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Theorem coe1mul3 22666
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 18506 . . . 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 1226 . . 3  |-  ( ph  ->  (coe1 `  ( F  .xb  G ) )  =  ( x  e.  NN0  |->  ( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( x  -  y
) ) ) ) ) ) )
109fveq1d 5850 . 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 10852 . . 3  |-  ( ph  ->  ( I  +  J
)  e.  NN0 )
14 oveq2 6278 . . . . . 6  |-  ( x  =  ( I  +  J )  ->  (
0 ... x )  =  ( 0 ... (
I  +  J ) ) )
15 oveq1 6277 . . . . . . . 8  |-  ( x  =  ( I  +  J )  ->  (
x  -  y )  =  ( ( I  +  J )  -  y ) )
1615fveq2d 5852 . . . . . . 7  |-  ( x  =  ( I  +  J )  ->  (
(coe1 `  G ) `  ( x  -  y
) )  =  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )
1716oveq2d 6286 . . . . . 6  |-  ( x  =  ( I  +  J )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( x  -  y
) ) )  =  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) )
1814, 17mpteq12dv 4517 . . . . 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 6286 . . . 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 2454 . . . 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 6298 . . . 4  |-  ( R 
gsumg  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) )  e.  _V
2219, 20, 21fvmpt 5931 . . 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 2454 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
25 eqid 2454 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
26 ringmnd 17402 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
271, 26syl 16 . . . 4  |-  ( ph  ->  R  e.  Mnd )
28 ovex 6298 . . . . 5  |-  ( 0 ... ( I  +  J ) )  e. 
_V
2928a1i 11 . . . 4  |-  ( ph  ->  ( 0 ... (
I  +  J ) )  e.  _V )
3011nn0red 10849 . . . . . 6  |-  ( ph  ->  I  e.  RR )
31 nn0addge1 10838 . . . . . 6  |-  ( ( I  e.  RR  /\  J  e.  NN0 )  ->  I  <_  ( I  +  J ) )
3230, 12, 31syl2anc 659 . . . . 5  |-  ( ph  ->  I  <_  ( I  +  J ) )
33 fznn0 11774 . . . . . 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 920 . . . 4  |-  ( ph  ->  I  e.  ( 0 ... ( I  +  J ) ) )
361adantr 463 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  R  e.  Ring )
37 eqid 2454 . . . . . . . . 9  |-  (coe1 `  F
)  =  (coe1 `  F
)
3837, 7, 4, 24coe1f 18445 . . . . . . . 8  |-  ( F  e.  B  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
392, 38syl 16 . . . . . . 7  |-  ( ph  ->  (coe1 `  F ) : NN0 --> ( Base `  R
) )
40 elfznn0 11775 . . . . . . 7  |-  ( y  e.  ( 0 ... ( I  +  J
) )  ->  y  e.  NN0 )
41 ffvelrn 6005 . . . . . . 7  |-  ( ( (coe1 `  F ) : NN0 --> ( Base `  R
)  /\  y  e.  NN0 )  ->  ( (coe1 `  F ) `  y
)  e.  ( Base `  R ) )
4239, 40, 41syl2an 475 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
(coe1 `  F ) `  y )  e.  (
Base `  R )
)
43 eqid 2454 . . . . . . . . 9  |-  (coe1 `  G
)  =  (coe1 `  G
)
4443, 7, 4, 24coe1f 18445 . . . . . . . 8  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
453, 44syl 16 . . . . . . 7  |-  ( ph  ->  (coe1 `  G ) : NN0 --> ( Base `  R
) )
46 fznn0sub 11720 . . . . . . 7  |-  ( y  e.  ( 0 ... ( I  +  J
) )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
47 ffvelrn 6005 . . . . . . 7  |-  ( ( (coe1 `  G ) : NN0 --> ( Base `  R
)  /\  ( (
I  +  J )  -  y )  e. 
