Step |
Hyp |
Ref |
Expression |
1 |
|
coe1tm.z |
|- .0. = ( 0g ` R ) |
2 |
|
coe1tm.k |
|- K = ( Base ` R ) |
3 |
|
coe1tm.p |
|- P = ( Poly1 ` R ) |
4 |
|
coe1tm.x |
|- X = ( var1 ` R ) |
5 |
|
coe1tm.m |
|- .x. = ( .s ` P ) |
6 |
|
coe1tm.n |
|- N = ( mulGrp ` P ) |
7 |
|
coe1tm.e |
|- .^ = ( .g ` N ) |
8 |
|
coe1tmmul.b |
|- B = ( Base ` P ) |
9 |
|
coe1tmmul.t |
|- .xb = ( .r ` P ) |
10 |
|
coe1tmmul.u |
|- .X. = ( .r ` R ) |
11 |
|
coe1tmmul.a |
|- ( ph -> A e. B ) |
12 |
|
coe1tmmul.r |
|- ( ph -> R e. Ring ) |
13 |
|
coe1tmmul.c |
|- ( ph -> C e. K ) |
14 |
|
coe1tmmul.d |
|- ( ph -> D e. NN0 ) |
15 |
|
coe1tmmul2fv.y |
|- ( ph -> Y e. NN0 ) |
16 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
coe1tmmul2 |
|- ( ph -> ( coe1 ` ( A .xb ( C .x. ( D .^ X ) ) ) ) = ( x e. NN0 |-> if ( D <_ x , ( ( ( coe1 ` A ) ` ( x - D ) ) .X. C ) , .0. ) ) ) |
17 |
16
|
fveq1d |
|- ( ph -> ( ( coe1 ` ( A .xb ( C .x. ( D .^ X ) ) ) ) ` ( D + Y ) ) = ( ( x e. NN0 |-> if ( D <_ x , ( ( ( coe1 ` A ) ` ( x - D ) ) .X. C ) , .0. ) ) ` ( D + Y ) ) ) |
18 |
14 15
|
nn0addcld |
|- ( ph -> ( D + Y ) e. NN0 ) |
19 |
|
breq2 |
|- ( x = ( D + Y ) -> ( D <_ x <-> D <_ ( D + Y ) ) ) |
20 |
|
fvoveq1 |
|- ( x = ( D + Y ) -> ( ( coe1 ` A ) ` ( x - D ) ) = ( ( coe1 ` A ) ` ( ( D + Y ) - D ) ) ) |
21 |
20
|
oveq1d |
|- ( x = ( D + Y ) -> ( ( ( coe1 ` A ) ` ( x - D ) ) .X. C ) = ( ( ( coe1 ` A ) ` ( ( D + Y ) - D ) ) .X. C ) ) |
22 |
19 21
|
ifbieq1d |
|- ( x = ( D + Y ) -> if ( D <_ x , ( ( ( coe1 ` A ) ` ( x - D ) ) .X. C ) , .0. ) = if ( D <_ ( D + Y ) , ( ( ( coe1 ` A ) ` ( ( D + Y ) - D ) ) .X. C ) , .0. ) ) |
23 |
|
eqid |
|- ( x e. NN0 |-> if ( D <_ x , ( ( ( coe1 ` A ) ` ( x - D ) ) .X. C ) , .0. ) ) = ( x e. NN0 |-> if ( D <_ x , ( ( ( coe1 ` A ) ` ( x - D ) ) .X. C ) , .0. ) ) |
24 |
|
ovex |
|- ( ( ( coe1 ` A ) ` ( ( D + Y ) - D ) ) .X. C ) e. _V |
25 |
1
|
fvexi |
|- .0. e. _V |
26 |
24 25
|
ifex |
|- if ( D <_ ( D + Y ) , ( ( ( coe1 ` A ) ` ( ( D + Y ) - D ) ) .X. C ) , .0. ) e. _V |
27 |
22 23 26
|
fvmpt |
|- ( ( D + Y ) e. NN0 -> ( ( x e. NN0 |-> if ( D <_ x , ( ( ( coe1 ` A ) ` ( x - D ) ) .X. C ) , .0. ) ) ` ( D + Y ) ) = if ( D <_ ( D + Y ) , ( ( ( coe1 ` A ) ` ( ( D + Y ) - D ) ) .X. C ) , .0. ) ) |
28 |
18 27
|
syl |
|- ( ph -> ( ( x e. NN0 |-> if ( D <_ x , ( ( ( coe1 ` A ) ` ( x - D ) ) .X. C ) , .0. ) ) ` ( D + Y ) ) = if ( D <_ ( D + Y ) , ( ( ( coe1 ` A ) ` ( ( D + Y ) - D ) ) .X. C ) , .0. ) ) |
29 |
14
|
nn0red |
|- ( ph -> D e. RR ) |
30 |
|
nn0addge1 |
|- ( ( D e. RR /\ Y e. NN0 ) -> D <_ ( D + Y ) ) |
31 |
29 15 30
|
syl2anc |
|- ( ph -> D <_ ( D + Y ) ) |
32 |
31
|
iftrued |
|- ( ph -> if ( D <_ ( D + Y ) , ( ( ( coe1 ` A ) ` ( ( D + Y ) - D ) ) .X. C ) , .0. ) = ( ( ( coe1 ` A ) ` ( ( D + Y ) - D ) ) .X. C ) ) |
33 |
14
|
nn0cnd |
|- ( ph -> D e. CC ) |
34 |
15
|
nn0cnd |
|- ( ph -> Y e. CC ) |
35 |
33 34
|
pncan2d |
|- ( ph -> ( ( D + Y ) - D ) = Y ) |
36 |
35
|
fveq2d |
|- ( ph -> ( ( coe1 ` A ) ` ( ( D + Y ) - D ) ) = ( ( coe1 ` A ) ` Y ) ) |
37 |
36
|
oveq1d |
|- ( ph -> ( ( ( coe1 ` A ) ` ( ( D + Y ) - D ) ) .X. C ) = ( ( ( coe1 ` A ) ` Y ) .X. C ) ) |
38 |
28 32 37
|
3eqtrd |
|- ( ph -> ( ( x e. NN0 |-> if ( D <_ x , ( ( ( coe1 ` A ) ` ( x - D ) ) .X. C ) , .0. ) ) ` ( D + Y ) ) = ( ( ( coe1 ` A ) ` Y ) .X. C ) ) |
39 |
17 38
|
eqtrd |
|- ( ph -> ( ( coe1 ` ( A .xb ( C .x. ( D .^ X ) ) ) ) ` ( D + Y ) ) = ( ( ( coe1 ` A ) ` Y ) .X. C ) ) |