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Theorem coe1tmmul 16353
Description: Coefficient vector of a polynomial multiplied on the left by a term. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypotheses
Ref Expression
coe1tm.z  |-  .0.  =  ( 0g `  R )
coe1tm.k  |-  K  =  ( Base `  R
)
coe1tm.p  |-  P  =  (Poly1 `  R )
coe1tm.x  |-  X  =  (var1 `  R )
coe1tm.m  |-  .x.  =  ( .s `  P )
coe1tm.n  |-  N  =  (mulGrp `  P )
coe1tm.e  |-  .^  =  (.g
`  N )
coe1tmmul.b  |-  B  =  ( Base `  P
)
coe1tmmul.t  |-  .xb  =  ( .r `  P )
coe1tmmul.u  |-  .X.  =  ( .r `  R )
coe1tmmul.a  |-  ( ph  ->  A  e.  B )
coe1tmmul.r  |-  ( ph  ->  R  e.  Ring )
coe1tmmul.c  |-  ( ph  ->  C  e.  K )
coe1tmmul.d  |-  ( ph  ->  D  e.  NN0 )
Assertion
Ref Expression
coe1tmmul  |-  ( ph  ->  (coe1 `  ( ( C 
.x.  ( D  .^  X ) )  .xb  A ) )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  ( C  .X.  ( (coe1 `  A ) `  ( x  -  D
) ) ) ,  .0.  ) ) )
Distinct variable groups:    x,  .0.    x, C    x, D    x, K    x,  .^    x, A    x, N    x, P    x, X    ph, x    x, R    x,  .x.    x,  .X.    x,  .xb
Allowed substitution hint:    B( x)

Proof of Theorem coe1tmmul
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 coe1tmmul.r . . 3  |-  ( ph  ->  R  e.  Ring )
2 coe1tmmul.c . . . 4  |-  ( ph  ->  C  e.  K )
3 coe1tmmul.d . . . 4  |-  ( ph  ->  D  e.  NN0 )
4 coe1tm.k . . . . 5  |-  K  =  ( Base `  R
)
5 coe1tm.p . . . . 5  |-  P  =  (Poly1 `  R )
6 coe1tm.x . . . . 5  |-  X  =  (var1 `  R )
7 coe1tm.m . . . . 5  |-  .x.  =  ( .s `  P )
8 coe1tm.n . . . . 5  |-  N  =  (mulGrp `  P )
9 coe1tm.e . . . . 5  |-  .^  =  (.g
`  N )
10 coe1tmmul.b . . . . 5  |-  B  =  ( Base `  P
)
114, 5, 6, 7, 8, 9, 10ply1tmcl 16348 . . . 4  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( C  .x.  ( D  .^  X ) )  e.  B )
121, 2, 3, 11syl3anc 1182 . . 3  |-  ( ph  ->  ( C  .x.  ( D  .^  X ) )  e.  B )
13 coe1tmmul.a . . 3  |-  ( ph  ->  A  e.  B )
14 coe1tmmul.t . . . 4  |-  .xb  =  ( .r `  P )
15 coe1tmmul.u . . . 4  |-  .X.  =  ( .r `  R )
165, 14, 15, 10coe1mul 16347 . . 3  |-  ( ( R  e.  Ring  /\  ( C  .x.  ( D  .^  X ) )  e.  B  /\  A  e.  B )  ->  (coe1 `  ( ( C  .x.  ( D  .^  X ) )  .xb  A )
)  =  ( x  e.  NN0  |->  ( R 
gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) ) ) )
171, 12, 13, 16syl3anc 1182 . 2  |-  ( ph  ->  (coe1 `  ( ( C 
.x.  ( D  .^  X ) )  .xb  A ) )  =  ( x  e.  NN0  |->  ( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) ) ) )
18 eqeq2 2292 . . . 4  |-  ( ( C  .X.  ( (coe1 `  A ) `  (
x  -  D ) ) )  =  if ( D  <_  x ,  ( C  .X.  ( (coe1 `  A ) `  ( x  -  D
) ) ) ,  .0.  )  ->  (
( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) )  =  ( C 
.X.  ( (coe1 `  A
) `  ( x  -  D ) ) )  <-> 
( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) )  =  if ( D  <_  x , 
( C  .X.  (
(coe1 `  A ) `  ( x  -  D
) ) ) ,  .0.  ) ) )
19 eqeq2 2292 . . . 4  |-  (  .0.  =  if ( D  <_  x ,  ( C  .X.  ( (coe1 `  A ) `  (
x  -  D ) ) ) ,  .0.  )  ->  ( ( R 
gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) )  =  .0.  <->  ( R  gsumg  ( y  e.  ( 0 ... x )  |->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  .X.  ( (coe1 `  A ) `  (
x  -  y ) ) ) ) )  =  if ( D  <_  x ,  ( C  .X.  ( (coe1 `  A ) `  (
x  -  D ) ) ) ,  .0.  ) ) )
20 coe1tm.z . . . . . 6  |-  .0.  =  ( 0g `  R )
211ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  R  e.  Ring )
