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Theorem cply1mul 18880
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 2450 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
4 cply1mul.b . . . . . . . . . 10  |-  B  =  ( Base `  P
)
51, 2, 3, 4coe1mul 18856 . . . . . . . . 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 1208 . . . . . . . 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 467 . . . . . . 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 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 )  ->  (coe1 `  ( F  .X.  G ) )  =  ( s  e.  NN0  |->  ( R  gsumg  ( k  e.  ( 0 ... s ) 
|->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( s  -  k ) ) ) ) ) ) )
9 oveq2 6296 . . . . . . . . 9  |-  ( s  =  n  ->  (
0 ... s )  =  ( 0 ... n
) )
10 oveq1 6295 . . . . . . . . . . 11  |-  ( s  =  n  ->  (
s  -  k )  =  ( n  -  k ) )
1110fveq2d 5867 . . . . . . . . . 10  |-  ( s  =  n  ->  (
(coe1 `  G ) `  ( s  -  k
) )  =  ( (coe1 `  G ) `  ( n  -  k
) ) )
1211oveq2d 6304 . . . . . . . . 9  |-  ( s  =  n  ->  (
( (coe1 `  F ) `  k ) ( .r
`  R ) ( (coe1 `  G ) `  ( s  -  k
) ) )  =  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) )
139, 12mpteq12dv 4480 . . . . . . . 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 6304 . . . . . . 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 468 . . . . . 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 10873 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  NN0 )
1716adantl 468 . . . . . 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 6316 . . . . . . 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 5952 . . . . 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 2916 . . . . . . . . . 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 6296 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  0  ->  (
n  -  k )  =  ( n  - 
0 ) )
23 nncn 10614 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  e.  NN  ->  n  e.  CC )
2423subid1d 9972 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  NN  ->  (
n  -  0 )  =  n )
2524adantr 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( n  - 
0 )  =  n )
2622, 25sylan9eqr 2506 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( n  -  k )  =  n )
27 simpll 759 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  n  e.  NN )
2826, 27eqeltrd 2528 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( n  -  k )  e.  NN )
29 fveq2 5863 . . . . . . . . . . . . . . . . . . . 20  |-  ( c  =  ( n  -  k )  ->  (
(coe1 `  G ) `  c )  =  ( (coe1 `  G ) `  ( n  -  k
) ) )
3029eqeq1d 2452 . . . . . . . . . . . . . . . . . . 19  |-  ( c  =  ( n  -  k )  ->  (
( (coe1 `  G ) `  c )  =  .0.  <->  ( (coe1 `  G ) `  ( n  -  k
) )  =  .0.  ) )
3130rspcv 3145 . . . . . . . . . . . . . . . . . 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 6296 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( (coe1 `  G ) `  ( n  -  k
) )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  ( ( (coe1 `  F ) `  k ) ( .r
`  R )  .0.  ) )
34 simpll 759 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  (
( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 ) )  ->  R  e.  Ring )
35 simpl 459 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( F  e.  B  /\  G  e.  B )  ->  F  e.  B )
3635adantl 468 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  F  e.  B )
37 elfznn0 11884 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  e.  ( 0 ... n )  ->  k  e.  NN0 )
3837adantl 468 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  k  e.  NN0 )
3938adantr 467 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  k  e.  NN0 )
40 eqid 2450 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (coe1 `  F
)  =  (coe1 `  F
)
41 eqid 2450 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( Base `  R )  =  (
Base `  R )
4240, 4, 1, 41coe1fvalcl 18798 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( F  e.  B  /\  k  e.  NN0 )  -> 
( (coe1 `  F ) `  k )  e.  (
Base `  R )
)
4336, 39, 42syl2an 480 . . . . . . . . . . . . . . . . . . . . . 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 17811 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( R  e.  Ring  /\  (
(coe1 `  F ) `  k )  e.  (
Base `  R )
)  ->  ( (
(coe1 `  F ) `  k ) ( .r
`  R )  .0.  )  =  .0.  )
4634, 43, 45syl2anc 666 . . . . . . . . . . . . . . . . . . . . 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 2506 . . . . . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . . . . . . 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 437 . . . . . . . . . . . . . . . . . 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 438 . . . . . . . . . . . . . 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 468 . . . . . . . . . . . 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 2623 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =/=  0  <->  -.  k  =  0 )
5857biimpri 210 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  k  =  0  -> 
k  =/=  0 )
5958, 37anim12ci 570 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  ( k  e.  NN0  /\  k  =/=  0 ) )
60 elnnne0 10880 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  e.  NN  <->  ( k  e.  NN0  /\  k  =/=  0 ) )
6159, 60sylibr 216 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  k  e.  NN )
