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Theorem coe1mul3 19583
Description: The coefficient vector of multiplication in the univariate polynomial ring, at indices high enough that at most one component can be active in the sum. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Hypotheses
Ref Expression
coe1mul3.s  |-  Y  =  (Poly1 `  R )
coe1mul3.t  |-  .xb  =  ( .r `  Y )
coe1mul3.u  |-  .x.  =  ( .r `  R )
coe1mul3.b  |-  B  =  ( Base `  Y
)
coe1mul3.d  |-  D  =  ( deg1  `  R )
coe1mul3.r  |-  ( ph  ->  R  e.  Ring )
coe1mul3.f1  |-  ( ph  ->  F  e.  B )
coe1mul3.f2  |-  ( ph  ->  I  e.  NN0 )
coe1mul3.f3  |-  ( ph  ->  ( D `  F
)  <_  I )
coe1mul3.g1  |-  ( ph  ->  G  e.  B )
coe1mul3.g2  |-  ( ph  ->  J  e.  NN0 )
coe1mul3.g3  |-  ( ph  ->  ( D `  G
)  <_  J )
Assertion
Ref Expression
coe1mul3  |-  ( ph  ->  ( (coe1 `  ( F  .xb  G ) ) `  ( I  +  J
) )  =  ( ( (coe1 `  F ) `  I )  .x.  (
(coe1 `  G ) `  J ) ) )

Proof of Theorem coe1mul3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coe1mul3.r . . . 4  |-  ( ph  ->  R  e.  Ring )
2 coe1mul3.f1 . . . 4  |-  ( ph  ->  F  e.  B )
3 coe1mul3.g1 . . . 4  |-  ( ph  ->  G  e.  B )
4 coe1mul3.s . . . . 5  |-  Y  =  (Poly1 `  R )
5 coe1mul3.t . . . . 5  |-  .xb  =  ( .r `  Y )
6 coe1mul3.u . . . . 5  |-  .x.  =  ( .r `  R )
7 coe1mul3.b . . . . 5  |-  B  =  ( Base `  Y
)
84, 5, 6, 7coe1mul 16440 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .xb  G ) )  =  ( x  e.  NN0  |->  ( R 
gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( x  -  y
) ) ) ) ) ) )
91, 2, 3, 8syl3anc 1182 . . 3  |-  ( ph  ->  (coe1 `  ( F  .xb  G ) )  =  ( x  e.  NN0  |->  ( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( x  -  y
) ) ) ) ) ) )
109fveq1d 5607 . 2  |-  ( ph  ->  ( (coe1 `  ( F  .xb  G ) ) `  ( I  +  J
) )  =  ( ( x  e.  NN0  |->  ( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( x  -  y
) ) ) ) ) ) `  (
I  +  J ) ) )
11 coe1mul3.f2 . . . 4  |-  ( ph  ->  I  e.  NN0 )
12 coe1mul3.g2 . . . 4  |-  ( ph  ->  J  e.  NN0 )
1311, 12nn0addcld 10111 . . 3  |-  ( ph  ->  ( I  +  J
)  e.  NN0 )
14 oveq2 5950 . . . . . 6  |-  ( x  =  ( I  +  J )  ->  (
0 ... x )  =  ( 0 ... (
I  +  J ) ) )
15 oveq1 5949 . . . . . . . 8  |-  ( x  =  ( I  +  J )  ->  (
x  -  y )  =  ( ( I  +  J )  -  y ) )
1615fveq2d 5609 . . . . . . 7  |-  ( x  =  ( I  +  J )  ->  (
(coe1 `  G ) `  ( x  -  y
) )  =  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )
1716oveq2d 5958 . . . . . 6  |-  ( x  =  ( I  +  J )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( x  -  y
) ) )  =  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) )
1814, 17mpteq12dv 4177 . . . . 5  |-  ( x  =  ( I  +  J )  ->  (
y  e.  ( 0 ... x )  |->  ( ( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( x  -  y
) ) ) )  =  ( y  e.  ( 0 ... (
I  +  J ) )  |->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) ) ) )
1918oveq2d 5958 . . . 4  |-  ( x  =  ( I  +  J )  ->  ( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( x  -  y
) ) ) ) )  =  ( R 
gsumg  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) ) )
20 eqid 2358 . . . 4  |-  ( x  e.  NN0  |->  ( R 
gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( x  -  y
) ) ) ) ) )  =  ( x  e.  NN0  |->  ( R 
gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( x  -  y
) ) ) ) ) )
21 ovex 5967 . . . 4  |-  ( R 
gsumg  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) )  e.  _V
2219, 20, 21fvmpt 5682 . . 3  |-  ( ( I  +  J )  e.  NN0  ->  ( ( x  e.  NN0  |->  ( R 
gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( x  -  y
) ) ) ) ) ) `  (
I  +  J ) )  =  ( R 
gsumg  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) ) )
2313, 22syl 15 . 2  |-  ( ph  ->  ( ( x  e. 
