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Theorem cply1mul 18103
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 2467 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
4 cply1mul.b . . . . . . . . . 10  |-  B  =  ( Base `  P
)
51, 2, 3, 4coe1mul 18079 . . . . . . . . 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 6290 . . . . . . . . 9  |-  ( s  =  n  ->  (
0 ... s )  =  ( 0 ... n
) )
10 oveq1 6289 . . . . . . . . . . 11  |-  ( s  =  n  ->  (
s  -  k )  =  ( n  -  k ) )
1110fveq2d 5868 . . . . . . . . . 10  |-  ( s  =  n  ->  (
(coe1 `  G ) `  ( s  -  k
) )  =  ( (coe1 `  G ) `  ( n  -  k
) ) )
1211oveq2d 6298 . . . . . . . . 9  |-  ( s  =  n  ->  (
( (coe1 `  F ) `  k ) ( .r
`  R ) ( (coe1 `  G ) `  ( s  -  k
) ) )  =  ( ( (coe1 `  F
) `  k )
( .r `  R
) ( (coe1 `  G
) `  ( n  -  k ) ) ) )
139, 12mpteq12dv 4525 . . . . . . . 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 6298 . . . . . . 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 10798 . . . . . . 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 6307 . . . . . . 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 5953 . . . . 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 2989 . . . . . . . . . . . . 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 6290 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  0  ->  (
n  -  k )  =  ( n  - 
0 ) )
23 nncn 10540 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( n  e.  NN  ->  n  e.  CC )
2423subid1d 9915 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( n  e.  NN  ->  (
n  -  0 )  =  n )
2524adantr 465 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  ->  ( n  - 
0 )  =  n )
2622, 25sylan9eqr 2530 . . . . . . . . . . . . . . . . . . . . . 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 2555 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( n  e.  NN  /\  k  e.  ( 0 ... n ) )  /\  k  =  0 )  ->  ( n  -  k )  e.  NN )
30 fveq2 5864 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( c  =  ( n  -  k )  ->  (
(coe1 `  G ) `  c )  =  ( (coe1 `  G ) `  ( n  -  k
) ) )
3130eqeq1d 2469 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  =  ( n  -  k )  ->  (
( (coe1 `  G ) `  c )  =  .0.  <->  ( (coe1 `  G ) `  ( n  -  k
) )  =  .0.  ) )
3231rspcva 3212 . . . . . . . . . . . . . . . . . . . . . 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 6290 . . . . . . . . . . . . . . . . . . . . . . . 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 11766 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  (coe1 `  F
)  =  (coe1 `  F
)
44 eqid 2467 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( Base `  R )  =  (
Base `  R )
4543, 4, 1, 44coe1fvalcl 18019 . . . . . . . . . . . . . . . . . . . . . . . . . 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 17020 . . . . . . . . . . . . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . . . . . . . . . . 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 2664 . . . . . . . . . . . . . . . . . . . . . . . . . 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 10805 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  e.  NN  <->  ( k  e.  NN0  /\  k  =/=  0 ) )
6462, 63sylibr 212 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( -.  k  =  0  /\  k  e.  ( 0 ... n ) )  ->  k  e.  NN )
65 fveq2 5864 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( c  =  k  ->  (
(coe1 `  F ) `  c )  =  ( (coe1 `  F ) `  k ) )
6665eqeq1d 2469 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( c  =  k  ->  (
( (coe1 `  F ) `  c )  =  .0.  <->  ( (coe1 `  F ) `  k )  =  .0.  ) )
6766rspcva 3212 . . . . . . . . . . . . . . . . . . . . . . . 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 6289 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( k  e.  ( 0 ... n )  ->  (
n  -  k )  e.  NN0 )
77 eqid 2467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  (coe1 `  G
)  =  (coe1 `  G
)
78 eqid 2467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( Base `  P )  =  (
Base `  P )
7977, 78, 1, 44coe1fvalcl 18019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 17019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4533 . . . . . . 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 6298 . . . . . 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 16992 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
107 ovex 6307 . . . . . . . . . . 11  |-  ( 0 ... n )  e. 
_V
108107a1i 11 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( 0 ... n )  e. 
_V )
10947gsumz 15821 . . . . . . . . . 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 2508 . . . . 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 2508 . . . 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 2878 . . 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 5864 . . . . 5  |-  ( c  =  n  ->  (
(coe1 `  ( F  .X.  G ) ) `  c )  =  ( (coe1 `  ( F  .X.  G ) ) `  n ) )
118117eqeq1d 2469 . . . 4  |-  ( c  =  n  ->  (
( (coe1 `  ( F  .X.  G ) ) `  c )  =  .0.  <->  ( (coe1 `  ( F  .X.  G ) ) `  n )  =  .0.  ) )
119118cbvralv 3088 . . 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282   0cc0 9488    - cmin 9801   NNcn 10532   NN0cn0 10791   ...cfz 11668   Basecbs 14483   .rcmulr 14549   0gc0g 14688    gsumg cgsu 14689   Mndcmnd 15719   Ringcrg 16983  Poly1cpl1 17984  coe1cco1 17985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-ofr 6523  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12071  df-hash 12368  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-sca 14564  df-vsca 14565  df-tset 14567  df-ple 14568  df-0g 14690  df-gsum 14691  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-mhm 15774  df-submnd 15775  df-grp 15855  df-minusg 15856  df-mulg 15858  df-ghm 16057  df-cntz 16147  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-rng 16985  df-psr 17773  df-mpl 17775  df-opsr 17777  df-psr1 17987  df-ply1 17989  df-coe1 17990
This theorem is referenced by:  cpmatmcllem  18983
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