Step |
Hyp |
Ref |
Expression |
1 |
|
coe1sclmul.p |
|- P = ( Poly1 ` R ) |
2 |
|
coe1sclmul.b |
|- B = ( Base ` P ) |
3 |
|
coe1sclmul.k |
|- K = ( Base ` R ) |
4 |
|
coe1sclmul.a |
|- A = ( algSc ` P ) |
5 |
|
coe1sclmul.t |
|- .xb = ( .r ` P ) |
6 |
|
coe1sclmul.u |
|- .x. = ( .r ` R ) |
7 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
8 |
|
eqid |
|- ( var1 ` R ) = ( var1 ` R ) |
9 |
|
eqid |
|- ( .s ` P ) = ( .s ` P ) |
10 |
|
eqid |
|- ( mulGrp ` P ) = ( mulGrp ` P ) |
11 |
|
eqid |
|- ( .g ` ( mulGrp ` P ) ) = ( .g ` ( mulGrp ` P ) ) |
12 |
|
simp3 |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> Y e. B ) |
13 |
|
simp1 |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> R e. Ring ) |
14 |
|
simp2 |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> X e. K ) |
15 |
|
0nn0 |
|- 0 e. NN0 |
16 |
15
|
a1i |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> 0 e. NN0 ) |
17 |
7 3 1 8 9 10 11 2 5 6 12 13 14 16
|
coe1tmmul2 |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> ( coe1 ` ( Y .xb ( X ( .s ` P ) ( 0 ( .g ` ( mulGrp ` P ) ) ( var1 ` R ) ) ) ) ) = ( x e. NN0 |-> if ( 0 <_ x , ( ( ( coe1 ` Y ) ` ( x - 0 ) ) .x. X ) , ( 0g ` R ) ) ) ) |
18 |
3 1 8 9 10 11 4
|
ply1scltm |
|- ( ( R e. Ring /\ X e. K ) -> ( A ` X ) = ( X ( .s ` P ) ( 0 ( .g ` ( mulGrp ` P ) ) ( var1 ` R ) ) ) ) |
19 |
18
|
3adant3 |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> ( A ` X ) = ( X ( .s ` P ) ( 0 ( .g ` ( mulGrp ` P ) ) ( var1 ` R ) ) ) ) |
20 |
19
|
oveq2d |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> ( Y .xb ( A ` X ) ) = ( Y .xb ( X ( .s ` P ) ( 0 ( .g ` ( mulGrp ` P ) ) ( var1 ` R ) ) ) ) ) |
21 |
20
|
fveq2d |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> ( coe1 ` ( Y .xb ( A ` X ) ) ) = ( coe1 ` ( Y .xb ( X ( .s ` P ) ( 0 ( .g ` ( mulGrp ` P ) ) ( var1 ` R ) ) ) ) ) ) |
22 |
|
nn0ex |
|- NN0 e. _V |
23 |
22
|
a1i |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> NN0 e. _V ) |
24 |
|
fvexd |
|- ( ( ( R e. Ring /\ X e. K /\ Y e. B ) /\ x e. NN0 ) -> ( ( coe1 ` Y ) ` x ) e. _V ) |
25 |
|
simpl2 |
|- ( ( ( R e. Ring /\ X e. K /\ Y e. B ) /\ x e. NN0 ) -> X e. K ) |
26 |
|
eqid |
|- ( coe1 ` Y ) = ( coe1 ` Y ) |
27 |
26 2 1 3
|
coe1f |
|- ( Y e. B -> ( coe1 ` Y ) : NN0 --> K ) |
28 |
27
|
feqmptd |
|- ( Y e. B -> ( coe1 ` Y ) = ( x e. NN0 |-> ( ( coe1 ` Y ) ` x ) ) ) |
29 |
28
|
3ad2ant3 |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> ( coe1 ` Y ) = ( x e. NN0 |-> ( ( coe1 ` Y ) ` x ) ) ) |
30 |
|
fconstmpt |
|- ( NN0 X. { X } ) = ( x e. NN0 |-> X ) |
31 |
30
|
a1i |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> ( NN0 X. { X } ) = ( x e. NN0 |-> X ) ) |
32 |
23 24 25 29 31
|
offval2 |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> ( ( coe1 ` Y ) oF .x. ( NN0 X. { X } ) ) = ( x e. NN0 |-> ( ( ( coe1 ` Y ) ` x ) .x. X ) ) ) |
33 |
|
nn0ge0 |
|- ( x e. NN0 -> 0 <_ x ) |
34 |
33
|
iftrued |
|- ( x e. NN0 -> if ( 0 <_ x , ( ( ( coe1 ` Y ) ` ( x - 0 ) ) .x. X ) , ( 0g ` R ) ) = ( ( ( coe1 ` Y ) ` ( x - 0 ) ) .x. X ) ) |
35 |
|
nn0cn |
|- ( x e. NN0 -> x e. CC ) |
36 |
35
|
subid1d |
|- ( x e. NN0 -> ( x - 0 ) = x ) |
37 |
36
|
fveq2d |
|- ( x e. NN0 -> ( ( coe1 ` Y ) ` ( x - 0 ) ) = ( ( coe1 ` Y ) ` x ) ) |
38 |
37
|
oveq1d |
|- ( x e. NN0 -> ( ( ( coe1 ` Y ) ` ( x - 0 ) ) .x. X ) = ( ( ( coe1 ` Y ) ` x ) .x. X ) ) |
39 |
34 38
|
eqtrd |
|- ( x e. NN0 -> if ( 0 <_ x , ( ( ( coe1 ` Y ) ` ( x - 0 ) ) .x. X ) , ( 0g ` R ) ) = ( ( ( coe1 ` Y ) ` x ) .x. X ) ) |
40 |
39
|
mpteq2ia |
|- ( x e. NN0 |-> if ( 0 <_ x , ( ( ( coe1 ` Y ) ` ( x - 0 ) ) .x. X ) , ( 0g ` R ) ) ) = ( x e. NN0 |-> ( ( ( coe1 ` Y ) ` x ) .x. X ) ) |
41 |
32 40
|
eqtr4di |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> ( ( coe1 ` Y ) oF .x. ( NN0 X. { X } ) ) = ( x e. NN0 |-> if ( 0 <_ x , ( ( ( coe1 ` Y ) ` ( x - 0 ) ) .x. X ) , ( 0g ` R ) ) ) ) |
42 |
17 21 41
|
3eqtr4d |
|- ( ( R e. Ring /\ X e. K /\ Y e. B ) -> ( coe1 ` ( Y .xb ( A ` X ) ) ) = ( ( coe1 ` Y ) oF .x. ( NN0 X. { X } ) ) ) |