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Theorem cply1mul 18815
Description: The product of two constant polynomials is a constant polynomial. (Contributed by AV, 18-Nov-2019.)
Hypotheses
Ref Expression
cply1mul.p  |-  P  =  (Poly1 `  R )
cply1mul.b  |-  B  =  ( Base `  P
)
cply1mul.0  |-  .0.  =  ( 0g `  R )
cply1mul.m  |-  .X.  =  ( .r `  P )
Assertion
Ref Expression
cply1mul  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  )  ->  A. c  e.  NN  ( (coe1 `  ( F  .X.  G ) ) `
 c )  =  .0.  ) )
Distinct variable groups:    F, c    G, c    .X. , c    .0. , c
Allowed substitution hints:    B( c)    P( c)    R( c)

Proof of Theorem cply1mul
Dummy variables  k  n  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cply1mul.p . . . . . . . . . 10  |-  P  =  (Poly1 `  R )
2 cply1mul.m . . . . . . . . . 10  |-  .X.  =  ( .r `  P )
3 eqid 2420 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
4 cply1mul.b . . . . . . . . . 10  |-  B  =  ( Base `  P
)
51, 2, 3, 4coe1mul 18791 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .X.  G ) )  =  ( s  e.  NN0  |->  ( R 
gsumg  ( k  e.  ( 0 ... s ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( s  -  k ) ) ) ) ) ) )
653expb 1206 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  (coe1 `  ( F  .X.  G ) )  =  ( s  e. 
NN0  |->  ( R  gsumg  ( k  e.  ( 0 ... s )  |->  ( ( (coe1 `  F ) `  k ) ( .r
`  R ) ( (coe1 `  G ) `  ( s  -  k
) ) ) ) ) ) )
76adantr 466 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  -> 
(coe1 `  ( F  .X.  G ) )  =  ( s  e.  NN0  |->  ( R  gsumg  ( k  e.  ( 0 ... s ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( s  -  k ) ) ) ) ) ) )
87adantr 466 . . . . . 6  |-  ( ( ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  /\  n  e.  NN )  ->  (coe1 `  ( F  .X.  G ) )  =  ( s  e.  NN0  |->  ( R  gsumg  ( k  e.  ( 0 ... s ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( s  -  k ) ) ) ) ) ) )
9 oveq2 6304 . . . . . . . . 9  |-  ( s  =  n  ->  (
0 ... s )  =  ( 0 ... n
) )
10 oveq1 6303 . . . . . . . . . . 11  |-  ( s  =  n  ->  (
s  -  k )  =  ( n  -  k ) )
1110fveq2d 5876 . . . . . . . . . 10  |-  ( s  =  n  ->  (
(coe1 `  G ) `  ( s  -  k
) )  =  ( (coe1 `  G ) `  ( n  -  k
) ) )
1211oveq2d 6312 . . . . . . . . 9  |-  ( s  =  n  ->  (
( (coe1 `  F ) `  k ) ( .r
`  R ) ( (coe1 `  G ) `  ( s  -  k
) ) )  =  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) )
139, 12mpteq12dv 4495 . . . . . . . 8  |-  ( s  =  n  ->  (
k  e.  ( 0 ... s )  |->  ( ( (coe1 `  F ) `  k ) ( .r
`  R ) ( (coe1 `  G ) `  ( s  -  k
) ) ) )  =  ( k  e.  ( 0 ... n
)  |->  ( ( (coe1 `  F ) `  k
) ( .r `  R ) ( (coe1 `  G ) `  (
n  -  k ) ) ) ) )
1413oveq2d 6312 . . . . . . 7  |-  ( s  =  n  ->  ( R  gsumg  ( k  e.  ( 0 ... s ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( s  -  k ) ) ) ) )  =  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) ) ) )
1514adantl 467 . . . . . 6  |-  ( ( ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  /\  A. c  e.  NN  ( ( (coe1 `  F ) `  c
)  =  .0.  /\  ( (coe1 `  G ) `  c )  =  .0.  ) )  /\  n  e.  NN )  /\  s  =  n )  ->  ( R  gsumg  ( k  e.  ( 0 ... s ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( s  -  k ) ) ) ) )  =  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) ) ) )
16 nnnn0 10865 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  NN0 )
1716adantl 467 . . . . . 6  |-  ( ( ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  /\  n  e.  NN )  ->  n  e.  NN0 )
18 ovex 6324 . . . . . . 7  |-  ( R 
gsumg  ( k  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) ) )  e. 
