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Theorem coe1mul3 20053
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 16694 . . . 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 1185 . . 3  |-  ( ph  ->  (coe1 `  ( F  .xb  G ) )  =  ( x  e.  NN0  |->  ( R  gsumg  ( y  e.  ( 0 ... x ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( x  -  y
) ) ) ) ) ) )
109fveq1d 5759 . 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 10309 . . 3  |-  ( ph  ->  ( I  +  J
)  e.  NN0 )
14 oveq2 6118 . . . . . 6  |-  ( x  =  ( I  +  J )  ->  (
0 ... x )  =  ( 0 ... (
I  +  J ) ) )
15 oveq1 6117 . . . . . . . 8  |-  ( x  =  ( I  +  J )  ->  (
x  -  y )  =  ( ( I  +  J )  -  y ) )
1615fveq2d 5761 . . . . . . 7  |-  ( x  =  ( I  +  J )  ->  (
(coe1 `  G ) `  ( x  -  y
) )  =  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )
1716oveq2d 6126 . . . . . 6  |-  ( x  =  ( I  +  J )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( x  -  y
) ) )  =  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) )
1814, 17mpteq12dv 4312 . . . . 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 6126 . . . 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 2442 . . . 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 6135 . . . 4  |-  ( R 
gsumg  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) )  e.  _V
2219, 20, 21fvmpt 5835 . . 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 2442 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
25 eqid 2442 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
26 rngmnd 15704 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
271, 26syl 16 . . . 4  |-  ( ph  ->  R  e.  Mnd )
28 ovex 6135 . . . . 5  |-  ( 0 ... ( I  +  J ) )  e. 
_V
2928a1i 11 . . . 4  |-  ( ph  ->  ( 0 ... (
I  +  J ) )  e.  _V )
3011nn0red 10306 . . . . . 6  |-  ( ph  ->  I  e.  RR )
31 nn0addge1 10297 . . . . . 6  |-  ( ( I  e.  RR  /\  J  e.  NN0 )  ->  I  <_  ( I  +  J ) )
3230, 12, 31syl2anc 644 . . . . 5  |-  ( ph  ->  I  <_  ( I  +  J ) )
33 fznn0 11144 . . . . . 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 890 . . . 4  |-  ( ph  ->  I  e.  ( 0 ... ( I  +  J ) ) )
361adantr 453 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  R  e.  Ring )
37 eqid 2442 . . . . . . . . 9  |-  (coe1 `  F
)  =  (coe1 `  F
)
3837, 7, 4, 24coe1f 16640 . . . . . . . 8  |-  ( F  e.  B  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
392, 38syl 16 . . . . . . 7  |-  ( ph  ->  (coe1 `  F ) : NN0 --> ( Base `  R
) )
40 elfznn0 11114 . . . . . . 7  |-  ( y  e.  ( 0 ... ( I  +  J
) )  ->  y  e.  NN0 )
41 ffvelrn 5897 . . . . . . 7  |-  ( ( (coe1 `  F ) : NN0 --> ( Base `  R
)  /\  y  e.  NN0 )  ->  ( (coe1 `  F ) `  y
)  e.  ( Base `  R ) )
4239, 40, 41syl2an 465 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
(coe1 `  F ) `  y )  e.  (
Base `  R )
)
43 eqid 2442 . . . . . . . . 9  |-  (coe1 `  G
)  =  (coe1 `  G
)
4443, 7, 4, 24coe1f 16640 . . . . . . . 8  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
453, 44syl 16 . . . . . . 7  |-  ( ph  ->  (coe1 `  G ) : NN0 --> ( Base `  R
) )
46 fznn0sub 11116 . . . . . . 7  |-  ( y  e.  ( 0 ... ( I  +  J
) )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
47 ffvelrn 5897 . . . . . . 7  |-  ( ( (coe1 `  G ) : NN0 --> ( Base `  R
)  /\  ( (
I  +  J )  -  y )  e. 
