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Theorem cply1mul 18461
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 2457 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
4 cply1mul.b . . . . . . . . . 10  |-  B  =  ( Base `  P
)
51, 2, 3, 4coe1mul 18437 . . . . . . . . 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 1197 . . . . . . . 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 465 . . . . . . 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 465 . . . . . 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 4535 . . . . . . . 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 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 )  /\  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 10823 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  NN0 )
1716adantl 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 )  ->  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 2984 . . . . . . . . . 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 10564 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  e.  NN  ->  n  e.  CC )
2423subid1d 9939 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  NN  ->  (
n  -  0 )  =  n )
2524adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( n  - 
0 )  =  n )
2622, 25sylan9eqr 2520 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( n  -  k )  =  n )
27 simpll 753 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  n  e.  NN )
2826, 27eqeltrd 2545 . . . . . . . . . . . . . . . . . 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 2459 . . . . . . . . . . . . . . . . . . 19  |-  ( c  =  ( n  -  k )  ->  (
( (coe1 `  G ) `  c )  =  .0.  <->  ( (coe1 `  G ) `  ( n  -  k
) )  =  .0.  ) )
3130rspcv 3206 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  -  k )  e.  NN  ->  ( A. c  e.  NN  ( (coe1 `  G ) `  c )  =  .0. 
->  ( (coe1 `  G ) `  ( n  -  k
) )  =  .0.  ) )
3228, 31syl 16 . . . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  (
( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 ) )  ->  R  e.  Ring )
35 simpl 457 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( F  e.  B  /\  G  e.  B )  ->  F  e.  B )
3635adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  F  e.  B )
37 elfznn0 11796 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  e.  ( 0 ... n )  ->  k  e.  NN0 )
3837adantl 466 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  k  e.  NN0 )
3938adantr 465 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  k  e.  NN0 )
40 eqid 2457 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (coe1 `  F
)  =  (coe1 `  F
)
41 eqid 2457 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( Base `  R )  =  (
Base `  R )
4240, 4, 1, 41coe1fvalcl 18377 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( F  e.  B  /\  k  e.  NN0 )  -> 
( (coe1 `  F ) `  k )  e.  (
Base `  R )
)
4336, 39, 42syl2an 477 . . . . . . . . . . . . . . . . . . . . . 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 17362 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( R  e.  Ring  /\  (
(coe1 `  F ) `  k )  e.  (
Base `  R )
)  ->  ( (
(coe1 `  F ) `  k ) ( .r
`  R )  .0.  )  =  .0.  )
4634, 43, 45syl2anc 661 . . . . . . . . . . . . . . . . . . . . 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 2520 . . . . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . . . . . 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 78 . . . . . . . . . . . . . . . . 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 44 . . . . . . . . . . . . . . . 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 31 . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . 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 87 . . . . . . . . . . . . 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 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.  ) ) ) )
5655com13 80 . . . . . . . . . . 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 2654 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =/=  0  <->  -.  k  =  0 )
5857biimpri 206 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  k  =  0  -> 
k  =/=  0 )
5958, 37anim12ci 567 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  ( k  e.  NN0  /\  k  =/=  0 ) )
60 elnnne0 10830 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  e.  NN  <->  ( k  e.  NN0  /\  k  =/=  0 ) )
6159, 60sylibr 212 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  k  e.  NN )
62 fveq2 5872 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( c  =  k  ->  (
(coe1 `  F ) `  c )  =  ( (coe1 `  F ) `  k ) )
6362eqeq1d 2459 . . . . . . . . . . . . . . . . . . . . 21  |-  ( c  =  k  ->  (
( (coe1 `  F ) `  c )  =  .0.  <->  ( (coe1 `  F ) `  k )  =  .0.  ) )
6463rspcv 3206 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  e.  NN  ->  ( A. c  e.  NN  ( (coe1 `  F ) `  c )  =  .0. 