NN0  |->  ( R  gsumg  ( y  e.  ( 0 ... x )  |->  ( ( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( x  -  y
) ) ) ) ) ) `  (
I  +  J ) )  =  ( R 
gsumg  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) ) )
24 eqid 2358 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
25 eqid 2358 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
26 rngmnd 15443 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
271, 26syl 15 . . . 4  |-  ( ph  ->  R  e.  Mnd )
28 ovex 5967 . . . . 5  |-  ( 0 ... ( I  +  J ) )  e. 
_V
2928a1i 10 . . . 4  |-  ( ph  ->  ( 0 ... (
I  +  J ) )  e.  _V )
3011nn0red 10108 . . . . . 6  |-  ( ph  ->  I  e.  RR )
31 nn0addge1 10099 . . . . . 6  |-  ( ( I  e.  RR  /\  J  e.  NN0 )  ->  I  <_  ( I  +  J ) )
3230, 12, 31syl2anc 642 . . . . 5  |-  ( ph  ->  I  <_  ( I  +  J ) )
33 fznn0 10940 . . . . . 6  |-  ( ( I  +  J )  e.  NN0  ->  ( I  e.  ( 0 ... ( I  +  J
) )  <->  ( I  e.  NN0  /\  I  <_ 
( I  +  J
) ) ) )
3413, 33syl 15 . . . . 5  |-  ( ph  ->  ( I  e.  ( 0 ... ( I  +  J ) )  <-> 
( I  e.  NN0  /\  I  <_  ( I  +  J ) ) ) )
3511, 32, 34mpbir2and 888 . . . 4  |-  ( ph  ->  I  e.  ( 0 ... ( I  +  J ) ) )
361adantr 451 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  R  e.  Ring )
37 eqid 2358 . . . . . . . . 9  |-  (coe1 `  F
)  =  (coe1 `  F
)
3837, 7, 4, 24coe1f 16385 . . . . . . . 8  |-  ( F  e.  B  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
392, 38syl 15 . . . . . . 7  |-  ( ph  ->  (coe1 `  F ) : NN0 --> ( Base `  R
) )
40 elfznn0 10911 . . . . . . 7  |-  ( y  e.  ( 0 ... ( I  +  J
) )  ->  y  e.  NN0 )
41 ffvelrn 5743 . . . . . . 7  |-  ( ( (coe1 `  F ) : NN0 --> ( Base `  R
)  /\  y  e.  NN0 )  ->  ( (coe1 `  F ) `  y
)  e.  ( Base `  R ) )
4239, 40, 41syl2an 463 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
(coe1 `  F ) `  y )  e.  (
Base `  R )
)
43 eqid 2358 . . . . . . . . 9  |-  (coe1 `  G
)  =  (coe1 `  G
)
4443, 7, 4, 24coe1f 16385 . . . . . . . 8  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
453, 44syl 15 . . . . . . 7  |-  ( ph  ->  (coe1 `  G ) : NN0 --> ( Base `  R
) )
46 fznn0sub 10913 . . . . . . 7  |-  ( y  e.  ( 0 ... ( I  +  J
) )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
47 ffvelrn 5743 . . . . . . 7  |-  ( ( (coe1 `  G ) : NN0 --> ( Base `  R
)  /\  ( (
I  +  J )  -  y )  e. 