_V
1918a1i 11 . . . . . 6  |-  ( ( ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  /\  n  e.  NN )  ->  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) ) )  e. 
_V )
208, 15, 17, 19fvmptd 5961 . . . . 5  |-  ( ( ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  /\  n  e.  NN )  ->  ( (coe1 `  ( F  .X.  G ) ) `  n )  =  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) ) ) )
21 r19.26 2953 . . . . . . . . . 10  |-  ( A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  )  <->  ( A. c  e.  NN  (
(coe1 `  F ) `  c )  =  .0. 
/\  A. c  e.  NN  ( (coe1 `  G ) `  c )  =  .0.  ) )
22 oveq2 6304 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  0  ->  (
n  -  k )  =  ( n  - 
0 ) )
23 nncn 10606 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  e.  NN  ->  n  e.  CC )
2423subid1d 9964 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  NN  ->  (
n  -  0 )  =  n )
2524adantr 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( n  - 
0 )  =  n )
2622, 25sylan9eqr 2483 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( n  -  k )  =  n )
27 simpll 758 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  n  e.  NN )
2826, 27eqeltrd 2508 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( n  -  k )  e.  NN )
29 fveq2 5872 . . . . . . . . . . . . . . . . . . . 20  |-  ( c  =  ( n  -  k )  ->  (
(coe1 `  G ) `  c )  =  ( (coe1 `  G ) `  ( n  -  k
) ) )
3029eqeq1d 2422 . . . . . . . . . . . . . . . . . . 19  |-  ( c  =  ( n  -  k )  ->  (
( (coe1 `  G ) `  c )  =  .0.  <->  ( (coe1 `  G ) `  ( n  -  k
) )  =  .0.  ) )
3130rspcv 3175 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  -  k )  e.  NN  ->  ( A. c  e.  NN  ( (coe1 `  G ) `  c )  =  .0. 
->  ( (coe1 `  G ) `  ( n  -  k
) )  =  .0.  ) )
3228, 31syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( A. c  e.  NN  (
(coe1 `  G ) `  c )  =  .0. 
->  ( (coe1 `  G ) `  ( n  -  k
) )  =  .0.  ) )
33 oveq2 6304 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( (coe1 `  G ) `  ( n  -  k
) )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  ( ( (coe1 `  F ) `  k ) ( .r
`  R )  .0.  ) )
34 simpll 758 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  (
( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 ) )  ->  R  e.  Ring )
35 simpl 458 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( F  e.  B  /\  G  e.  B )  ->  F  e.  B )
3635adantl 467 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  F  e.  B )
37 elfznn0 11874 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  e.  ( 0 ... n )  ->  k  e.  NN0 )
3837adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  k  e.  NN0 )
3938adantr 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  k  e.  NN0 )
40 eqid 2420 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (coe1 `  F
)  =  (coe1 `  F
)
41 eqid 2420 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( Base `  R )  =  (
Base `  R )
4240, 4, 1, 41coe1fvalcl 18733 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( F  e.  B  /\  k  e.  NN0 )  -> 
( (coe1 `  F ) `  k )  e.  (
Base `  R )
)
4336, 39, 42syl2an 479 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  (
( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 ) )  ->  (
(coe1 `  F ) `  k )  e.  (
Base `  R )
)
44 cply1mul.0 . . . . . . . . . . . . . . . . . . . . . . 23  |-  .0.  =  ( 0g `  R )
4541, 3, 44ringrz 17746 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( R  e.  Ring  /\  (
(coe1 `  F ) `  k )  e.  (
Base `  R )
)  ->  ( (
(coe1 `  F ) `  k ) ( .r
`  R )  .0.  )  =  .0.  )
