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Theorem coe1fzgsumd 30838
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 2917 . . . . . . 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 4372 . . . . . . . . . 10  |-  ( n  =  (/)  ->  ( x  e.  n  |->  M )  =  ( x  e.  (/)  |->  M ) )
65oveq2d 6107 . . . . . . . . 9  |-  ( n  =  (/)  ->  ( P 
gsumg  ( x  e.  n  |->  M ) )  =  ( P  gsumg  ( x  e.  (/)  |->  M ) ) )
76fveq2d 5695 . . . . . . . 8  |-  ( n  =  (/)  ->  (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) )  =  (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) ) )
87fveq1d 5693 . . . . . . 7  |-  ( n  =  (/)  ->  ( (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `
 K )  =  ( (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) ) `  K ) )
9 mpteq1 4372 . . . . . . . 8  |-  ( n  =  (/)  ->  ( x  e.  n  |->  ( (coe1 `  M ) `  K
) )  =  ( x  e.  (/)  |->  ( (coe1 `  M ) `  K
) ) )
109oveq2d 6107 . . . . . . 7  |-  ( n  =  (/)  ->  ( R 
gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) )  =  ( R  gsumg  ( x  e.  (/)  |->  ( (coe1 `  M
) `  K )
) ) )
118, 10eqeq12d 2457 . . . . . 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 2917 . . . . . . 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 4372 . . . . . . . . . 10  |-  ( n  =  m  ->  (
x  e.  n  |->  M )  =  ( x  e.  m  |->  M ) )
1615oveq2d 6107 . . . . . . . . 9  |-  ( n  =  m  ->  ( P  gsumg  ( x  e.  n  |->  M ) )  =  ( P  gsumg  ( x  e.  m  |->  M ) ) )
1716fveq2d 5695 . . . . . . . 8  |-  ( n  =  m  ->  (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) )  =  (coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) )
1817fveq1d 5693 . . . . . . 7  |-  ( n  =  m  ->  (
(coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `  K
)  =  ( (coe1 `  ( P  gsumg  ( x  e.  m  |->  M ) ) ) `
 K ) )
19 mpteq1 4372 . . . . . . . 8  |-  ( n  =  m  ->  (
x  e.  n  |->  ( (coe1 `  M ) `  K ) )  =  ( x  e.  m  |->  ( (coe1 `  M ) `  K ) ) )
2019oveq2d 6107 . . . . . . 7  |-  ( n  =  m  ->  ( R  gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) )  =  ( R  gsumg  ( x  e.  m  |->  ( (coe1 `  M ) `  K
) ) ) )
2118, 20eqeq12d 2457 . . . . . 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 2917 . . . . . . 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 4372 . . . . . . . . . 10  |-  ( n  =  ( m  u. 
{ a } )  ->  ( x  e.  n  |->  M )  =  ( x  e.  ( m  u.  { a } )  |->  M ) )
2625oveq2d 6107 . . . . . . . . 9  |-  ( n  =  ( m  u. 
{ a } )  ->  ( P  gsumg  ( x  e.  n  |->  M ) )  =  ( P 
gsumg  ( x  e.  (
m  u.  { a } )  |->  M ) ) )
2726fveq2d 5695 . . . . . . . 8  |-  ( n  =  ( m  u. 
{ a } )  ->  (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) )  =  (coe1 `  ( P  gsumg  ( x  e.  ( m  u.  { a } )  |->  M ) ) ) )
2827fveq1d 5693 . . . . . . 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 4372 . . . . . . . 8  |-  ( n  =  ( m  u. 
{ a } )  ->  ( x  e.  n  |->  ( (coe1 `  M
) `  K )
)  =  ( x  e.  ( m  u. 