NN0 )  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  e.  (
Base `  R )
)
4845, 46, 47syl2an 463 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  e.  (
Base `  R )
)
4924, 6rngcl 15447 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
(coe1 `  F ) `  y )  e.  (
Base `  R )  /\  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) )  e.  (
Base `  R )
)  ->  ( (
(coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  e.  ( Base `  R
) )
5036, 42, 48, 49syl3anc 1182 . . . . 5  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  e.  ( Base `  R
) )
51 eqid 2358 . . . . 5  |-  ( y  e.  ( 0 ... ( I  +  J
) )  |->  ( ( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) )  =  ( y  e.  ( 0 ... (
I  +  J ) )  |->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) ) )
5250, 51fmptd 5764 . . . 4  |-  ( ph  ->  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) : ( 0 ... ( I  +  J
) ) --> ( Base `  R ) )
53 eldifsn 3825 . . . . . 6  |-  ( y  e.  ( ( 0 ... ( I  +  J ) )  \  { I } )  <-> 
( y  e.  ( 0 ... ( I  +  J ) )  /\  y  =/=  I
) )
5440adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  NN0 )
5554nn0red 10108 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  RR )
5630adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  I  e.  RR )
5755, 56lttri2d 9045 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  =/=  I  <->  ( y  <  I  \/  I  < 
y ) ) )
583ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  G  e.  B )
5946adantl 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
6059adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
61 coe1mul3.d . . . . . . . . . . . . . . . . 17  |-  D  =  ( deg1  `  R )
6261, 4, 7deg1xrcl 19566 . . . . . . . . . . . . . . . 16  |-  ( G  e.  B  ->  ( D `  G )  e.  RR* )
633, 62syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  G
)  e.  RR* )
6463ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  e.  RR* )
6512nn0red 10108 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  J  e.  RR )
6665rexrd 8968 . . . . . . . . . . . . . . 15  |-  ( ph  ->  J  e.  RR* )
6766ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  J  e.  RR* )
6813nn0red 10108 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( I  +  J
)  e.  RR )
6968adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
I  +  J )  e.  RR )
7069, 55resubcld 9298 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  RR )
7170rexrd 8968 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  RR* )
7271adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( I  +  J
)  -  y )  e.  RR* )
73 coe1mul3.g3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  G
)  <_  J )
7473ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  <_  J )
7565adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  J  e.  RR )
7655, 56, 75ltadd1d 9452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  <  I  <->  ( y  +  J )  <  (
I  +  J ) ) )
77 ltaddsub2 9336 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  RR  /\  J  e.  RR  /\  (
I  +  J )  e.  RR )  -> 
( ( y  +  J )  <  (
I  +  J )  <-> 
J  <  ( (
I  +  J )  -  y ) ) )
7855, 75, 69, 77syl3anc 1182 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( y  +  J
)  <  ( I  +  J )  <->  J  <  ( ( I  +  J
)  -  y ) ) )
7976, 78bitrd 244 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  <  I  <->  J  <  ( ( I  +  J
)  -  y ) ) )
8079biimpa 470 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  J  <  ( ( I  +  J )  -  y
) )
8164, 67, 72, 74, 80xrlelttrd 10580 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  <  ( ( I  +  J )  -  y
) )
8261, 4, 7, 25, 43deg1lt 19581 . . . . . . . . . . . . 13  |-  ( ( G  e.  B  /\  ( ( I  +  J )  -  y
)  e.  NN0  /\  ( D `  G )  <  ( ( I  +  J )  -  y ) )  -> 
( (coe1 `  G ) `  ( ( I  +  J )  -  y
) )  =  ( 0g `  R ) )
8358, 60, 81, 82syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  =  ( 0g `  R ) )
8483oveq2d 5958 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( ( (coe1 `  F
) `  y )  .x.  ( 0g `  R
) ) )
8524, 6, 25rngrz 15471 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  (
(coe1 `  F ) `  y )  e.  (
Base `  R )
)  ->  ( (
(coe1 `  F ) `  y )  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
8636, 42, 85syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( (coe1 `  F ) `  y )  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
8786adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( (coe1 `  F ) `  y )  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
8884, 87eqtrd 2390 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
892ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  F  e.  B )
9054adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  y  e.  NN0 )
9161, 4, 7deg1xrcl 19566 . . . . . . . . . . . . . . . 16  |-  ( F  e.  B  ->  ( D `  F )  e.  RR* )
922, 91syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  F
)  e.  RR* )
9392ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  e.  RR* )
9430rexrd 8968 . . . . . . . . . . . . . . 15  |-  ( ph  ->  I  e.  RR* )
9594ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  I  e.  RR* )
9655rexrd 8968 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  RR* )
9796adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  y  e.  RR* )
98 coe1mul3.f3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  F
)  <_  I )
9998ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  <_  I )
100 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  I  <  y )
