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Theorem coe1mul3 19485
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 16347 . . . 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 5527 . 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 10022 . . 3  |-  ( ph  ->  ( I  +  J
)  e.  NN0 )
14 oveq2 5866 . . . . . 6  |-  ( x  =  ( I  +  J )  ->  (
0 ... x )  =  ( 0 ... (
I  +  J ) ) )
15 oveq1 5865 . . . . . . . 8  |-  ( x  =  ( I  +  J )  ->  (
x  -  y )  =  ( ( I  +  J )  -  y ) )
1615fveq2d 5529 . . . . . . 7  |-  ( x  =  ( I  +  J )  ->  (
(coe1 `  G ) `  ( x  -  y
) )  =  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )
1716oveq2d 5874 . . . . . 6  |-  ( x  =  ( I  +  J )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( x  -  y
) ) )  =  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) )
1814, 17mpteq12dv 4098 . . . . 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 5874 . . . 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 2283 . . . 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 5883 . . . 4  |-  ( R 
gsumg  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) )  e.  _V
2219, 20, 21fvmpt 5602 . . 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 2283 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
25 eqid 2283 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
26 rngmnd 15350 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
271, 26syl 15 . . . 4  |-  ( ph  ->  R  e.  Mnd )
28 ovex 5883 . . . . 5  |-  ( 0 ... ( I  +  J ) )  e. 
_V
2928a1i 10 . . . 4  |-  ( ph  ->  ( 0 ... (
I  +  J ) )  e.  _V )
3011nn0red 10019 . . . . . 6  |-  ( ph  ->  I  e.  RR )
31 nn0addge1 10010 . . . . . 6  |-  ( ( I  e.  RR  /\  J  e.  NN0 )  ->  I  <_  ( I  +  J ) )
3230, 12, 31syl2anc 642 . . . . 5  |-  ( ph  ->  I  <_  ( I  +  J ) )
33 fznn0 10851 . . . . . 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 2283 . . . . . . . . 9  |-  (coe1 `  F
)  =  (coe1 `  F
)
3837, 7, 4, 24coe1f 16292 . . . . . . . 8  |-  ( F  e.  B  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
392, 38syl 15 . . . . . . 7  |-  ( ph  ->  (coe1 `  F ) : NN0 --> ( Base `  R
) )
40 elfznn0 10822 . . . . . . 7  |-  ( y  e.  ( 0 ... ( I  +  J
) )  ->  y  e.  NN0 )
41 ffvelrn 5663 . . . . . . 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 2283 . . . . . . . . 9  |-  (coe1 `  G
)  =  (coe1 `  G
)
4443, 7, 4, 24coe1f 16292 . . . . . . . 8  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
453, 44syl 15 . . . . . . 7  |-  ( ph  ->  (coe1 `  G ) : NN0 --> ( Base `  R
) )
46 fznn0sub 10824 . . . . . . 7  |-  ( y  e.  ( 0 ... ( I  +  J
) )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
47 ffvelrn 5663 . . . . . . 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 15354 . . . . . 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 2283 . . . . 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 5684 . . . 4  |-  ( ph  ->  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) : ( 0 ... ( I  +  J
) ) --> ( Base `  R ) )
53 eldifsn 3749 . . . . . 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 10019 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  RR )
5630adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  I  e.  RR )
5755, 56lttri2d 8958 . . . . . . . 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 19468 . . . . . . . . . . . . . . . 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 10019 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  J  e.  RR )
6665rexrd 8881 . . . . . . . . . . . . . . 15  |-  ( ph  ->  J  e.  RR* )
6766ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  J  e.  RR* )
6813nn0red 10019 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( I  +  J
)  e.  RR )
6968adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
I  +  J )  e.  RR )
7069, 55resubcld 9211 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  RR )
7170rexrd 8881 . . . . . . . . . . . . . . 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 9365 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  <  I  <->  ( y  +  J )  <  (
I  +  J ) ) )
77 ltaddsub2 9249 . . . . . . . . . . . . . . . . 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 10491 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  <  ( ( I  +  J )  -  y
) )
8261, 4, 7, 25, 43deg1lt 19483 . . . . . . . . . . . . 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 5874 . . . . . . . . . . 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 15378 . . . . . . . . . . . . 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 2315 . . . . . . . . . 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 19468 . . . . . . . . . . . . . . . 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 8881 . . . . . . . . . . . . . . 15  |-  ( ph  ->  I  e.  RR* )
9594ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  I  e.  RR* )
9655rexrd 8881 . . . . . . . . . . . . . . 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 10491 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  <  y )
10261, 4, 7, 25, 37deg1lt 19483 . . . . . . . . . . . . 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 5873 . . . . . . . . . . 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 15377 . . . . . . . . . . . . 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 2315 . . . . . . . . . 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 6073 . . . 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 15222 . . 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 5525 . . . . . 6  |-  ( y  =  I  ->  (
(coe1 `  F ) `  y )  =  ( (coe1 `  F ) `  I ) )
117 oveq2 5866 . . . . . . 7  |-  ( y  =  I  ->  (
( I  +  J
)  -  y )  =  ( ( I  +  J )  -  I ) )
118117fveq2d 5529 . . . . . 6  |-  ( y  =  I  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  =  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )
119116, 118oveq12d 5876 . . . . 5  |-  ( y  =  I  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) ) )
120 ovex 5883 . . . . 5  |-  ( ( (coe1 `  F ) `  I )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )  e. 
_V
121119, 51, 120fvmpt 5602 . . . 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 10020 . . . . . 6  |-  ( ph  ->  I  e.  CC )
12412nn0cnd 10020 . . . . . 6  |-  ( ph  ->  J  e.  CC )
125123, 124pncan2d 9159 . . . . 5  |-  ( ph  ->  ( ( I  +  J )  -  I
)  =  J )
126125fveq2d 5529 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) )  =  ( (coe1 `  G ) `  J ) )
127126oveq2d 5874 . . 3  |-  ( ph  ->  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )  =  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  J ) ) )
128115, 122, 1273eqtrd 2319 . 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 2319 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149   {csn 3640   class class class wbr 4023    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737    + caddc 8740   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   NN0cn0 9965   ...cfz 10782   Basecbs 13148   .rcmulr 13209   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361   Ringcrg 15337  Poly1cpl1 16252  coe1cco1 16255   deg1 cdg1 19440
This theorem is referenced by:  coe1mul4  19486
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  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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-sup 7194  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-dec 10125  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-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  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-mulg 14492  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-psr 16098  df-mpl 16100  df-opsr 16106  df-psr1 16257  df-ply1 16259  df-coe1 16262  df-cnfld 16378  df-mdeg 19441  df-deg1 19442
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