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Theorem cply1mul 31006
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 2454 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
4 cply1mul.b . . . . . . . . . 10  |-  B  =  ( Base `  P
)
51, 2, 3, 4coe1mul 17848 . . . . . . . . 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 1189 . . . . . . . 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 6209 . . . . . . . . 9  |-  ( s  =  n  ->  (
0 ... s )  =  ( 0 ... n
) )
10 oveq1 6208 . . . . . . . . . . 11  |-  ( s  =  n  ->  (
s  -  k )  =  ( n  -  k ) )
1110fveq2d 5804 . . . . . . . . . 10  |-  ( s  =  n  ->  (
(coe1 `  G ) `  ( s  -  k
) )  =  ( (coe1 `  G ) `  ( n  -  k
) ) )
1211oveq2d 6217 . . . . . . . . 9  |-  ( s  =  n  ->  (
( (coe1 `  F ) `  k ) ( .r
`  R ) ( (coe1 `  G ) `  ( s  -  k
) ) )  =  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) )
139, 12mpteq12dv 4479 . . . . . . . 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 6217 . . . . . . 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 10698 . . . . . . 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 6226 . . . . . . 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 5889 . . . . 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 2955 . . . . . . . . . . . . 13  |-  ( 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 6209 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  0  ->  (
n  -  k )  =  ( n  - 
0 ) )
23 nncn 10442 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( n  e.  NN  ->  n  e.  CC )
2423subid1d 9820 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( n  e.  NN  ->  (
n  -  0 )  =  n )
2524adantr 465 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( n  - 
0 )  =  n )
2622, 25sylan9eqr 2517 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( n  -  k )  =  n )
27 simpl 457 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  n  e.  NN )
2827adantr 465 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  n  e.  NN )
2926, 28eqeltrd 2542 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( n  -  k )  e.  NN )
30 fveq2 5800 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( c  =  ( n  -  k )  ->  (
(coe1 `  G ) `  c )  =  ( (coe1 `  G ) `  ( n  -  k
) ) )
3130eqeq1d 2456 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  =  ( n  -  k )  ->  (
( (coe1 `  G ) `  c )  =  .0.  <->  ( (coe1 `  G ) `  ( n  -  k
) )  =  .0.  ) )
3231rspcva 3177 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( n  -  k
)  e.  NN  /\  A. c  e.  NN  (
(coe1 `  G ) `  c )  =  .0.  )  ->  ( (coe1 `  G ) `  (
n  -  k ) )  =  .0.  )
3332ex 434 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( n  -  k )  e.  NN  ->  ( A. c  e.  NN  ( (coe1 `  G ) `  c )  =  .0. 
->  ( (coe1 `  G ) `  ( n  -  k
) )  =  .0.  ) )
3429, 33syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( A. c  e.  NN  (
(coe1 `  G ) `  c )  =  .0. 
->  ( (coe1 `  G ) `  ( n  -  k
) )  =  .0.  ) )
35 oveq2 6209 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( (coe1 `  G ) `  ( n  -  k
) )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  ( ( (coe1 `  F ) `  k ) ( .r
`  R )  .0.  ) )
36 simpl 457 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  R  e.  Ring )
3736adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  (
( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 ) )  ->  R  e.  Ring )
38 simpl 457 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( F  e.  B  /\  G  e.  B )  ->  F  e.  B )
3938adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  F  e.  B )
40 elfznn0 11599 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( k  e.  ( 0 ... n )  ->  k  e.  NN0 )
4140adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  k  e.  NN0 )
4241adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  k  e.  NN0 )
43 eqid 2454 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  (coe1 `  F
)  =  (coe1 `  F
)
44 eqid 2454 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( Base `  R )  =  (
Base `  R )
4543, 4, 1, 44coe1fvalcl 30984 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( F  e.  B  /\  k  e.  NN0 )  -> 
( (coe1 `  F ) `  k )  e.  (
Base `  R )
)
4639, 42, 45syl2an 477 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  (
( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 ) )  ->  (
(coe1 `  F ) `  k )  e.  (
Base `  R )
)
47 cply1mul.0 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  .0.  =  ( 0g `  R )
4844, 3, 47rngrz 16806 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( R  e.  Ring  /\  (
(coe1 `  F ) `  k )  e.  (
Base `  R )
)  ->  ( (
(coe1 `  F ) `  k ) ( .r
`  R )  .0.  )  =  .0.  )
4937, 46, 48syl2anc 661 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( 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.  )
5035, 49sylan9eqr 2517 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( 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.  )