NN0 )  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  e.  (
Base `  R )
)
4845, 46, 47syl2an 465 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  e.  (
Base `  R )
)
4924, 6rngcl 15708 . . . . . 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 1185 . . . . 5  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  e.  ( Base `  R
) )
51 eqid 2442 . . . . 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 5922 . . . 4  |-  ( ph  ->  ( y  e.  ( 0 ... ( I  +  J ) ) 
|->  ( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) ) ) : ( 0 ... ( I  +  J
) ) --> ( Base `  R ) )
53 eldifsn 3951 . . . . . 6  |-  ( y  e.  ( ( 0 ... ( I  +  J ) )  \  { I } )  <-> 
( y  e.  ( 0 ... ( I  +  J ) )  /\  y  =/=  I
) )
5440adantl 454 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  NN0 )
5554nn0red 10306 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  RR )
5630adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  I  e.  RR )
5755, 56lttri2d 9243 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  =/=  I  <->  ( y  <  I  \/  I  < 
y ) ) )
583ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  G  e.  B )
5946adantl 454 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
6059adantr 453 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( I  +  J
)  -  y )  e.  NN0 )
61 coe1mul3.d . . . . . . . . . . . . . . . . 17  |-  D  =  ( deg1  `  R )
6261, 4, 7deg1xrcl 20036 . . . . . . . . . . . . . . . 16  |-  ( G  e.  B  ->  ( D `  G )  e.  RR* )
633, 62syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  G
)  e.  RR* )
6463ad2antrr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  e.  RR* )
6512nn0red 10306 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  J  e.  RR )
6665rexrd 9165 . . . . . . . . . . . . . . 15  |-  ( ph  ->  J  e.  RR* )
6766ad2antrr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  J  e.  RR* )
6813nn0red 10306 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( I  +  J
)  e.  RR )
6968adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
I  +  J )  e.  RR )
7069, 55resubcld 9496 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  RR )
7170rexrd 9165 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( I  +  J
)  -  y )  e.  RR* )
7271adantr 453 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( I  +  J
)  -  y )  e.  RR* )
73 coe1mul3.g3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  G
)  <_  J )
7473ad2antrr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  <_  J )
7565adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  J  e.  RR )
7655, 56, 75ltadd1d 9650 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  <  I  <->  ( y  +  J )  <  (
I  +  J ) ) )
7755, 75, 69ltaddsub2d 9658 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( y  +  J
)  <  ( I  +  J )  <->  J  <  ( ( I  +  J
)  -  y ) ) )
7876, 77bitrd 246 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  <  I  <->  J  <  ( ( I  +  J
)  -  y ) ) )
7978biimpa 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  J  <  ( ( I  +  J )  -  y
) )
8064, 67, 72, 74, 79xrlelttrd 10781 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  ( D `  G )  <  ( ( I  +  J )  -  y
) )
8161, 4, 7, 25, 43deg1lt 20051 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  =  ( 0g `  R ) )
8382oveq2d 6126 . . . . . . . . . . 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 15732 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  (
(coe1 `  F ) `  y )  e.  (
Base `  R )
)  ->  ( (
(coe1 `  F ) `  y )  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
8536, 42, 84syl2anc 644 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( (coe1 `  F ) `  y )  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
8685adantr 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( (coe1 `  F ) `  y )  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
8783, 86eqtrd 2474 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  y  <  I )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
882ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  F  e.  B )
8954adantr 453 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  y  e.  NN0 )
9061, 4, 7deg1xrcl 20036 . . . . . . . . . . . . . . . 16  |-  ( F  e.  B  ->  ( D `  F )  e.  RR* )
912, 90syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  F
)  e.  RR* )
9291ad2antrr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  e.  RR* )
9330rexrd 9165 . . . . . . . . . . . . . . 15  |-  ( ph  ->  I  e.  RR* )
9493ad2antrr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  I  e.  RR* )
9555rexrd 9165 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  y  e.  RR* )
9695adantr 453 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  y  e.  RR* )
97 coe1mul3.f3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D `  F
)  <_  I )
9897ad2antrr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  <_  I )
99 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  I  <  y )
