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Theorem coe1fzgsumd 18214
Description: Value of an evaluated coefficient in a finite group sum of polynomials. (Contributed by AV, 8-Oct-2019.)
Hypotheses
Ref Expression
coe1fzgsumd.p  |-  P  =  (Poly1 `  R )
coe1fzgsumd.b  |-  B  =  ( Base `  P
)
coe1fzgsumd.r  |-  ( ph  ->  R  e.  Ring )
coe1fzgsumd.k  |-  ( ph  ->  K  e.  NN0 )
coe1fzgsumd.m  |-  ( ph  ->  A. x  e.  N  M  e.  B )
coe1fzgsumd.n  |-  ( ph  ->  N  e.  Fin )
Assertion
Ref Expression
coe1fzgsumd  |-  ( ph  ->  ( (coe1 `  ( P  gsumg  ( x  e.  N  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  N  |->  ( (coe1 `  M ) `  K ) ) ) )
Distinct variable groups:    x, B    x, K    x, N
Allowed substitution hints:    ph( x)    P( x)    R( x)    M( x)

Proof of Theorem coe1fzgsumd
Dummy variables  a  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coe1fzgsumd.m . 2  |-  ( ph  ->  A. x  e.  N  M  e.  B )
2 coe1fzgsumd.n . . 3  |-  ( ph  ->  N  e.  Fin )
3 raleq 3063 . . . . . . 7  |-  ( n  =  (/)  ->  ( A. x  e.  n  M  e.  B  <->  A. x  e.  (/)  M  e.  B ) )
43anbi2d 703 . . . . . 6  |-  ( n  =  (/)  ->  ( (
ph  /\  A. x  e.  n  M  e.  B )  <->  ( ph  /\ 
A. x  e.  (/)  M  e.  B ) ) )
5 mpteq1 4533 . . . . . . . . . 10  |-  ( n  =  (/)  ->  ( x  e.  n  |->  M )  =  ( x  e.  (/)  |->  M ) )
65oveq2d 6311 . . . . . . . . 9  |-  ( n  =  (/)  ->  ( P 
gsumg  ( x  e.  n  |->  M ) )  =  ( P  gsumg  ( x  e.  (/)  |->  M ) ) )
76fveq2d 5876 . . . . . . . 8  |-  ( n  =  (/)  ->  (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) )  =  (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) ) )
87fveq1d 5874 . . . . . . 7  |-  ( n  =  (/)  ->  ( (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `
 K )  =  ( (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) ) `  K ) )
9 mpteq1 4533 . . . . . . . 8  |-  ( n  =  (/)  ->  ( x  e.  n  |->  ( (coe1 `  M ) `  K
) )  =  ( x  e.  (/)  |->  ( (coe1 `  M ) `  K
) ) )
109oveq2d 6311 . . . . . . 7  |-  ( n  =  (/)  ->  ( R 
gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) )  =  ( R  gsumg  ( x  e.  (/)  |->  ( (coe1 `  M
) `  K )
) ) )
118, 10eqeq12d 2489 . . . . . 6  |-  ( n  =  (/)  ->  ( ( (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) )  <-> 
( (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) ) `  K )  =  ( R  gsumg  ( x  e.  (/)  |->  ( (coe1 `  M
) `  K )
) ) ) )
124, 11imbi12d 320 . . . . 5  |-  ( n  =  (/)  ->  ( ( ( ph  /\  A. x  e.  n  M  e.  B )  ->  (
(coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) ) )  <->  ( ( ph  /\ 
A. x  e.  (/)  M  e.  B )  -> 
( (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) ) `  K )  =  ( R  gsumg  ( x  e.  (/)  |->  ( (coe1 `  M
) `  K )
) ) ) ) )
13 raleq 3063 . . . . . . 7  |-  ( n  =  m  ->  ( A. x  e.  n  M  e.  B  <->  A. x  e.  m  M  e.  B ) )
1413anbi2d 703 . . . . . 6  |-  ( n  =  m  ->  (
( ph  /\  A. x  e.  n  M  e.  B )  <->  ( ph  /\ 
A. x  e.  m  M  e.  B )
) )
15 mpteq1 4533 . . . . . . . . . 10  |-  ( n  =  m  ->  (
x  e.  n  |->  M )  =  ( x  e.  m  |->  M ) )