62 fveq2 5863 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( c  =  k  ->  (
(coe1 `  F ) `  c )  =  ( (coe1 `  F ) `  k ) )
6362eqeq1d 2452 . . . . . . . . . . . . . . . . . . . . 21  |-  ( c  =  k  ->  (
( (coe1 `  F ) `  c )  =  .0.  <->  ( (coe1 `  F ) `  k )  =  .0.  ) )
6463rspcv 3145 . . . . . . . . . . . . . . . . . . . 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 6295 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( (coe1 `  F ) `  k )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  (  .0.  ( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) )
67 simpll 759 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  R  e.  Ring )
684eleq2i 2520 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( G  e.  B  <->  G  e.  ( Base `  P )
)
6968biimpi 198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( G  e.  B  ->  G  e.  ( Base `  P
) )
7069adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( F  e.  B  /\  G  e.  B )  ->  G  e.  ( Base `  P ) )
7170adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  G  e.  ( Base `  P )
)
72 fznn0sub 11828 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( k  e.  ( 0 ... n )  ->  (
n  -  k )  e.  NN0 )
73 eqid 2450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  (coe1 `  G
)  =  (coe1 `  G
)
74 eqid 2450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( Base `  P )  =  (
Base `  P )
7573, 74, 1, 41coe1fvalcl 18798 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( G  e.  ( Base `  P )  /\  (
n  -  k )  e.  NN0 )  -> 
( (coe1 `  G ) `  ( n  -  k
) )  e.  (
Base `  R )
)
7671, 72, 75syl2an 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  (
(coe1 `  G ) `  ( n  -  k
) )  e.  (
Base `  R )
)
7741, 3, 44ringlz 17810 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( R  e.  Ring  /\  (
(coe1 `  G ) `  ( n  -  k
) )  e.  (
Base `  R )
)  ->  (  .0.  ( .r `  R ) ( (coe1 `  G ) `  ( n  -  k
) ) )  =  .0.  )
7867, 76, 77syl2anc 666 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  (  .0.  ( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
7966, 78sylan9eqr 2506 . . . . . . . . . . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . . . 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 431 . . . . . . . . . . . . . 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 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.  ) ) ) )
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 168 . . . . . . . . . 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 221 . . . . . . . . 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 431 . . . . . . . 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 625 . . . . . . 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 4488 . . . . . 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 6304 . . . . 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 17782 . . . . . . . . 9  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
101 ovex 6316 . . . . . . . . . 10  |-  ( 0 ... n )  e. 
_V
102101a1i 11 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( 0 ... n )  e. 
_V )
10344gsumz 16614 . . . . . . . . 9  |-  ( ( R  e.  Mnd  /\  ( 0 ... n
)  e.  _V )  ->  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) )  =  .0.  )
104100, 102, 103syl2anc 666 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( R 
gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) )  =  .0.  )
105104adantr 467 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( R  gsumg  ( k  e.  ( 0 ... n )  |->  .0.  ) )  =  .0.  )
106105adantr 467 . . . . . 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 467 . . . . 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 2488 . . . 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 2801 . . 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 5863 . . . . 5  |-  ( c  =  n  ->  (
(coe1 `  ( F  .X.  G ) ) `  c )  =  ( (coe1 `  ( F  .X.  G ) ) `  n ) )
111110eqeq1d 2452 . . . 4  |-  ( c  =  n  ->  (
( (coe1 `  ( F  .X.  G ) ) `  c )  =  .0.  <->  ( (coe1 `  ( F  .X.  G ) ) `  n )  =  .0.  ) )
112111cbvralv 3018 . . 3  |-  ( A. c  e.  NN  (
(coe1 `  ( F  .X.  G ) ) `  c )  =  .0.  <->  A. n  e.  NN  (
(coe1 `  ( F  .X.  G ) ) `  n )  =  .0.  )
113109, 112sylibr 216 . 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 436 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 371    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   _Vcvv 3044    |-> cmpt 4460   ` cfv 5581  (class class class)co 6288   0cc0 9536    - cmin 9857   NNcn 10606   NN0cn0 10866   ...cfz 11781   Basecbs 15114   .rcmulr 15184   0gc0g 15331    gsumg cgsu 15332   Mndcmnd 16528   Ringcrg 17773  Poly1cpl1 18763  coe1cco1 18764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-ofr 6529  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-fzo 11913  df-seq 12211  df-hash 12513  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-sca 15199  df-vsca 15200  df-tset 15202  df-ple 15203  df-0g 15333  df-gsum 15334  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-submnd 16576  df-grp 16666  df-minusg 16667  df-mulg 16669  df-ghm 16874  df-cntz 16964  df-cmn 17425  df-abl 17426  df-mgp 17717  df-ur 17729  df-ring 17775  df-psr 18573  df-mpl 18575  df-opsr 18577  df-psr1 18766  df-ply1 18768  df-coe1 18769
This theorem is referenced by:  cpmatmcllem  19735
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