10092, 94, 96, 98, 99xrlelttrd 10781 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  ( D `  F )  <  y )
10161, 4, 7, 25, 37deg1lt 20051 . . . . . . . . . . . . 13  |-  ( ( F  e.  B  /\  y  e.  NN0  /\  ( D `  F )  <  y )  ->  (
(coe1 `  F ) `  y )  =  ( 0g `  R ) )
10288, 89, 100, 101syl3anc 1185 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
(coe1 `  F ) `  y )  =  ( 0g `  R ) )
103102oveq1d 6125 . . . . . . . . . . 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 15731 . . . . . . . . . . . . 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 644 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( 0g `  R
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
106105adantr 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
( 0g `  R
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
107103, 106eqtrd 2474 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  I  <  y )  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
10887, 107jaodan 762 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  /\  (
y  <  I  \/  I  <  y ) )  ->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
109108ex 425 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
( y  <  I  \/  I  <  y )  ->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) ) )
11057, 109sylbid 208 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( 0 ... (
I  +  J ) ) )  ->  (
y  =/=  I  -> 
( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) ) )
111110impr 604 . . . . . 6  |-  ( (
ph  /\  ( y  e.  ( 0 ... (
I  +  J ) )  /\  y  =/=  I ) )  -> 
( ( (coe1 `  F
) `  y )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( 0g `  R
) )
11253, 111sylan2b 463 . . . . 5  |-  ( (
ph  /\  y  e.  ( ( 0 ... ( I  +  J
) )  \  {
I } ) )  ->  ( ( (coe1 `  F ) `  y
)  .x.  ( (coe1 `  G ) `  (
( I  +  J
)  -  y ) ) )  =  ( 0g `  R ) )
113112suppss2 6329 . . . 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 15576 . . 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 5757 . . . . . 6  |-  ( y  =  I  ->  (
(coe1 `  F ) `  y )  =  ( (coe1 `  F ) `  I ) )
116 oveq2 6118 . . . . . . 7  |-  ( y  =  I  ->  (
( I  +  J
)  -  y )  =  ( ( I  +  J )  -  I ) )
117116fveq2d 5761 . . . . . 6  |-  ( y  =  I  ->  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) )  =  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )
118115, 117oveq12d 6128 . . . . 5  |-  ( y  =  I  ->  (
( (coe1 `  F ) `  y )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  y
) ) )  =  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) ) )
119 ovex 6135 . . . . 5  |-  ( ( (coe1 `  F ) `  I )  .x.  (
(coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )  e. 
_V
120118, 51, 119fvmpt 5835 . . . 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 10307 . . . . . 6  |-  ( ph  ->  I  e.  CC )
12312nn0cnd 10307 . . . . . 6  |-  ( ph  ->  J  e.  CC )
124122, 123pncan2d 9444 . . . . 5  |-  ( ph  ->  ( ( I  +  J )  -  I
)  =  J )
125124fveq2d 5761 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) )  =  ( (coe1 `  G ) `  J ) )
126125oveq2d 6126 . . 3  |-  ( ph  ->  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  ( ( I  +  J )  -  I
) ) )  =  ( ( (coe1 `  F
) `  I )  .x.  ( (coe1 `  G ) `  J ) ) )
127114, 121, 1263eqtrd 2478 . 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 2478 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 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   _Vcvv 2962    \ cdif 3303   {csn 3838   class class class wbr 4237    e. cmpt 4291   -->wf 5479   ` cfv 5483  (class class class)co 6110   RRcr 9020   0cc0 9021    + caddc 9024   RR*cxr 9150    < clt 9151    <_ cle 9152    - cmin 9322   NN0cn0 10252   ...cfz 11074   Basecbs 13500   .rcmulr 13561   0gc0g 13754    gsumg cgsu 13755   Mndcmnd 14715   Ringcrg 15691  Poly1cpl1 16602  coe1cco1 16605   deg1 cdg1 20008
This theorem is referenced by:  coe1mul4  20054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099  ax-addf 9100  ax-mulf 9101
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-ofr 6335  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-2o 6754  df-oadd 6757  df-er 6934  df-map 7049  df-pm 7050  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-sup 7475  df-oi 7508  df-card 7857  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-fz 11075  df-fzo 11167  df-seq 11355  df-hash 11650  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-mulr 13574  df-starv 13575  df-sca 13576  df-vsca 13577  df-tset 13579  df-ple 13580  df-ds 13582  df-unif 13583  df-0g 13758  df-gsum 13759  df-mre 13842  df-mrc 13843  df-acs 13845  df-mnd 14721  df-mhm 14769  df-submnd 14770  df-grp 14843  df-minusg 14844  df-mulg 14846  df-ghm 15035  df-cntz 15147  df-cmn 15445  df-abl 15446  df-mgp 15680  df-rng 15694  df-cring 15695  df-ur 15696  df-psr 16448  df-mpl 16450  df-opsr 16456  df-psr1 16607  df-ply1 16609  df-coe1 16612  df-cnfld 16735  df-mdeg 20009  df-deg1 20010
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