1615oveq2d 6311 . . . . . . . . 9  |-  ( n  =  m  ->  ( P  gsumg  ( x  e.  n  |->  M ) )  =  ( P  gsumg  ( x  e.  m  |->  M ) ) )
1716fveq2d 5876 . . . . . . . 8  |-  ( n  =  m  ->  (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) )  =  (coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) )
1817fveq1d 5874 . . . . . . 7  |-  ( n  =  m  ->  (
(coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `  K
)  =  ( (coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) `
 K ) )
19 mpteq1 4533 . . . . . . . 8  |-  ( n  =  m  ->  (
x  e.  n  |->  ( (coe1 `  M ) `  K ) )  =  ( x  e.  m  |->  ( (coe1 `  M ) `  K ) ) )
2019oveq2d 6311 . . . . . . 7  |-  ( n  =  m  ->  ( R  gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) )  =  ( R  gsumg  ( x  e.  m  |->  ( (coe1 `  M ) `  K
) ) ) )
2118, 20eqeq12d 2489 . . . . . 6  |-  ( n  =  m  ->  (
( (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) )  <-> 
( (coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  m  |->  ( (coe1 `  M ) `  K ) ) ) ) )
2214, 21imbi12d 320 . . . . 5  |-  ( n  =  m  ->  (
( ( ph  /\  A. x  e.  n  M  e.  B )  -> 
( (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) ) )  <->  ( ( ph  /\ 
A. x  e.  m  M  e.  B )  ->  ( (coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  m  |->  ( (coe1 `  M ) `  K ) ) ) ) ) )
23 raleq 3063 . . . . . . 7  |-  ( n  =  ( m  u. 
{ a } )  ->  ( A. x  e.  n  M  e.  B 
<-> 
A. x  e.  ( m  u.  { a } ) M  e.  B ) )
2423anbi2d 703 . . . . . 6  |-  ( n  =  ( m  u. 
{ a } )  ->  ( ( ph  /\ 
A. x  e.  n  M  e.  B )  <->  (
ph  /\  A. x  e.  ( m  u.  {
a } ) M  e.  B ) ) )
25 mpteq1 4533 . . . . . . . . . 10  |-  ( n  =  ( m  u. 
{ a } )  ->  ( x  e.  n  |->  M )  =  ( x  e.  ( m  u.  { a } )  |->  M ) )
2625oveq2d 6311 . . . . . . . . 9  |-  ( n  =  ( m  u. 
{ a } )  ->  ( P  gsumg  ( x  e.  n  |->  M ) )  =  ( P 
gsumg  ( x  e.  (
m  u.  { a } )  |->  M ) ) )
2726fveq2d 5876 . . . . . . . 8  |-  ( n  =  ( m  u. 
{ a } )  ->  (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) )  =  (coe1 `  ( P  gsumg  ( x  e.  ( m  u.  { a } )  |->  M ) ) ) )
2827fveq1d 5874 . . . . . . 7  |-  ( n  =  ( m  u. 
{ a } )  ->  ( (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `
 K )  =  ( (coe1 `  ( P  gsumg  ( x  e.  ( m  u. 
{ a } ) 
|->  M ) ) ) `
 K ) )
29 mpteq1 4533 . . . . . . . 8  |-  ( n  =  ( m  u. 
{ a } )  ->  ( x  e.  n  |->  ( (coe1 `  M
) `  K )
)  =  ( x  e.  ( m  u. 
{ a } ) 
|->  ( (coe1 `  M ) `  K ) ) )
3029oveq2d 6311 . . . . . . 7  |-  ( n  =  ( m  u. 
{ a } )  ->  ( R  gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K
) ) )  =  ( R  gsumg  ( x  e.  ( m  u.  { a } )  |->  ( (coe1 `  M ) `  K
) ) ) )
3128, 30eqeq12d 2489 . . . . . 6  |-  ( n  =  ( m  u. 
{ a } )  ->  ( ( (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `
 K )  =  ( R  gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) )  <-> 
( (coe1 `  ( P  gsumg  ( x  e.  ( m  u. 
{ a } ) 
|->  M ) ) ) `
 K )  =  ( R  gsumg  ( x  e.  ( m  u.  { a } )  |->  ( (coe1 `  M ) `  K
) ) ) ) )
3224, 31imbi12d 320 . . . . 5  |-  ( n  =  ( m  u. 