22 rngmnd 15350 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
2321, 22syl 15 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  R  e.  Mnd )
24 ovex 5883 . . . . . . 7  |-  ( 0 ... x )  e. 
_V
2524a1i 10 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  (
0 ... x )  e. 
_V )
263ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  D  e.  NN0 )
27 simpr 447 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  D  <_  x )
28 fznn0 10851 . . . . . . . 8  |-  ( x  e.  NN0  ->  ( D  e.  ( 0 ... x )  <->  ( D  e.  NN0  /\  D  <_  x ) ) )
2928ad2antlr 707 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  ( D  e.  ( 0 ... x )  <->  ( D  e.  NN0  /\  D  <_  x ) ) )
3026, 27, 29mpbir2and 888 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  D  e.  ( 0 ... x
) )
311ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  y  e.  ( 0 ... x
) )  ->  R  e.  Ring )
32 eqid 2283 . . . . . . . . . . . . 13  |-  (coe1 `  ( C  .x.  ( D  .^  X ) ) )  =  (coe1 `  ( C  .x.  ( D  .^  X ) ) )
3332, 10, 5, 4coe1f 16292 . . . . . . . . . . . 12  |-  ( ( C  .x.  ( D 
.^  X ) )  e.  B  ->  (coe1 `  ( C  .x.  ( D 
.^  X ) ) ) : NN0 --> K )
3412, 33syl 15 . . . . . . . . . . 11  |-  ( ph  ->  (coe1 `  ( C  .x.  ( D  .^  X ) ) ) : NN0 --> K )
3534adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN0 )  ->  (coe1 `  ( C  .x.  ( D  .^  X ) ) ) : NN0 --> K )
36 elfznn0 10822 . . . . . . . . . 10  |-  ( y  e.  ( 0 ... x )  ->  y  e.  NN0 )
37 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) : NN0 --> K  /\  y  e.  NN0 )  ->  ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  e.  K )
3835, 36, 37syl2an 463 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  y  e.  ( 0 ... x
) )  ->  (
(coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  e.  K )
39 eqid 2283 . . . . . . . . . . . . 13  |-  (coe1 `  A
)  =  (coe1 `  A
)
4039, 10, 5, 4coe1f 16292 . . . . . . . . . . . 12  |-  ( A  e.  B  ->  (coe1 `  A ) : NN0 --> K )
4113, 40syl 15 . . . . . . . . . . 11  |-  ( ph  ->  (coe1 `  A ) : NN0 --> K )
4241adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN0 )  ->  (coe1 `  A
) : NN0 --> K )
43 fznn0sub 10824 . . . . . . . . . 10  |-  ( y  e.  ( 0 ... x )  ->  (
x  -  y )  e.  NN0 )
44 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( (coe1 `  A ) : NN0 --> K  /\  (
x  -  y )  e.  NN0 )  -> 
( (coe1 `  A ) `  ( x  -  y
) )  e.  K
)
4542, 43, 44syl2an 463 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  y  e.  ( 0 ... x
) )  ->  (
(coe1 `  A ) `  ( x  -  y
) )  e.  K
)
464, 15rngcl 15354 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
(coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  e.  K  /\  ( (coe1 `  A ) `  ( x  -  y
) )  e.  K
)  ->  ( (
(coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  .X.  ( (coe1 `  A ) `  (
x  -  y ) ) )  e.  K
)
4731, 38, 45, 46syl3anc 1182 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  y  e.  ( 0 ... x
) )  ->  (
( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  .X.  ( (coe1 `  A ) `  (
x  -  y ) ) )  e.  K
)
48 eqid 2283 . . . . . . . 8  |-  ( y  e.  ( 0 ... x )  |->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  .X.  ( (coe1 `  A ) `  (
x  -  y ) ) ) )  =  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) )
4947, 48fmptd 5684 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN0 )  ->  ( y  e.  ( 0 ... x
)  |->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y ) 