5150ex 434 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( 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.  )
)
5251expcom 435 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 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.  )
) )
5352com23 78 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 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.  )
) )
5434, 53syld 44 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 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.  )
) )
5554com12 31 . . . . . . . . . . . . . . . . . 18  |-  ( 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.  )
) )
5655expd 436 . . . . . . . . . . . . . . . . 17  |-  ( 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.  )
) ) )
5756com24 87 . . . . . . . . . . . . . . . 16  |-  ( 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.  ) ) ) )
5857adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( 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.  ) ) ) )
5958com13 80 . . . . . . . . . . . . . 14  |-  ( 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.  ) ) ) )
60 df-ne 2650 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( k  =/=  0  <->  -.  k  =  0 )
6160biimpri 206 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -.  k  =  0  -> 
k  =/=  0 )
6261, 40anim12ci 567 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  ( k  e.  NN0  /\  k  =/=  0 ) )
63 elnnne0 10705 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  e.  NN  <->  ( k  e.  NN0  /\  k  =/=  0 ) )
6462, 63sylibr 212 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  k  e.  NN )
65 fveq2 5800 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( c  =  k  ->  (
(coe1 `  F ) `  c )  =  ( (coe1 `  F ) `  k ) )
6665eqeq1d 2456 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( c  =  k  ->  (
( (coe1 `  F ) `  c )  =  .0.  <->  ( (coe1 `  F ) `  k )  =  .0.  ) )
6766rspcva 3177 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( k  e.  NN  /\  A. c  e.  NN  (
(coe1 `  F ) `  c )  =  .0.  )  ->  ( (coe1 `  F ) `  k
)  =  .0.  )
6867ex 434 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  e.  NN  ->  ( A. c  e.  NN  ( (coe1 `  F ) `  c )  =  .0. 
->  ( (coe1 `  F ) `  k )  =  .0.  ) )
6964, 68syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  ( A. c  e.  NN  (
(coe1 `  F ) `  c )  =  .0. 
->  ( (coe1 `  F ) `  k )  =  .0.  ) )
70 oveq1 6208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( (coe1 `  F ) `  k )  =  .0. 
->  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  (  .0.  ( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) )
7136adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  R  e.  Ring )
724eleq2i 2532 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( G  e.  B  <->  G  e.  ( Base `  P )
)
7372biimpi 194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( G  e.  B  ->  G  e.  ( Base `  P
) )
7473adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( F  e.  B  /\  G  e.  B )  ->  G  e.  ( Base `  P ) )
7574adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  G  e.  ( Base `  P )
)
76 fznn0sub 11605 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( k  e.  ( 0 ... n )  ->  (
n  -  k )  e.  NN0 )
77 eqid 2454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  (coe1 `  G
)  =  (coe1 `  G
)
78 eqid 2454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( Base `  P )  =  (
Base `  P )
7977, 78, 1, 44coe1fvalcl 30984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( G  e.  ( Base `  P )  /\  (
n  -  k )  e.  NN0 )  -> 
( (coe1 `  G ) `  ( n  -  k
) )  e.  (
Base `  R )
)
8075, 76, 79syl2an 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  (
(coe1 `  G ) `  ( n  -  k
) )  e.  (
Base `  R )
)
8144, 3, 47rnglz 16805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( R  e.  Ring  /\  (
(coe1 `  G ) `  ( n  -  k
) )  e.  (
Base `  R )
)  ->  (  .0.  ( .r `  R ) ( (coe1 `  G ) `  ( n  -  k
) ) )  =  .0.  )
8271, 80, 81syl2anc 661 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B
) )  /\  k  e.  ( 0 ... n
) )  ->  (  .0.  ( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) )  =  .0.  )
8370, 82sylan9eqr 2517 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( 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.  )
8483ex 434 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( 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.  )
)
8584ex 434 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( 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.  )
) )
8685com23 78 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( 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.  ) ) )
8786a1dd 46 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( 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.  ) ) ) )
8887com14 88 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 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.  )