4634, 43, 45syl2anc 665 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  (
( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 ) )  ->  (
( (coe1 `  F ) `  k ) ( .r
`  R )  .0.  )  =  .0.  )
4733, 46sylan9eqr 2483 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  /\  ( ( n  e.  NN  /\  k  e.  ( 0 ... n
) )  /\  k  =  0 ) )  /\  ( (coe1 `  G
) `  ( n  -  k ) )  =  .0.  )  -> 
( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
4847ex 435 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  (
( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 ) )  ->  (
( (coe1 `  G ) `  ( n  -  k
) )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
)
4948expcom 436 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( (
(coe1 `  G ) `  ( n  -  k
) )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) )
5049com23 81 . . . . . . . . . . . . . . . . 17  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( (
(coe1 `  G ) `  ( n  -  k
) )  =  .0. 
->  ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) )
5132, 50syld 45 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( A. c  e.  NN  (
(coe1 `  G ) `  c )  =  .0. 
->  ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) )
5251com12 32 . . . . . . . . . . . . . . 15  |-  ( A. c  e.  NN  (
(coe1 `  G ) `  c )  =  .0. 
->  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n
) )  /\  k  =  0 )  -> 
( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) )
5352expd 437 . . . . . . . . . . . . . 14  |-  ( A. c  e.  NN  (
(coe1 `  G ) `  c )  =  .0. 
->  ( ( n  e.  NN  /\  k  e.  ( 0 ... n
) )  ->  (
k  =  0  -> 
( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) ) )
5453com24 90 . . . . . . . . . . . . 13  |-  ( A. c  e.  NN  (
(coe1 `  G ) `  c )  =  .0. 
->  ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( k  =  0  ->  ( ( n  e.  NN  /\  k  e.  ( 0 ... n
) )  ->  (
( (coe1 `  F ) `  k ) ( .r
`  R ) ( (coe1 `  G ) `  ( n  -  k
) ) )  =  .0.  ) ) ) )
5554adantl 467 . . . . . . . . . . . 12  |-  ( ( A. c  e.  NN  ( (coe1 `  F ) `  c )  =  .0. 
/\  A. c  e.  NN  ( (coe1 `  G ) `  c )  =  .0.  )  ->  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( k  =  0  ->  (
( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( ( (coe1 `  F ) `  k
) ( .r `  R ) ( (coe1 `  G ) `  (
n  -  k ) ) )  =  .0.  ) ) ) )
5655com13 83 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  ->  (
( A. c  e.  NN  ( (coe1 `  F
) `  c )  =  .0.  /\  A. c  e.  NN  ( (coe1 `  G
) `  c )  =  .0.  )  ->  (
( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( ( (coe1 `  F ) `  k
) ( .r `  R ) ( (coe1 `  G ) `  (
n  -  k ) ) )  =  .0.  ) ) ) )
57 df-ne 2618 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =/=  0  <->  -.  k  =  0 )
5857biimpri 209 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  k  =  0  -> 
k  =/=  0 )
5958, 37anim12ci 569 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  ( k  e.  NN0  /\  k  =/=  0 ) )
60 elnnne0 10872 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  e.  NN  <->  ( k  e.  NN0  /\  k  =/=  0 ) )
6159, 60sylibr 215 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  k  e.  NN )
62 fveq2 5872 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( c  =  k  ->  (
(coe1 `  F ) `  c )  =  ( (coe1 `  F ) `  k ) )
6362eqeq1d 2422 . . . . . . . . . . . . . . . . . . . . 21  |-  ( c  =  k  ->  (
( (coe1 `  F ) `  c )  =  .0.  <->  ( (coe1 `  F ) `  k )  =  .0.  ) )
6463rspcv 3175 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  e.  NN  ->  ( A. c  e.  NN  ( (coe1 `  F ) `  c )  =  .0. 