.X.  ( (coe1 `  A
) `  ( x  -  y ) ) ) ) : ( 0 ... x ) --> K )
5049adantr 451 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  (
y  e.  ( 0 ... x )  |->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  .X.  ( (coe1 `  A ) `  (
x  -  y ) ) ) ) : ( 0 ... x
) --> K )
511ad3antrrr 710 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  /\  y  e.  ( (
0 ... x )  \  { D } ) )  ->  R  e.  Ring )
522ad3antrrr 710 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  /\  y  e.  ( (
0 ... x )  \  { D } ) )  ->  C  e.  K
)
533ad3antrrr 710 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  /\  y  e.  ( (
0 ... x )  \  { D } ) )  ->  D  e.  NN0 )
54 eldifi 3298 . . . . . . . . . . . 12  |-  ( y  e.  ( ( 0 ... x )  \  { D } )  -> 
y  e.  ( 0 ... x ) )
5554, 36syl 15 . . . . . . . . . . 11  |-  ( y  e.  ( ( 0 ... x )  \  { D } )  -> 
y  e.  NN0 )
5655adantl 452 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  /\  y  e.  ( (
0 ... x )  \  { D } ) )  ->  y  e.  NN0 )
57 eldifsni 3750 . . . . . . . . . . . 12  |-  ( y  e.  ( ( 0 ... x )  \  { D } )  -> 
y  =/=  D )
5857necomd 2529 . . . . . . . . . . 11  |-  ( y  e.  ( ( 0 ... x )  \  { D } )  ->  D  =/=  y )
5958adantl 452 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  /\  y  e.  ( (
0 ... x )  \  { D } ) )  ->  D  =/=  y
)
6020, 4, 5, 6, 7, 8, 9, 51, 52, 53, 56, 59coe1tmfv2 16351 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  /\  y  e.  ( (
0 ... x )  \  { D } ) )  ->  ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  =  .0.  )
6160oveq1d 5873 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  /\  y  e.  ( (
0 ... x )  \  { D } ) )  ->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y ) 
.X.  ( (coe1 `  A
) `  ( x  -  y ) ) )  =  (  .0.  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) )
624, 15, 20rnglz 15377 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  (
(coe1 `  A ) `  ( x  -  y
) )  e.  K
)  ->  (  .0.  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) )  =  .0.  )
6331, 45, 62syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  y  e.  ( 0 ... x
) )  ->  (  .0.  .X.  ( (coe1 `  A
) `  ( x  -  y ) ) )  =  .0.  )
6454, 63sylan2 460 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  y  e.  ( ( 0 ... x )  \  { D } ) )  -> 
(  .0.  .X.  (
(coe1 `  A ) `  ( x  -  y
) ) )  =  .0.  )
6564adantlr 695 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  /\  y  e.  ( (
0 ... x )  \  { D } ) )  ->  (  .0.  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) )  =  .0.  )
6661, 65eqtrd 2315 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  /\  y  e.  ( (
0 ... x )  \  { D } ) )  ->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y ) 
.X.  ( (coe1 `  A
) `  ( x  -  y ) ) )  =  .0.  )
6766suppss2 6073 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  ( `' ( y  e.  ( 0 ... x
)  |->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y ) 
.X.  ( (coe1 `  A
) `  ( x  -  y ) ) ) ) " ( _V  \  {  .0.  }
) )  C_  { D } )
684, 20, 23, 25, 30, 50, 67gsumpt 15222 . . . . 5  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  ( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) )  =  ( ( y  e.  ( 0 ... x )  |->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  .X.  ( (coe1 `  A ) `  (
x  -  y ) ) ) ) `  D ) )
69 fveq2 5525 . . . . . . . . 9  |-  ( y  =  D  ->  (
(coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  =  ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  D ) )
70 oveq2 5866 . . . . . . . . . 10  |-  ( y  =  D  ->  (
x  -  y )  =  ( x  -  D ) )
7170fveq2d 5529 . . . . . . . . 9  |-  ( y  =  D  ->  (
(coe1 `  A ) `  ( x  -  y
) )  =  ( (coe1 `  A ) `  ( x  -  D
) ) )
7269, 71oveq12d 5876 . . . . . . . 8  |-  ( y  =  D  ->  (
( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  .X.  ( (coe1 `  A ) `  (
x  -  y ) ) )  =  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  D
)  .X.  ( (coe1 `  A ) `  (
x  -  D ) ) ) )
73 ovex 5883 . . . . . . . 8  |-  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  D
)  .X.  ( (coe1 `  A ) `  (
x  -  D ) ) )  e.  _V
7472, 48, 73fvmpt 5602 . . . . . . 7  |-  ( D  e.  ( 0 ... x )  ->  (
( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) `
 D )  =  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 D )  .X.  ( (coe1 `  A ) `  ( x  -  D
) ) ) )
7530, 74syl 15 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  (
( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) `
 D )  =  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 D )  .X.  ( (coe1 `  A ) `  ( x  -  D
) ) ) )
7620, 4, 5, 6, 7, 8, 9coe1tmfv1 16350 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
(coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  D
)  =  C )
771, 2, 3, 76syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  D
)  =  C )
7877ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  (
(coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  D
)  =  C )
7978oveq1d 5873 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  (
( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  D
)  .X.  ( (coe1 `  A ) `  (
x  -  D ) ) )  =  ( C  .X.  ( (coe1 `  A ) `  (
x  -  D ) ) ) )
8075, 79eqtrd 2315 . . . . 5  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  (
( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) `
 D )  =  ( C  .X.  (
(coe1 `  A ) `  ( x  -  D
) ) ) )
8168, 80eqtrd 2315 . . . 4  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  D  <_  x )  ->  ( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) )  =  ( C 
.X.  ( (coe1 `  A
) `  ( x  -  D ) ) ) )
821ad3antrrr 710 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  /\  y  e.  ( 0 ... x ) )  ->  R  e.  Ring )
832ad3antrrr 710 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  /\  y  e.  ( 0 ... x ) )  ->  C  e.  K )
843ad3antrrr 710 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  /\  y  e.  ( 0 ... x ) )  ->  D  e.  NN0 )
8536adantl 452 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  /\  y  e.  ( 0 ... x ) )  ->  y  e.  NN0 )
86 elfzle2 10800 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( 0 ... x )  ->  y  <_  x )
8786adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  y  e.  ( 0 ... x
) )  ->  y  <_  x )
88 breq1 4026 . . . . . . . . . . . . . 14  |-  ( D  =  y  ->  ( D  <_  x  <->  y  <_  x ) )
8987, 88syl5ibrcom 213 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  y  e.  ( 0 ... x
) )  ->  ( D  =  y  ->  D  <_  x ) )
9089necon3bd 2483 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  y  e.  ( 0 ... x
) )  ->  ( -.  D  <_  x  ->  D  =/=  y ) )
9190imp 418 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  y  e.  ( 0 ... x ) )  /\  -.  D  <_  x )  ->  D  =/=  y )
9291an32s 779 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  /\  y  e.  ( 0 ... x ) )  ->  D  =/=  y )
