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Theorem coe1mul3 19983
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 16626 . . . 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 1184 . . 3  |-  ( ph  ->  (coe1 `  ( F  .xb  G ) )  =  ( x  e.  NN0  |->  ( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( x  -  y
) ) ) ) ) ) )
109fveq1d 5697 . 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 10242 . . 3  |-  ( ph  ->  ( I  +  J
)  e.  NN0 )
14 oveq2 6056 . . . . . 6  |-  ( x  =  ( I  +  J )  ->  (
0 ... x )  =  ( 0 ... (
I  +  J ) ) )
15 oveq1 6055 . . . . . . . 8  |-  ( x  =  ( I  +  J )  ->  (
x  -  y )  =  ( ( I  +  J )  -  y ) )
1615fveq2d 5699 . . . . . . 7  |-  ( x  =  ( I  +  J )  ->  (
(coe1 `  G ) `  ( x  -  y
) )  =  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )
1716oveq2d 6064 . . . . . 6  |-  ( x  =  ( I  +  J )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( x  -  y
) ) )  =  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) )
1814, 17mpteq12dv 4255 . . . . 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 6064 . . . 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 2412 . . . 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 6073 . . . 4  |-  ( R 
gsumg  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) )  e.  _V
2219, 20, 21fvmpt 5773 . . 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 16 . 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 2412 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
25 eqid 2412 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
26 rngmnd 15636 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
271, 26syl 16 . . . 4  |-  ( ph  ->  R  e.  Mnd )
28 ovex 6073 . . . . 5  |-  ( 0 ... ( I  +  J ) )  e. 
_V
2928a1i 11 . . . 4  |-  ( ph  ->  ( 0 ... (
I  +  J ) )  e.  _V )
3011nn0red 10239 . . . . . 6  |-  ( ph  ->  I  e.  RR )
31 nn0addge1 10230 . . . . . 6  |-  ( ( I  e.  RR  /\  J  e.  NN0 )  ->  I  <_  ( I  +  J ) )
3230, 12, 31syl2anc 643 . . . . 5  |-  ( ph  ->  I  <_  ( I  +  J ) )
33 fznn0 11077 . . . . . 6  |-  ( ( I  +  J )  e.  NN0  ->  ( I  e.  ( 0 ... ( I  +  J
) )  <->  ( I  e.  NN0  /\  I  <_ 
( I  +  J
) ) ) )
3413, 33syl 16 . . . . 5  |-  ( ph  ->  ( I  e.  ( 0 ... ( I  +  J ) )  <-> 
( I  e.  NN0  /\  I  <_  ( I  +  J ) ) ) )
3511, 32, 34mpbir2and 889 . . . 4  |-  ( ph  ->  I  e.  ( 0 ... ( I  +  J ) ) )
361adantr 452 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  R  e.  Ring )
37 eqid 2412 . . . . . . . . 9  |-  (coe1 `  F
)  =  (coe1 `  F
)
3837, 7, 4, 24coe1f 16572 . . . . . . . 8  |-  ( F  e.  B  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
392, 38syl 16 . . . . . . 7  |-  ( ph  ->  (coe1 `  F ) : NN0 --> ( Base `  R
) )
40 elfznn0 11047 . . . . . . 7  |-  ( y  e.  ( 0 ... ( I  +  J
) )  ->  y  e.  NN0 )
41 ffvelrn 5835 . . . . . . 7  |-  ( ( (coe1 `  F ) : NN0 --> ( Base `  R
)  /\  y  e.  NN0 )  ->  ( (coe1 `  F ) `  y
)  e.  ( Base `  R ) )
4239, 40, 41syl2an 464 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
(coe1 `  F ) `  y )  e.  (
Base `  R )
)
43 eqid 2412 . . . . . . . . 9  |-  (coe1 `  G
)  =  (coe1 `  G
)
4443, 7, 4, 24coe1f 16572 . . . . . . . 8  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
453, 44syl 16 . . . . . . 7  |-  ( ph  ->  (coe1 `  G ) : NN0 --> ( Base `  R
) )
46 fznn0sub 11049 . . . . . . 7  |-  ( y  e.  ( 0 ... ( I  +  J
) )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
47 ffvelrn 5835 . . . . . . 7  |-  ( ( (coe1 `  G ) : NN0 --> ( Base `  R
)  /\  ( (
I  +  J )  -  y )  e. 
