtr2d 2321 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  o F  x.  G ) ) `
 ( M  +  N ) )  =  ( ( A `  M )  x.  ( B `  N )
) )
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    \ cdif 3149   {csn 3640   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    <_ cle 8868    - cmin 9037   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   sum_csu 12158  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  dgrmul  19651  plymul0or  19661  plydivlem4  19676  vieta1lem2  19691
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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