{ a } )  ->  ( ( (
ph  /\  A. x  e.  n  M  e.  B )  ->  (
(coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) ) )  <->  ( ( ph  /\ 
A. x  e.  ( m  u.  { a } ) M  e.  B )  ->  (
(coe1 `  ( P  gsumg  ( x  e.  ( m  u. 
{ a } ) 
|->  M ) ) ) `
 K )  =  ( R  gsumg  ( x  e.  ( m  u.  { a } )  |->  ( (coe1 `  M ) `  K
) ) ) ) ) )
33 raleq 3063 . . . . . . 7  |-  ( n  =  N  ->  ( A. x  e.  n  M  e.  B  <->  A. x  e.  N  M  e.  B ) )
3433anbi2d 703 . . . . . 6  |-  ( n  =  N  ->  (
( ph  /\  A. x  e.  n  M  e.  B )  <->  ( ph  /\ 
A. x  e.  N  M  e.  B )
) )
35 mpteq1 4533 . . . . . . . . . 10  |-  ( n  =  N  ->  (
x  e.  n  |->  M )  =  ( x  e.  N  |->  M ) )
3635oveq2d 6311 . . . . . . . . 9  |-  ( n  =  N  ->  ( P  gsumg  ( x  e.  n  |->  M ) )  =  ( P  gsumg  ( x  e.  N  |->  M ) ) )
3736fveq2d 5876 . . . . . . . 8  |-  ( n  =  N  ->  (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) )  =  (coe1 `  ( P  gsumg  ( x  e.  N  |->  M ) ) ) )
3837fveq1d 5874 . . . . . . 7  |-  ( n  =  N  ->  (
(coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `  K
)  =  ( (coe1 `  ( P  gsumg  ( x  e.  N  |->  M ) ) ) `
 K ) )
39 mpteq1 4533 . . . . . . . 8  |-  ( n  =  N  ->  (
x  e.  n  |->  ( (coe1 `  M ) `  K ) )  =  ( x  e.  N  |->  ( (coe1 `  M ) `  K ) ) )
4039oveq2d 6311 . . . . . . 7  |-  ( n  =  N  ->  ( R  gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) )  =  ( R  gsumg  ( x  e.  N  |->  ( (coe1 `  M ) `  K
) ) ) )
4138, 40eqeq12d 2489 . . . . . 6  |-  ( n  =  N  ->  (
( (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) )  <-> 
( (coe1 `  ( P  gsumg  ( x  e.  N  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  N  |->  ( (coe1 `  M ) `  K ) ) ) ) )
4234, 41imbi12d 320 . . . . 5  |-  ( n  =  N  ->  (
( ( ph  /\  A. x  e.  n  M  e.  B )  -> 
( (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) ) )  <->  ( ( ph  /\ 
A. x  e.  N  M  e.  B )  ->  ( (coe1 `  ( P  gsumg  ( x  e.  N  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  N  |->  ( (coe1 `  M ) `  K ) ) ) ) ) )
43 mpt0 5714 . . . . . . . . . . . . 13  |-  ( x  e.  (/)  |->  M )  =  (/)
4443oveq2i 6306 . . . . . . . . . . . 12  |-  ( P 
gsumg  ( x  e.  (/)  |->  M ) )  =  ( P 
gsumg  (/) )
45 eqid 2467 . . . . . . . . . . . . 13  |-  ( 0g
`  P )  =  ( 0g `  P
)
4645gsum0 15779 . . . . . . . . . . . 12  |-  ( P 
gsumg  (/) )  =  ( 0g
`  P )
4744, 46eqtri 2496 . . . . . . . . . . 11  |-  ( P 
gsumg  ( x  e.  (/)  |->  M ) )  =  ( 0g
`  P )
4847fveq2i 5875 . . . . . . . . . 10  |-  (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) )  =  (coe1 `  ( 0g `  P ) )
4948a1i 11 . . . . . . . . 9  |-  ( ph  ->  (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) )  =  (coe1 `  ( 0g `  P ) ) )
5049fveq1d 5874 . . . . . . . 8  |-  ( ph  ->  ( (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) ) `  K )  =  ( (coe1 `  ( 0g `  P ) ) `
 K ) )
51 coe1fzgsumd.r . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
52 coe1fzgsumd.p . . . . . . . . . . 11  |-  P  =  (Poly1 `  R )