NN0 )  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  e.  (
Base `  R )
)
4845, 46, 47syl2an 464 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  e.  (
Base `  R )
)
4924, 6rngcl 15640 . . . . . 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 1184 . . . . 5  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  e.  ( Base `  R
) )
51 eqid 2412 . . . . 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 5860 . . . 4  |-  ( ph  ->  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) : ( 0 ... ( I  +  J
) ) --> ( Base `  R ) )
53 eldifsn 3895 . . . . . 6  |-  ( y  e.  ( ( 0 ... ( I  +  J ) )  \  { I } )  <-> 
( y  e.  ( 0 ... ( I  +  J ) )  /\  y  =/=  I
) )
5440adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  NN0 )
5554nn0red 10239 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  RR )
5630adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  I  e.  RR )
5755, 56lttri2d 9176 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  =/=  I  <->  ( y  <  I  \/  I  < 
y ) ) )
583ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  G  e.  B )
5946adantl 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
6059adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
61 coe1mul3.d . . . . . . . . . . . . . . . . 17  |-  D  =  ( deg1  `  R )
6261, 4, 7deg1xrcl 19966 . . . . . . . . . . . . . . . 16  |-  ( G  e.  B  ->  ( D `  G )  e.  RR* )
633, 62syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  G
)  e.  RR* )
6463ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  e.  RR* )
6512nn0red 10239 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  J  e.  RR )
6665rexrd 9098 . . . . . . . . . . . . . . 15  |-  ( ph  ->  J  e.  RR* )
6766ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  J  e.  RR* )
6813nn0red 10239 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( I  +  J
)  e.  RR )
6968adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
I  +  J )  e.  RR )
7069, 55resubcld 9429 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  RR )
7170rexrd 9098 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  RR* )
7271adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( I  +  J
)  -  y )  e.  RR* )
73 coe1mul3.g3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  G
)  <_  J )
7473ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  <_  J )
7565adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  J  e.  RR )
7655, 56, 75ltadd1d 9583 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  <  I  <->  ( y  +  J )  <  (
I  +  J ) ) )
7755, 75, 69ltaddsub2d 9591 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( y  +  J
)  <  ( I  +  J )  <->  J  <  ( ( I  +  J
)  -  y ) ) )
7876, 77bitrd 245 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  <  I  <->  J  <  ( ( I  +  J
)  -  y ) ) )
7978biimpa 471 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  J  <  ( ( I  +  J )  -  y
) )
8064, 67, 72, 74, 79xrlelttrd 10714 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  <  ( ( I  +  J )  -  y
) )
8161, 4, 7, 25, 43deg1lt 19981 . . . . . . . . . . . . 13  |-  ( ( G  e.  B  /\  ( ( I  +  J )  -  y
)  e.  NN0  /\  ( D `  G )  <  ( ( I  +  J )  -  y ) )  -> 
( (coe1 `  G ) `  ( ( I  +  J )  -  y
) )  =  ( 0g `  R ) )
8258, 60, 80, 81syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  =  ( 0g `  R ) )
8382oveq2d 6064 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( ( (coe1 `  F
) `  y )  .x.  ( 0g `  R
) ) )
8424, 6, 25rngrz 15664 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  (
(coe1 `  F ) `  y )  e.  (
Base `  R )
)  ->  ( (
(coe1 `  F ) `  y )  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
8536, 42, 84syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( (coe1 `  F ) `  y )  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
8685adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( (coe1 `  F ) `  y )  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
8783, 86eqtrd 2444 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
882ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  F  e.  B )
8954adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  y  e.  NN0 )
9061, 4, 7deg1xrcl 19966 . . . . . . . . . . . . . . . 16  |-  ( F  e.  B  ->  ( D `  F )  e.  RR* )
912, 90syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  F
)  e.  RR* )
9291ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  e.  RR* )
9330rexrd 9098 . . . . . . . . . . . . . . 15  |-  ( ph  ->  I  e.  RR* )
9493ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  I  e.  RR* )
9555rexrd 9098 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  RR* )
9695adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  y  e.  RR* )
97 coe1mul3.f3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  F