->  ( (coe1 `  F ) `  k )  =  .0.  ) )
6561, 64syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  ( A. c  e.  NN  (
(coe1 `  F ) `  c )  =  .0. 
->  ( (coe1 `  F ) `  k )  =  .0.  ) )
66 oveq1 6303 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( (coe1 `  F ) `  k )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  (  .0.  ( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) )
67 simpll 758 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  R  e.  Ring )
684eleq2i 2498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( G  e.  B  <->  G  e.  ( Base `  P )
)
6968biimpi 197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( G  e.  B  ->  G  e.  ( Base `  P
) )
7069adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( F  e.  B  /\  G  e.  B )  ->  G  e.  ( Base `  P ) )
7170adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  G  e.  ( Base `  P )
)
72 fznn0sub 11818 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( k  e.  ( 0 ... n )  ->  (
n  -  k )  e.  NN0 )
73 eqid 2420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  (coe1 `  G
)  =  (coe1 `  G
)
74 eqid 2420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( Base `  P )  =  (
Base `  P )
7573, 74, 1, 41coe1fvalcl 18733 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( G  e.  ( Base `  P )  /\  (
n  -  k )  e.  NN0 )  -> 
( (coe1 `  G ) `  ( n  -  k
) )  e.  (
Base `  R )
)
7671, 72, 75syl2an 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  (
(coe1 `  G ) `  ( n  -  k
) )  e.  (
Base `  R )
)
7741, 3, 44ringlz 17745 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( R  e.  Ring  /\  (
(coe1 `  G ) `  ( n  -  k
) )  e.  (
Base `  R )
)  ->  (  .0.  ( .r `  R ) ( (coe1 `  G ) `  ( n  -  k
) ) )  =  .0.  )
7867, 76, 77syl2anc 665 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  (  .0.  ( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
7966, 78sylan9eqr 2483 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  /\  k  e.  ( 0 ... n ) )  /\  ( (coe1 `  F
) `  k )  =  .0.  )  ->  (
( (coe1 `  F ) `  k ) ( .r
`  R ) ( (coe1 `  G ) `  ( n  -  k
) ) )  =  .0.  )
8079ex 435 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  (
( (coe1 `  F ) `  k )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
)
8180ex 435 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( k  e.  ( 0 ... n
)  ->  ( (
(coe1 `  F ) `  k )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) )
8281com23 81 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( (
(coe1 `  F ) `  k )  =  .0. 
->  ( k  e.  ( 0 ... n )  ->  ( ( (coe1 `  F ) `  k
) ( .r `  R ) ( (coe1 `  G ) `  (
n  -  k ) ) )  =  .0.  ) ) )
8382a1dd 47 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( (
(coe1 `  F ) `  k )  =  .0. 
->  ( n  e.  NN  ->  ( k  e.  ( 0 ... n )  ->  ( ( (coe1 `  F ) `  k
) ( .r `  R ) ( (coe1 `  G ) `  (
n  -  k ) ) )  =  .0.  ) ) ) )
8483com14 91 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  e.  ( 0 ... n )  ->  (
( (coe1 `  F ) `  k )  =  .0. 
->  ( n  e.  NN  ->  ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) ) )
8584adantl 467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  ( (
(coe1 `  F ) `  k )  =  .0. 
->  ( n  e.  NN  ->  ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) ) )
8665, 85syld 45 . . . . . . . . . . . . . . . . . 18  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  ( A. c  e.  NN  (
(coe1 `  F ) `  c )  =  .0. 