->  ( (coe1 `  F ) `  k )  =  .0.  ) )
6561, 64syl 16 . . . . . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  R  e.  Ring )
684eleq2i 2535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( G  e.  B  <->  G  e.  ( Base `  P )
)
6968biimpi 194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( G  e.  B  ->  G  e.  ( Base `  P
) )
7069adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( F  e.  B  /\  G  e.  B )  ->  G  e.  ( Base `  P ) )
7170adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  G  e.  ( Base `  P )
)
72 fznn0sub 11741 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( k  e.  ( 0 ... n )  ->  (
n  -  k )  e.  NN0 )
73 eqid 2457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  (coe1 `  G
)  =  (coe1 `  G
)
74 eqid 2457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( Base `  P )  =  (
Base `  P )
7573, 74, 1, 41coe1fvalcl 18377 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( G  e.  ( Base `  P )  /\  (
n  -  k )  e.  NN0 )  -> 
( (coe1 `  G ) `  ( n  -  k
) )  e.  (
Base `  R )
)
7671, 72, 75syl2an 477 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  (
(coe1 `  G ) `  ( n  -  k
) )  e.  (
Base `  R )
)
7741, 3, 44ringlz 17361 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( R  e.  Ring  /\  (
(coe1 `  G ) `  ( n  -  k
) )  e.  (
Base `  R )
)  ->  (  .0.  ( .r `  R ) ( (coe1 `  G ) `  ( n  -  k
) ) )  =  .0.  )
7867, 76, 77syl2anc 661 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  (  .0.  ( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
7966, 78sylan9eqr 2520 . . . . . . . . . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . . . . . . . . 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 78 . . . . . . . . . . . . . . . . . . . . . 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 46 . . . . . . . . . . . . . . . . . . . . 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 88 . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . 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 44 . . . . . . . . . . . . . . . . . 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 87 . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . 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 88 . . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . . 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 88 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 80 . . . . . . . . . . 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 164 . . . . . . . . . 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 217 . . . . . . . . 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 429 . . . . . . . 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 620 . . . . . . 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 4543 . . . . . 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 17333 . . . . . . . . 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 16131 . . . . . . . . 9  |-  ( ( R  e.  Mnd  /\  ( 0 ... n
)  e.  _V )  ->  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) )  =  .0.  )
104100, 102, 103syl2anc 661 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( R 
gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) )  =  .0.  )
105104adantr 465 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( R  gsumg  ( k  e.  ( 0 ... n )  |->  .0.  ) )  =  .0.  )
106105adantr 465 . . . . . 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 465 . . . . 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 2502 . . . 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 2871 . . 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 2459 . . . 4  |-  ( c  =  n  ->  (
( (coe1 `  ( F  .X.  G ) ) `  c )  =  .0.  <->  ( (coe1 `  ( F  .X.  G ) ) `  n )  =  .0.  ) )
112111cbvralv 3084 . . 3  |-  ( A. c  e.  NN  (
(coe1 `  ( F  .X.  G ) ) `  c )  =  .0.  <->  A. n  e.  NN  (
(coe1 `  ( F  .X.  G ) ) `  n )  =  .0.  )
113109, 112sylibr 212 . 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 434 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 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   _Vcvv 3109    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   0cc0 9509    - cmin 9824   NNcn 10556   NN0cn0 10816   ...cfz 11697   Basecbs 14643   .rcmulr 14712   0gc0g 14856    gsumg cgsu 14857   Mndcmnd 16045   Ringcrg 17324  Poly1cpl1 18342  coe1cco1 18343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-seq 12110  df-hash 12408  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-sca 14727  df-vsca 14728  df-tset 14730  df-ple 14731  df-0g 14858  df-gsum 14859  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-submnd 16093  df-grp 16183  df-minusg 16184  df-mulg 16186  df-ghm 16391  df-cntz 16481  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-psr 18131  df-mpl 18133  df-opsr 18135  df-psr1 18345  df-ply1 18347  df-coe1 18348
This theorem is referenced by:  cpmatmcllem  19345
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