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Theorem coemulhi 19651
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 19631 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
31, 2syl5eqel 2380 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
4 coemulhi.4 . . . . 5  |-  N  =  (deg `  G )
5 dgrcl 19631 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
64, 5syl5eqel 2380 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
7 nn0addcl 10015 . . . 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 19649 . . 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 10037 . . . . . 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 10035 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  RR )
1713nn0red 10035 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  RR )
1816, 17addge01d 9376 . . . . . 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 10278 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2115, 20syl6eleq 2386 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  ( ZZ>= `  0 )
)
228nn0zd 10131 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  +  N )  e.  ZZ )
23 elfz5 10806 . . . . . 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 3776 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  { M }  C_  ( 0 ... ( M  +  N
) ) )
27 elsni 3677 . . . . . 6  |-  ( k  e.  { M }  ->  k  =  M )
2827adantl 452 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  { M } )  -> 
k  =  M )
29 fveq2 5541 . . . . . 6  |-  ( k  =  M  ->  ( A `  k )  =  ( A `  M ) )
30 oveq2 5882 . . . . . . 7  |-  ( k  =  M  ->  (
( M  +  N
)  -  k )  =  ( ( M  +  N )  -  M ) )
3130fveq2d 5545 . . . . . 6  |-  ( k  =  M  ->  ( B `  ( ( M  +  N )  -  k ) )  =  ( B `  ( ( M  +  N )  -  M
) ) )
3229, 31oveq12d 5892 . . . . 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 8877 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  e.  CC )
3517recnd 8877 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  N  e.  CC )
3634, 35pncan2d 9175 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( M  +  N )  -  M )  =  N )
3736fveq2d 5545 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( B `  ( ( M  +  N )  -  M
) )  =  ( B `  N ) )
3837oveq2d 5890 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  (
( M  +  N
)  -  M ) ) )  =  ( ( A `  M
)  x.  ( B `
 N ) ) )
399coef3 19630 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
4039adantr 451 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
41 ffvelrn 5679 . . . . . . . 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 19630 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
4443adantl 452 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
45 ffvelrn 5679 . . . . . . . 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 8871 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( A `  M )  x.  ( B `  N
) )  e.  CC )
4838, 47eqeltrd 2370 . . . . 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 2370 . . 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 3311 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  e.  ( 0 ... ( M  +  N ) ) )
53 elfznn0 10838 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... ( M  +  N
) )  ->  k  e.  NN0 )
5452, 53syl 15 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... ( M  +  N ) )  \  { M } )  -> 
k  e.  NN0 )
559, 1dgrub 19632 . . . . . . . . . 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 2527 . . . . . . 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 5889 . . . . 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 10840 . . . . . . . 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 5679 . . . . . . 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 9026 . . . . 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 2328 . . . 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 10035 . . . . . . . . . . 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 9382 . . . . . . . . . 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 10035 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( M  +  N
)  e.  RR )
7776, 72, 73lesubadd2d 9387 . . . . . . . . . 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 19632 . . . . . . . . . . 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 2527 . . . . . . . 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 5890 . . . . 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 5679 . . . . . . 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 9027 . . . . 5  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  /\  k  e.  ( (
0 ... ( M  +  N ) )  \  { M } ) )  /\  -.  M  <_ 
k )  ->  (
( A `  k
)  x.  0 )  =  0 )
9589, 94eqtrd 2328 . . . 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 3763 . . . . . . 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 8977 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  k  e.  ( ( 0 ... ( M  +  N
) )  \  { M } ) )  -> 
( k  =  M  <-> 
( k  <_  M  /\  M  <_  k ) ) )
9998necon3abid 2492 . . . . . 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 11051 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( 0 ... ( M  +  N ) )  e. 
Fin )
10526, 50, 103, 104fsumss 12214 . 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 12229 . . . 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 2328 . 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 2334 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 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162   {csn 3653   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   sum_csu 12174  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem is referenced by:  dgrmul  19667  plymul0or  19677  plydivlem4  19692  vieta1lem2  19707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589
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