{ a } ) 
|->  ( (coe1 `  M ) `  K ) ) )
3029oveq2d 6107 . . . . . . 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 2457 . . . . . 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 2917 . . . . . . 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 4372 . . . . . . . . . 10  |-  ( n  =  N  ->  (
x  e.  n  |->  M )  =  ( x  e.  N  |->  M ) )
3635oveq2d 6107 . . . . . . . . 9  |-  ( n  =  N  ->  ( P  gsumg  ( x  e.  n  |->  M ) )  =  ( P  gsumg  ( x  e.  N  |->  M ) ) )
3736fveq2d 5695 . . . . . . . 8  |-  ( n  =  N  ->  (coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) )  =  (coe1 `  ( P  gsumg  ( x  e.  N  |->  M ) ) ) )
3837fveq1d 5693 . . . . . . 7  |-  ( n  =  N  ->  (
(coe1 `  ( P  gsumg  ( x  e.  n  |->  M ) ) ) `  K
)  =  ( (coe1 `  ( P  gsumg  ( x  e.  N  |->  M ) ) ) `
 K ) )
39 mpteq1 4372 . . . . . . . 8  |-  ( n  =  N  ->  (
x  e.  n  |->  ( (coe1 `  M ) `  K ) )  =  ( x  e.  N  |->  ( (coe1 `  M ) `  K ) ) )
4039oveq2d 6107 . . . . . . 7  |-  ( n  =  N  ->  ( R  gsumg  ( x  e.  n  |->  ( (coe1 `  M ) `  K ) ) )  =  ( R  gsumg  ( x  e.  N  |->  ( (coe1 `  M ) `  K
) ) ) )
4138, 40eqeq12d 2457 . . . . . 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 5538 . . . . . . . . . . . . 13  |-  ( x  e.  (/)  |->  M )  =  (/)
4443oveq2i 6102 . . . . . . . . . . . 12  |-  ( P 
gsumg  ( x  e.  (/)  |->  M ) )  =  ( P 
gsumg  (/) )
45 eqid 2443 . . . . . . . . . . . . 13  |-  ( 0g
`  P )  =  ( 0g `  P
)
4645gsum0 15510 . . . . . . . . . . . 12  |-  ( P 
gsumg  (/) )  =  ( 0g
`  P )
4744, 46eqtri 2463 . . . . . . . . . . 11  |-  ( P 
gsumg  ( x  e.  (/)  |->  M ) )  =  ( 0g
`  P )
4847fveq2i 5694 . . . . . . . . . 10  |-  (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) )  =  (coe1 `  ( 0g `  P ) )
4948a1i 11 . . . . . . . . 9  |-  ( ph  ->  (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) )  =  (coe1 `  ( 0g `  P ) ) )
5049fveq1d 5693 . . . . . . . 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 2443 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
5452, 45, 53coe1z 17717 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (coe1 `  ( 0g `  P ) )  =  ( NN0  X.  { ( 0g `  R ) } ) )
5551, 54syl 16 . . . . . . . . 9  |-  ( ph  ->  (coe1 `  ( 0g `  P ) )  =  ( NN0  X.  {
( 0g `  R
) } ) )
5655fveq1d 5693 . . . . . . . 8  |-  ( ph  ->  ( (coe1 `  ( 0g `  P ) ) `  K )  =  ( ( NN0  X.  {
( 0g `  R
) } ) `  K ) )
57 fvex 5701 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
58 coe1fzgsumd.k . . . . . . . . 9  |-  ( ph  ->  K  e.  NN0 )
59 fvconst2g 5931 . . . . . . . . 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 2479 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( P  gsumg  ( x  e.  (/)  |->  M ) ) ) `  K )  =  ( 0g `  R ) )
62 mpt0 5538 . . . . . . . . 9  |-  ( x  e.  (/)  |->  ( (coe1 `  M
) `  K )
)  =  (/)
6362oveq2i 6102 . . . . . . . 8  |-  ( R 
gsumg  ( x  e.  (/)  |->  ( (coe1 `  M ) `  K
) ) )  =  ( R  gsumg  (/) )
6453gsum0 15510 . . . . . . . 8  |-  ( R 
gsumg  (/) )  =  ( 0g
`  R )
6563, 64eqtri 2463 . . . . . . 7  |-  ( R 
gsumg  ( x  e.  (/)  |->  ( (coe1 `  M ) `  K
) ) )  =  ( 0g `  R
)
6661, 65syl6eqr 2493 . . . . . 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 30837 . . . . . . . 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 1189 . . . . . . 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 7553 . . . 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 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972    u. cun 3326   (/)c0 3637   {csn 3877    e. cmpt 4350    X. cxp 4838   ` cfv 5418  (class class class)co 6091   Fincfn 7310   NN0cn0 10579   Basecbs 14174   0gc0g 14378    gsumg cgsu 14379   Ringcrg 16645  Poly1cpl1 17633  coe1cco1 17634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-ofr 6321  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-tset 14257  df-ple 14258  df-0g 14380  df-gsum 14381  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-mhm 15464  df-submnd 15465  df-grp 15545  df-minusg 15546  df-mulg 15548  df-subg 15678  df-ghm 15745  df-cntz 15835  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-subrg 16863  df-psr 17423  df-mpl 17425  df-opsr 17427  df-psr1 17636  df-ply1 17638  df-coe1 17639
This theorem is referenced by:  gsummoncoe1  30843  pmatcollpw1dstlem1  30900  mp2pm2mplem4  30919
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