->  ( n  e.  NN  ->  ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) ) )
8786com24 90 . . . . . . . . . . . . . . . . 17  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( n  e.  NN  ->  ( A. c  e.  NN  (
(coe1 `  F ) `  c )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) ) )
8887ex 435 . . . . . . . . . . . . . . . 16  |-  ( -.  k  =  0  -> 
( k  e.  ( 0 ... n )  ->  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( n  e.  NN  ->  ( A. c  e.  NN  (
(coe1 `  F ) `  c )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) ) ) )
8988com14 91 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
k  e.  ( 0 ... n )  -> 
( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( -.  k  =  0  ->  ( A. c  e.  NN  (
(coe1 `  F ) `  c )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) ) ) )
9089imp 430 . . . . . . . . . . . . . 14  |-  ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( -.  k  =  0  ->  ( A. c  e.  NN  ( (coe1 `  F ) `  c )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
) ) )
9190com14 91 . . . . . . . . . . . . 13  |-  ( A. c  e.  NN  (
(coe1 `  F ) `  c )  =  .0. 
->  ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( -.  k  =  0  ->  ( (
n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( ( (coe1 `  F ) `  k
) ( .r `  R ) ( (coe1 `  G ) `  (
n  -  k ) ) )  =  .0.  ) ) ) )
9291adantr 466 . . . . . . . . . . . 12  |-  ( ( A. c  e.  NN  ( (coe1 `  F ) `  c )  =  .0. 
/\  A. c  e.  NN  ( (coe1 `  G ) `  c )  =  .0.  )  ->  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( -.  k  =  0  ->  ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( ( (coe1 `  F ) `  k
) ( .r `  R ) ( (coe1 `  G ) `  (
n  -  k ) ) )  =  .0.  ) ) ) )
9392com13 83 . . . . . . . . . . 11  |-  ( -.  k  =  0  -> 
( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  -> 
( ( A. c  e.  NN  ( (coe1 `  F
) `  c )  =  .0.  /\  A. c  e.  NN  ( (coe1 `  G
) `  c )  =  .0.  )  ->  (
( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( ( (coe1 `  F ) `  k
) ( .r `  R ) ( (coe1 `  G ) `  (
n  -  k ) ) )  =  .0.  ) ) ) )
9456, 93pm2.61i 167 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( ( A. c  e.  NN  ( (coe1 `  F ) `  c )  =  .0. 
/\  A. c  e.  NN  ( (coe1 `  G ) `  c )  =  .0.  )  ->  ( (
n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( ( (coe1 `  F ) `  k
) ( .r `  R ) ( (coe1 `  G ) `  (
n  -  k ) ) )  =  .0.  ) ) )
9521, 94syl5bi 220 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  )  ->  (
( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( ( (coe1 `  F ) `  k
) ( .r `  R ) ( (coe1 `  G ) `  (
n  -  k ) ) )  =  .0.  ) ) )
9695imp 430 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  -> 
( ( n  e.  NN  /\  k  e.  ( 0 ... n
) )  ->  (
( (coe1 `  F ) `  k ) ( .r
`  R ) ( (coe1 `  G ) `  ( n  -  k
) ) )  =  .0.  ) )
9796impl 624 . . . . . . 7  |-  ( ( ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  /\  A. c  e.  NN  ( ( (coe1 `  F ) `  c
)  =  .0.  /\  ( (coe1 `  G ) `  c )  =  .0.  ) )  /\  n  e.  NN )  /\  k  e.  ( 0 ... n
) )  ->  (
( (coe1 `  F ) `  k ) ( .r
`  R ) ( (coe1 `  G ) `  ( n  -  k
) ) )  =  .0.  )
9897mpteq2dva 4503 . . . . . 6  |-  ( ( ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  /\  n  e.  NN )  ->  ( k  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) )  =  ( k  e.  ( 0 ... n )  |->  .0.  ) )
9998oveq2d 6312 . . . . 5  |-  ( ( ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  /\  n  e.  NN )  ->  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) ) )  =  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) ) )
100 ringmnd 17717 . . . . . . . . 9  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
101 ovex 6324 . . . . . . . . . 10  |-  ( 0 ... n )  e. 