9320, 4, 5, 6, 7, 8, 9, 82, 83, 84, 85, 92coe1tmfv2 16351 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  /\  y  e.  ( 0 ... x ) )  ->  ( (coe1 `  ( C  .x.  ( D 
.^  X ) ) ) `  y )  =  .0.  )
9493oveq1d 5873 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  /\  y  e.  ( 0 ... x ) )  ->  ( (
(coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  .X.  ( (coe1 `  A ) `  (
x  -  y ) ) )  =  (  .0.  .X.  ( (coe1 `  A ) `  (
x  -  y ) ) ) )
9563adantlr 695 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  /\  y  e.  ( 0 ... x ) )  ->  (  .0.  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) )  =  .0.  )
9694, 95eqtrd 2315 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  /\  y  e.  ( 0 ... x ) )  ->  ( (
(coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  .X.  ( (coe1 `  A ) `  (
x  -  y ) ) )  =  .0.  )
9796mpteq2dva 4106 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  -> 
( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) )  =  ( y  e.  ( 0 ... x
)  |->  .0.  ) )
9897oveq2d 5874 . . . . 5  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  -> 
( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) )  =  ( R 
gsumg  ( y  e.  ( 0 ... x ) 
|->  .0.  ) ) )
991, 22syl 15 . . . . . . 7  |-  ( ph  ->  R  e.  Mnd )
10099ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  ->  R  e.  Mnd )
10124a1i 10 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  -> 
( 0 ... x
)  e.  _V )
10220gsumz 14458 . . . . . 6  |-  ( ( R  e.  Mnd  /\  ( 0 ... x
)  e.  _V )  ->  ( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  .0.  ) )  =  .0.  )
103100, 101, 102syl2anc 642 . . . . 5  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  -> 
( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  .0.  ) )  =  .0.  )
10498, 103eqtrd 2315 . . . 4  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  -.  D  <_  x )  -> 
( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) )  =  .0.  )
10518, 19, 81, 104ifbothda 3595 . . 3  |-  ( (
ph  /\  x  e.  NN0 )  ->  ( R  gsumg  ( y  e.  ( 0 ... x )  |->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  y
)  .X.  ( (coe1 `  A ) `  (
x  -  y ) ) ) ) )  =  if ( D  <_  x ,  ( C  .X.  ( (coe1 `  A ) `  (
x  -  D ) ) ) ,  .0.  ) )
106105mpteq2dva 4106 . 2  |-  ( ph  ->  ( x  e.  NN0  |->  ( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `
 y )  .X.  ( (coe1 `  A ) `  ( x  -  y
) ) ) ) ) )  =  ( x  e.  NN0  |->  if ( D  <_  x , 
( C  .X.  (
(coe1 `  A ) `  ( x  -  D
) ) ) ,  .0.  ) ) )
10717, 106eqtrd 2315 1  |-  ( ph  ->  (coe1 `  ( ( C 
.x.  ( D  .^  X ) )  .xb  A ) )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  ( C  .X.  ( (coe1 `  A ) `  ( x  -  D
) ) ) ,  .0.  ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149   ifcif 3565   {csn 3640   class class class wbr 4023    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   0cc0 8737    <_ cle 8868    - cmin 9037   NN0cn0 9965   ...cfz 10782   Basecbs 13148   .rcmulr 13209   .scvsca 13212   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361  .gcmg 14366  mulGrpcmgp 15325   Ringcrg 15337  var1cv1 16251  Poly1cpl1 16252  coe1cco1 16255
This theorem is referenced by:  coe1pwmul  16355  coe1sclmul  16358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-psr 16098  df-mvr 16099  df-mpl 16100  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-coe1 16262
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