53 eqid 2467 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
5452, 45, 53coe1z 18174 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (coe1 `  ( 0g `  P ) )  =  ( NN0  X.  { ( 0g `  R ) } ) )
5551, 54syl 16 . . . . . . . . 9  |-  ( ph  ->  (coe1 `  ( 0g `  P ) )  =  ( NN0  X.  {
( 0g `  R
) } ) )
5655fveq1d 5874 . . . . . . . 8  |-  ( ph  ->  ( (coe1 `  ( 0g `  P ) ) `  K )  =  ( ( NN0  X.  {
( 0g `  R
) } ) `  K ) )
57 fvex 5882 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
58 coe1fzgsumd.k . . . . . . . . 9  |-  ( ph  ->  K  e.  NN0 )
59 fvconst2g 6125 . . . . . . . . 9  |-  ( ( ( 0g `  R
)  e.  _V  /\  K  e.  NN0 )  -> 
( ( NN0  X.  { ( 0g `  R ) } ) `
 K )  =  ( 0g `  R
) )
6057, 58, 59sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( ( NN0  X.  { ( 0g `  R ) } ) `
 K )  =  ( 0g `  R
) )
6150, 56, 603eqtrd 2512 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) ) `  K )  =  ( 0g `  R ) )
62 mpt0 5714 . . . . . . . . 9  |-  ( x  e.  (/)  |->  ( (coe1 `  M
) `  K )
)  =  (/)
6362oveq2i 6306 . . . . . . . 8  |-  ( R 
gsumg  ( x  e.  (/)  |->  ( (coe1 `  M ) `  K
) ) )  =  ( R  gsumg  (/) )
6453gsum0 15779 . . . . . . . 8  |-  ( R 
gsumg  (/) )  =  ( 0g
`  R )
6563, 64eqtri 2496 . . . . . . 7  |-  ( R 
gsumg  ( x  e.  (/)  |->  ( (coe1 `  M ) `  K
) ) )  =  ( 0g `  R
)
6661, 65syl6eqr 2526 . . . . . 6  |-  ( ph  ->  ( (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) ) `  K )  =  ( R  gsumg  ( x  e.  (/)  |->  ( (coe1 `  M
) `  K )
) ) )
6766adantr 465 . . . . 5  |-  ( (
ph  /\  A. x  e.  (/)  M  e.  B
)  ->  ( (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) ) `
 K )  =  ( R  gsumg  ( x  e.  (/)  |->  ( (coe1 `  M ) `  K ) ) ) )
68 coe1fzgsumd.b . . . . . . . . 9  |-  B  =  ( Base `  P
)
6952, 68, 51, 58coe1fzgsumdlem 18213 . . . . . . . 8  |-  ( ( m  e.  Fin  /\  -.  a  e.  m  /\  ph )  ->  (
( A. x  e.  m  M  e.  B  ->  ( (coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  m  |->  ( (coe1 `  M ) `  K ) ) ) )  ->  ( A. x  e.  ( m  u.  { a } ) M  e.  B  -> 
( (coe1 `  ( P  gsumg  ( x  e.  ( m  u. 
{ a } ) 
|->  M ) ) ) `
 K )  =  ( R  gsumg  ( x  e.  ( m  u.  { a } )  |->  ( (coe1 `  M ) `  K
) ) ) ) ) )
70693expia 1198 . . . . . . 7  |-  ( ( m  e.  Fin  /\  -.  a  e.  m
)  ->  ( ph  ->  ( ( A. x  e.  m  M  e.  B  ->  ( (coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) `
 K )  =  ( R  gsumg  ( x  e.  m  |->  ( (coe1 `  M ) `  K ) ) ) )  ->  ( A. x  e.  ( m  u.  { a } ) M  e.  B  -> 
( (coe1 `  ( P  gsumg  ( x  e.  ( m  u. 
{ a } ) 
|->  M ) ) ) `
 K )  =  ( R  gsumg  ( x  e.  ( m  u.  { a } )  |->  ( (coe1 `  M ) `  K
) ) ) ) ) ) )
7170a2d 26 . . . . . 6  |-  ( ( m  e.  Fin  /\  -.  a  e.  m
)  ->  ( ( ph  ->  ( A. x  e.  m  M  e.  B  ->  ( (coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) `
 K )  =  ( R  gsumg  ( x  e.  m  |->  ( (coe1 `  M ) `  K ) ) ) ) )  ->  ( ph  ->  ( A. x  e.  ( m  u.  {
a } ) M  e.  B  ->  (
(coe1 `  ( P  gsumg  ( x  e.  ( m  u. 