_V
102101a1i 11 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( 0 ... n )  e. 
_V )
10344gsumz 16565 . . . . . . . . 9  |-  ( ( R  e.  Mnd  /\  ( 0 ... n
)  e.  _V )  ->  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) )  =  .0.  )
104100, 102, 103syl2anc 665 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( R 
gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) )  =  .0.  )
105104adantr 466 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( R  gsumg  ( k  e.  ( 0 ... n )  |->  .0.  ) )  =  .0.  )
106105adantr 466 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  -> 
( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) )  =  .0.  )
107106adantr 466 . . . . 5  |-  ( ( ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  /\  n  e.  NN )  ->  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) )  =  .0.  )
10820, 99, 1073eqtrd 2465 . . . 4  |-  ( ( ( ( R  e. 
Ring  /\  ( F  e.  B  /\  G  e.  B ) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  /\  n  e.  NN )  ->  ( (coe1 `  ( F  .X.  G ) ) `  n )  =  .0.  )
109108ralrimiva 2837 . . 3  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  ->  A. n  e.  NN  ( (coe1 `  ( F  .X.  G ) ) `  n )  =  .0.  )
110 fveq2 5872 . . . . 5  |-  ( c  =  n  ->  (
(coe1 `  ( F  .X.  G ) ) `  c )  =  ( (coe1 `  ( F  .X.  G ) ) `  n ) )
111110eqeq1d 2422 . . . 4  |-  ( c  =  n  ->  (
( (coe1 `  ( F  .X.  G ) ) `  c )  =  .0.  <->  ( (coe1 `  ( F  .X.  G ) ) `  n )  =  .0.  ) )
112111cbvralv 3053 . . 3  |-  ( A. c  e.  NN  (
(coe1 `  ( F  .X.  G ) ) `  c )  =  .0.  <->  A. n  e.  NN  (
(coe1 `  ( F  .X.  G ) ) `  n )  =  .0.  )
113109, 112sylibr 215 . 2  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  ) )  ->  A. c  e.  NN  ( (coe1 `  ( F  .X.  G ) ) `  c )  =  .0.  )
114113ex 435 1  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( A. c  e.  NN  (
( (coe1 `  F ) `  c )  =  .0. 
/\  ( (coe1 `  G
) `  c )  =  .0.  )  ->  A. c  e.  NN  ( (coe1 `  ( F  .X.  G ) ) `
 c )  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   A.wral 2773   _Vcvv 3078    |-> cmpt 4475   ` cfv 5592  (class class class)co 6296   0cc0 9528    - cmin 9849   NNcn 10598   NN0cn0 10858   ...cfz 11771   Basecbs 15073   .rcmulr 15143   0gc0g 15290    gsumg cgsu 15291   Mndcmnd 16479   Ringcrg 17708  Poly1cpl1 18698  coe1cco1 18699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-ofr 6537  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-fzo 11903  df-seq 12200  df-hash 12502  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-mulr 15156  df-sca 15158  df-vsca 15159  df-tset 15161  df-ple 15162  df-0g 15292  df-gsum 15293  df-mre 15436  df-mrc 15437  df-acs 15439  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-mhm 16526  df-submnd 16527  df-grp 16617  df-minusg 16618  df-mulg 16620  df-ghm 16825  df-cntz 16915  df-cmn 17360  df-abl 17361  df-mgp 17652  df-ur 17664  df-ring 17710  df-psr 18508  df-mpl 18510  df-opsr 18512  df-psr1 18701  df-ply1 18703  df-coe1 18704
This theorem is referenced by:  cpmatmcllem  19666
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