10193, 95, 97, 99, 100xrlelttrd 10580 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  <  y )
10261, 4, 7, 25, 37deg1lt 19581 . . . . . . . . . . . . 13  |-  ( ( F  e.  B  /\  y  e.  NN0  /\  ( D `  F )  <  y )  ->  (
(coe1 `  F ) `  y )  =  ( 0g `  R ) )
10389, 90, 101, 102syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
(coe1 `  F ) `  y )  =  ( 0g `  R ) )
104103oveq1d 5957 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( ( 0g `  R )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) )
10524, 6, 25rnglz 15470 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  e.  (
Base `  R )
)  ->  ( ( 0g `  R )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
10636, 48, 105syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( 0g `  R
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
107106adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
( 0g `  R
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
108104, 107eqtrd 2390 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
10988, 108jaodan 760 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  (
y  <  I  \/  I  <  y ) )  ->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
110109ex 423 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( y  <  I  \/  I  <  y )  ->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) ) )
11157, 110sylbid 206 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  =/=  I  -> 
( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) ) )
112111impr 602 . . . . . 6  |-  ( (
ph  /\  ( y  e.  ( 0 ... (
I  +  J ) )  /\  y  =/=  I ) )  -> 
( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
11353, 112sylan2b 461 . . . . 5  |-  ( (
ph  /\  y  e.  ( ( 0 ... ( I  +  J
) )  \  {
I } ) )  ->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
114113suppss2 6157 . . . 4  |-  ( ph  ->  ( `' ( y  e.  ( 0 ... ( I  +  J
) )  |->  ( ( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) )
" ( _V  \  { ( 0g `  R ) } ) )  C_  { I } )
11524, 25, 27, 29, 35, 52, 114gsumpt 15315 . . 3  |-  ( ph  ->  ( R  gsumg  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) )  =  ( ( y  e.  ( 0 ... ( I  +  J ) )  |->  ( ( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) `
 I ) )
116 fveq2 5605 . . . . . 6  |-  ( y  =  I  ->  (
(coe1 `  F ) `  y )  =  ( (coe1 `  F ) `  I ) )
117 oveq2 5950 . . . . . . 7  |-  ( y  =  I  ->  (
( I  +  J
)  -  y )  =  ( ( I  +  J )  -  I ) )
118117fveq2d 5609 . . . . . 6  |-  ( y  =  I  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  =  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )
119116, 118oveq12d 5960 . . . . 5  |-  ( y  =  I  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) ) )
120 ovex 5967 . . . . 5  |-  ( ( (coe1 `  F ) `  I )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )  e. 
_V
121119, 51, 120fvmpt 5682 . . . 4  |-  ( I  e.  ( 0 ... ( I  +  J
) )  ->  (
( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) `
 I )  =  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) ) )
12235, 121syl 15 . . 3  |-  ( ph  ->  ( ( y  e.  ( 0 ... (
I  +  J ) )  |->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) ) ) `  I )  =  ( ( (coe1 `  F ) `  I )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  I
) ) ) )
12311nn0cnd 10109 . . . . . 6  |-  ( ph  ->  I  e.  CC )
12412nn0cnd 10109 . . . . . 6  |-  ( ph  ->  J  e.  CC )
125123, 124pncan2d 9246 . . . . 5  |-  ( ph  ->  ( ( I  +  J )  -  I
)  =  J )
126125fveq2d 5609 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) )  =  ( (coe1 `  G ) `  J ) )
127126oveq2d 5958 . . 3  |-  ( ph  ->  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )  =  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  J ) ) )
128115, 122, 1273eqtrd 2394 . 2  |-  ( ph  ->  ( R  gsumg  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) )  =  ( ( (coe1 `  F ) `  I )  .x.  (
(coe1 `  G ) `  J ) ) )
12910, 23, 1283eqtrd 2394 1  |-  ( ph  ->  ( (coe1 `  ( F  .xb  G ) ) `  ( I  +  J
) )  =  ( ( (coe1 `  F ) `  I )  .x.  (
(coe1 `  G ) `  J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864    \ cdif 3225   {csn 3716   class class class wbr 4102    e. cmpt 4156   -->wf 5330   ` cfv 5334  (class class class)co 5942   RRcr 8823   0cc0 8824    + caddc 8827   RR*cxr 8953    < clt 8954    <_ cle 8955    - cmin 9124   NN0cn0 10054   ...cfz 10871   Basecbs 13239   .rcmulr 13300   0gc0g 13493    gsumg cgsu 13494   Mndcmnd 14454   Ringcrg 15430  Poly1cpl1 16345  coe1cco1 16348   deg1 cdg1 19538
This theorem is referenced by:  coe1mul4  19584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902  ax-addf 8903  ax-mulf 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-ofr 6163  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-map 6859  df-pm 6860  df-ixp 6903  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-oi 7312  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-fz 10872  df-fzo 10960  df-seq 11136  df-hash 11428  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-starv 13314  df-sca 13315  df-vsca 13316  df-tset 13318  df-ple 13319  df-ds 13321  df-unif 13322  df-0g 13497  df-gsum 13498  df-mre 13581  df-mrc 13582  df-acs 13584  df-mnd 14460  df-mhm 14508  df-submnd 14509  df-grp 14582  df-minusg 14583  df-mulg 14585  df-ghm 14774  df-cntz 14886  df-cmn 15184  df-abl 15185  df-mgp 15419  df-rng 15433  df-cring 15434  df-ur 15435  df-psr 16191  df-mpl 16193  df-opsr 16199  df-psr1 16350  df-ply1 16352  df-coe1 16355  df-cnfld 16477  df-mdeg 19539  df-deg1 19540
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