{ a } ) 
|->  M ) ) ) `
 K )  =  ( R  gsumg  ( x  e.  ( m  u.  { a } )  |->  ( (coe1 `  M ) `  K
) ) ) ) ) ) )
72 impexp 446 . . . . . 6  |-  ( ( ( ph  /\  A. x  e.  m  M  e.  B )  ->  (
(coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  m  |->  ( (coe1 `  M ) `  K ) ) ) )  <->  ( ph  ->  ( A. x  e.  m  M  e.  B  ->  ( (coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  m  |->  ( (coe1 `  M ) `  K ) ) ) ) ) )
73 impexp 446 . . . . . 6  |-  ( ( ( ph  /\  A. x  e.  ( m  u.  { a } ) M  e.  B )  ->  ( (coe1 `  ( P  gsumg  ( x  e.  ( m  u.  { a } )  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  (
m  u.  { a } )  |->  ( (coe1 `  M ) `  K
) ) ) )  <-> 
( ph  ->  ( A. x  e.  ( m  u.  { a } ) M  e.  B  -> 
( (coe1 `  ( P  gsumg  ( x  e.  ( m  u. 
{ a } ) 
|->  M ) ) ) `
 K )  =  ( R  gsumg  ( x  e.  ( m  u.  { a } )  |->  ( (coe1 `  M ) `  K
) ) ) ) ) )
7471, 72, 733imtr4g 270 . . . . 5  |-  ( ( m  e.  Fin  /\  -.  a  e.  m
)  ->  ( (
( ph  /\  A. x  e.  m  M  e.  B )  ->  (
(coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  m  |->  ( (coe1 `  M ) `  K ) ) ) )  ->  ( ( ph  /\  A. x  e.  ( m  u.  {
a } ) M  e.  B )  -> 
( (coe1 `  ( P  gsumg  ( x  e.  ( m  u. 
{ a } ) 
|->  M ) ) ) `
 K )  =  ( R  gsumg  ( x  e.  ( m  u.  { a } )  |->  ( (coe1 `  M ) `  K
) ) ) ) ) )
7512, 22, 32, 42, 67, 74findcard2s 7773 . . . 4  |-  ( N  e.  Fin  ->  (
( ph  /\  A. x  e.  N  M  e.  B )  ->  (
(coe1 `  ( P  gsumg  ( x  e.  N  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  N  |->  ( (coe1 `  M ) `  K ) ) ) ) )
7675expd 436 . . 3  |-  ( N  e.  Fin  ->  ( ph  ->  ( A. x  e.  N  M  e.  B  ->  ( (coe1 `  ( P  gsumg  ( x  e.  N  |->  M ) ) ) `
 K )  =  ( R  gsumg  ( x  e.  N  |->  ( (coe1 `  M ) `  K ) ) ) ) ) )
772, 76mpcom 36 . 2  |-  ( ph  ->  ( A. x  e.  N  M  e.  B  ->  ( (coe1 `  ( P  gsumg  ( x  e.  N  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  N  |->  ( (coe1 `  M ) `  K ) ) ) ) )
781, 77mpd 15 1  |-  ( ph  ->  ( (coe1 `  ( P  gsumg  ( x  e.  N  |->  M ) ) ) `  K
)  =  ( R 
gsumg  ( x  e.  N  |->  ( (coe1 `  M ) `  K ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    u. cun 3479   (/)c0 3790   {csn 4033    |-> cmpt 4511    X. cxp 5003   ` cfv 5594  (class class class)co 6295   Fincfn 7528   NN0cn0 10807   Basecbs 14507   0gc0g 14712    gsumg cgsu 14713   Ringcrg 17070  Poly1cpl1 18086  coe1cco1 18087
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-ofr 6536  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-tset 14591  df-ple 14592  df-0g 14714  df-gsum 14715  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-mulg 15932  df-subg 16070  df-ghm 16137  df-cntz 16227  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-subrg 17298  df-psr 17875  df-mpl 17877  df-opsr 17879  df-psr1 18089  df-ply1 18091  df-coe1 18092
This theorem is referenced by:  gsummoncoe1  18216  cpmatmcllem  19088  decpmatmullem  19141  mp2pm2mplem4  19179
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