NN0 )  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  e.  (
Base `  R )
)
4845, 46, 47syl2an 475 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  e.  (
Base `  R )
)
4924, 6ringcl 17407 . . . . . 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 1226 . . . . 5  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  e.  ( Base `  R
) )
51 eqid 2454 . . . . 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 6031 . . . 4  |-  ( ph  ->  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) : ( 0 ... ( I  +  J
) ) --> ( Base `  R ) )
53 eldifsn 4141 . . . . . 6  |-  ( y  e.  ( ( 0 ... ( I  +  J ) )  \  { I } )  <-> 
( y  e.  ( 0 ... ( I  +  J ) )  /\  y  =/=  I
) )
5440adantl 464 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  NN0 )
5554nn0red 10849 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  RR )
5630adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  I  e.  RR )
5755, 56lttri2d 9713 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  =/=  I  <->  ( y  <  I  \/  I  < 
y ) ) )
583ad2antrr 723 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  G  e.  B )
5946adantl 464 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
6059adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
61 coe1mul3.d . . . . . . . . . . . . . . . . 17  |-  D  =  ( deg1  `  R )
6261, 4, 7deg1xrcl 22648 . . . . . . . . . . . . . . . 16  |-  ( G  e.  B  ->  ( D `  G )  e.  RR* )
633, 62syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  G
)  e.  RR* )
6463ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  e.  RR* )
6512nn0red 10849 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  J  e.  RR )
6665rexrd 9632 . . . . . . . . . . . . . . 15  |-  ( ph  ->  J  e.  RR* )
6766ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  J  e.  RR* )
6813nn0red 10849 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( I  +  J
)  e.  RR )
6968adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
I  +  J )  e.  RR )
7069, 55resubcld 9983 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  RR )
7170rexrd 9632 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  RR* )
7271adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( I  +  J
)  -  y )  e.  RR* )
73 coe1mul3.g3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  G
)  <_  J )
7473ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  <_  J )
7565adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  J  e.  RR )
7655, 56, 75ltadd1d 10141 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  <  I  <->  ( y  +  J )  <  (
I  +  J ) ) )
7755, 75, 69ltaddsub2d 10149 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( y  +  J
)  <  ( I  +  J )  <->  J  <  ( ( I  +  J
)  -  y ) ) )
7876, 77bitrd 253 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  <  I  <->  J  <  ( ( I  +  J
)  -  y ) ) )
7978biimpa 482 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  J  <  ( ( I  +  J )  -  y
) )
8064, 67, 72, 74, 79xrlelttrd 11366 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  <  ( ( I  +  J )  -  y
) )
8161, 4, 7, 25, 43deg1lt 22664 . . . . . . . . . . . . 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 1226 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  =  ( 0g `  R ) )
8382oveq2d 6286 . . . . . . . . . . 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, 25ringrz 17431 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  (
(coe1 `  F ) `  y )  e.  (
Base `  R )
)  ->  ( (
(coe1 `  F ) `  y )  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
8536, 42, 84syl2anc 659 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( (coe1 `  F ) `  y )  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
8685adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( (coe1 `  F ) `  y )  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
8783, 86eqtrd 2495 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
882ad2antrr 723 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  F  e.  B )
8954adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  y  e.  NN0 )
9061, 4, 7deg1xrcl 22648 . . . . . . . . . . . . . . . 16  |-  ( F  e.  B  ->  ( D `  F )  e.  RR* )
912, 90syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  F
)  e.  RR* )
9291ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  e.  RR* )
9330rexrd 9632 . . . . . . . . . . . . . . 15  |-  ( ph  ->  I  e.  RR* )
9493ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  I  e.  RR* )
9555rexrd 9632 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  RR* )