) ) )
8988adantl 466 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( -.  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.  )
) ) )
9069, 89syld 44 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( -.  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.  )
) ) )
9190com24 87 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( -.  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.  )
) ) )
9291ex 434 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  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.  )
) ) ) )
9392com14 88 . . . . . . . . . . . . . . . . . 18  |-  ( 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.  )
) ) ) )
9493imp 429 . . . . . . . . . . . . . . . . 17  |-  ( ( 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.  )
) ) )
9594com14 88 . . . . . . . . . . . . . . . 16  |-  ( 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.  ) ) ) )
9695adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( 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.  ) ) ) )
9796com13 80 . . . . . . . . . . . . . 14  |-  ( -.  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.  ) ) ) )
9859, 97pm2.61i 164 . . . . . . . . . . . . 13  |-  ( ( 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.  ) ) )
9921, 98syl5bi 217 . . . . . . . . . . . 12  |-  ( ( 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.  ) ) )
10099imp 429 . . . . . . . . . . 11  |-  ( ( ( 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.  ) )
101100expd 436 . . . . . . . . . 10  |-  ( ( ( 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.  ) ) )
102101imp 429 . . . . . . . . 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.  ) )
103102imp 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.  )
104103mpteq2dva 4487 . . . . . . 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 ) ) ) )  =  ( k  e.  ( 0 ... n )  |->  .0.  ) )
105104oveq2d 6217 . . . . . 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 ) ) ) ) )  =  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) ) )
106 rngmnd 16778 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
107 ovex 6226 . . . . . . . . . . 11  |-  ( 0 ... n )  e. 
_V
108107a1i 11 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( 0 ... n )  e. 
_V )
10947gsumz 15631 . . . . . . . . . 10  |-  ( ( R  e.  Mnd  /\  ( 0 ... n
)  e.  _V )  ->  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) )  =  .0.  )
110106, 108, 109syl2anc 661 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( R 
gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) )  =  .0.  )
111110adantr 465 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( F  e.  B  /\  G  e.  B )
)  ->  ( R  gsumg  ( k  e.  ( 0 ... n )  |->  .0.  ) )  =  .0.  )
112111adantr 465 . . . . . . 7  |-  ( ( ( 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.  )
113112adantr 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 )  ->  ( R  gsumg  ( k  e.  ( 0 ... n ) 
|->  .0.  ) )  =  .0.  )
114105, 113eqtrd 2495 . . . . 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 ) ) ) ) )  =  .0.  )
11520, 114eqtrd 2495 . . . 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.  )
116115ralrimiva 2830 . . 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.  )
117 fveq2 5800 . . . . 5  |-  ( c  =  n  ->  (
(coe1 `  ( F  .X.  G ) ) `  c )  =  ( (coe1 `  ( F  .X.  G ) ) `  n ) )
118117eqeq1d 2456 . . . 4  |-  ( c  =  n  ->  (
( (coe1 `  ( F  .X.  G ) ) `  c )  =  .0.  <->  ( (coe1 `  ( F  .X.  G ) ) `  n )  =  .0.  ) )
119118cbvralv 3053 . . 3  |-  ( A. c  e.  NN  (
(coe1 `  ( F  .X.  G ) ) `  c )  =  .0.  <->  A. n  e.  NN  (
(coe1 `  ( F  .X.  G ) ) `  n )  =  .0.  )
120116, 119sylibr 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.  )
121120ex 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 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   _Vcvv 3078    |-> cmpt 4459   ` cfv 5527  (class class class)co 6201   0cc0 9394    - cmin 9707   NNcn 10434   NN0cn0 10691   ...cfz 11555   Basecbs 14293   .rcmulr 14359   0gc0g 14498    gsumg cgsu 14499   Mndcmnd 15529   Ringcrg 16769  Poly1cpl1 17758  coe1cco1 17759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-ofr 6432  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-ixp 7375  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-seq 11925  df-hash 12222  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-sca 14374  df-vsca 14375  df-tset 14377  df-ple 14378  df-0g 14500  df-gsum 14501  df-mre 14644  df-mrc 14645  df-acs 14647  df-mnd 15535  df-mhm 15584  df-submnd 15585  df-grp 15665  df-minusg 15666  df-mulg 15668  df-ghm 15865  df-cntz 15955  df-cmn 16401  df-abl 16402  df-mgp 16715  df-ur 16727  df-rng 16771  df-psr 17547  df-mpl 17549  df-opsr 17551  df-psr1 17761  df-ply1 17763  df-coe1 17764
This theorem is referenced by:  cpmatmcllem  31208
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