)  <_  I )
9897ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  <_  I )
99 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  I  <  y )
10092, 94, 96, 98, 99xrlelttrd 10714 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  <  y )
10161, 4, 7, 25, 37deg1lt 19981 . . . . . . . . . . . . 13  |-  ( ( F  e.  B  /\  y  e.  NN0  /\  ( D `  F )  <  y )  ->  (
(coe1 `  F ) `  y )  =  ( 0g `  R ) )
10288, 89, 100, 101syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
(coe1 `  F ) `  y )  =  ( 0g `  R ) )
103102oveq1d 6063 . . . . . . . . . . 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
) ) ) )
10424, 6, 25rnglz 15663 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  e.  (
Base `  R )
)  ->  ( ( 0g `  R )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
10536, 48, 104syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( 0g `  R
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
106105adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
( 0g `  R
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
107103, 106eqtrd 2444 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
10887, 107jaodan 761 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  (
y  <  I  \/  I  <  y ) )  ->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
109108ex 424 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( y  <  I  \/  I  <  y )  ->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) ) )
11057, 109sylbid 207 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  =/=  I  -> 
( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) ) )
111110impr 603 . . . . . 6  |-  ( (
ph  /\  ( y  e.  ( 0 ... (
I  +  J ) )  /\  y  =/=  I ) )  -> 
( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
11253, 111sylan2b 462 . . . . 5  |-  ( (
ph  /\  y  e.  ( ( 0 ... ( I  +  J
) )  \  {
I } ) )  ->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
113112suppss2 6267 . . . 4  |-  ( ph  ->  ( `' ( y  e.  ( 0 ... ( I  +  J
) )  |->  ( ( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) )
" ( _V  \  { ( 0g `  R ) } ) )  C_  { I } )
11424, 25, 27, 29, 35, 52, 113gsumpt 15508 . . 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 ) )
115 fveq2 5695 . . . . . 6  |-  ( y  =  I  ->  (
(coe1 `  F ) `  y )  =  ( (coe1 `  F ) `  I ) )
116 oveq2 6056 . . . . . . 7  |-  ( y  =  I  ->  (
( I  +  J
)  -  y )  =  ( ( I  +  J )  -  I ) )
117116fveq2d 5699 . . . . . 6  |-  ( y  =  I  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  =  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )
118115, 117oveq12d 6066 . . . . 5  |-  ( y  =  I  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) ) )
119 ovex 6073 . . . . 5  |-  ( ( (coe1 `  F ) `  I )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )  e. 
_V
120118, 51, 119fvmpt 5773 . . . 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
) ) ) )
12135, 120syl 16 . . 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
) ) ) )
12211nn0cnd 10240 . . . . . 6  |-  ( ph  ->  I  e.  CC )
12312nn0cnd 10240 . . . . . 6  |-  ( ph  ->  J  e.  CC )
124122, 123pncan2d 9377 . . . . 5  |-  ( ph  ->  ( ( I  +  J )  -  I
)  =  J )
125124fveq2d 5699 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) )  =  ( (coe1 `  G ) `  J ) )
126125oveq2d 6064 . . 3  |-  ( ph  ->  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )  =  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  J ) ) )
127114, 121, 1263eqtrd 2448 . 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 ) ) )
12810, 23, 1273eqtrd 2448 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 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   _Vcvv 2924    \ cdif 3285   {csn 3782   class class class wbr 4180    e. cmpt 4234   -->wf 5417   ` cfv 5421  (class class class)co 6048   RRcr 8953   0cc0 8954    + caddc 8957   RR*cxr 9083    < clt 9084    <_ cle 9085    - cmin 9255   NN0cn0 10185   ...cfz 11007   Basecbs 13432   .rcmulr 13493   0gc0g 13686    gsumg cgsu 13687   Mndcmnd 14647   Ringcrg 15623  Poly1cpl1 16534  coe1cco1 16537   deg1 cdg1 19938
This theorem is referenced by:  coe1mul4  19984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-ofr 6273  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-fz 11008  df-fzo 11099  df-seq 11287  df-hash 11582  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-0g 13690  df-gsum 13691  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-mhm 14701  df-submnd 14702  df-grp 14775  df-minusg 14776  df-mulg 14778  df-ghm 14967  df-cntz 15079  df-cmn 15377  df-abl 15378  df-mgp 15612  df-rng 15626  df-cring 15627  df-ur 15628  df-psr 16380  df-mpl 16382  df-opsr 16388  df-psr1 16539  df-ply1 16541  df-coe1 16544  df-cnfld 16667  df-mdeg 19939  df-deg1 19940
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