9695adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  y  e.  RR* )
97 coe1mul3.f3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  F
)  <_  I )
9897ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  <_  I )
99 simpr 459 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  I  <  y )
10092, 94, 96, 98, 99xrlelttrd 11366 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  <  y )
10161, 4, 7, 25, 37deg1lt 22664 . . . . . . . . . . . . 13  |-  ( ( F  e.  B  /\  y  e.  NN0  /\  ( D `  F )  <  y )  ->  (
(coe1 `  F ) `  y )  =  ( 0g `  R ) )
10288, 89, 100, 101syl3anc 1226 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
(coe1 `  F ) `  y )  =  ( 0g `  R ) )
103102oveq1d 6285 . . . . . . . . . . 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, 25ringlz 17430 . . . . . . . . . . . . 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 659 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( 0g `  R
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
106105adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
( 0g `  R
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
107103, 106eqtrd 2495 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
10887, 107jaodan 783 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  (
y  <  I  \/  I  <  y ) )  ->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
109108ex 432 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( y  <  I  \/  I  <  y )  ->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) ) )
11057, 109sylbid 215 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  =/=  I  -> 
( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) ) )
111110impr 617 . . . . . 6  |-  ( (
ph  /\  ( y  e.  ( 0 ... (
I  +  J ) )  /\  y  =/=  I ) )  -> 
( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
11253, 111sylan2b 473 . . . . 5  |-  ( (
ph  /\  y  e.  ( ( 0 ... ( I  +  J
) )  \  {
I } ) )  ->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
113112, 29suppss2 6926 . . . 4  |-  ( ph  ->  ( ( y  e.  ( 0 ... (
I  +  J ) )  |->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) ) ) supp  ( 0g `  R ) ) 
C_  { I }
)
11424, 25, 27, 29, 35, 52, 113gsumpt 17184 . . 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 5848 . . . . . 6  |-  ( y  =  I  ->  (
(coe1 `  F ) `  y )  =  ( (coe1 `  F ) `  I ) )
116 oveq2 6278 . . . . . . 7  |-  ( y  =  I  ->  (
( I  +  J
)  -  y )  =  ( ( I  +  J )  -  I ) )
117116fveq2d 5852 . . . . . 6  |-  ( y  =  I  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  =  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )
118115, 117oveq12d 6288 . . . . 5  |-  ( y  =  I  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) ) )
119 ovex 6298 . . . . 5  |-  ( ( (coe1 `  F ) `  I )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )  e. 
_V
120118, 51, 119fvmpt 5931 . . . 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 10850 . . . . . 6  |-  ( ph  ->  I  e.  CC )
12312nn0cnd 10850 . . . . . 6  |-  ( ph  ->  J  e.  CC )
124122, 123pncan2d 9924 . . . . 5  |-  ( ph  ->  ( ( I  +  J )  -  I
)  =  J )
125124fveq2d 5852 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) )  =  ( (coe1 `  G ) `  J ) )
126125oveq2d 6286 . . 3  |-  ( ph  ->  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )  =  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  J ) ) )
127114, 121, 1263eqtrd 2499 . 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 2499 1  |-  ( ph  ->  ( (coe1 `  ( F  .xb  G ) ) `  ( I  +  J
) )  =  ( ( (coe1 `  F ) `  I )  .x.  (
(coe1 `  G ) `  J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106    \ cdif 3458   {csn 4016   class class class wbr 4439    |-> cmpt 4497   -->wf 5566   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481    + caddc 9484   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9796   NN0cn0 10791   ...cfz 11675   Basecbs 14716   .rcmulr 14785   0gc0g 14929    gsumg cgsu 14930   Mndcmnd 16118   Ringcrg 17393  Poly1cpl1 18411  coe1cco1 18412   deg1 cdg1 22618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-ofr 6514  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-0g 14931  df-gsum 14932  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-grp 16256  df-minusg 16257  df-mulg 16259  df-ghm 16464  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-psr 18200  df-mpl 18202  df-opsr 18204  df-psr1 18414  df-ply1 18416  df-coe1 18417  df-cnfld 18616  df-mdeg 22619  df-deg1 22620
This theorem is referenced by:  coe1mul4  22667
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