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Metamath Proof Explorer

Theorem coe1tmfv1

Description: Nonzero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015)

Ref Expression
Hypotheses coe1tm.z 
|- .0. = ( 0g ` R )
coe1tm.k 
|- K = ( Base ` R )
coe1tm.p 
|- P = ( Poly1 ` R )
coe1tm.x 
|- X = ( var1 ` R )
coe1tm.m 
|- .x. = ( .s ` P )
coe1tm.n 
|- N = ( mulGrp ` P )
coe1tm.e 
|- .^ = ( .g ` N )
Assertion coe1tmfv1 
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( ( coe1 ` ( C .x. ( D .^ X ) ) ) ` D ) = C )

Proof

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 1 2 3 4 5 6 7 coe1tm 
 |-  ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( coe1 ` ( C .x. ( D .^ X ) ) ) = ( x e. NN0 |-> if ( x = D , C , .0. ) ) )
9 8 fveq1d 
 |-  ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( ( coe1 ` ( C .x. ( D .^ X ) ) ) ` D ) = ( ( x e. NN0 |-> if ( x = D , C , .0. ) ) ` D ) )
10 eqid 
 |-  ( x e. NN0 |-> if ( x = D , C , .0. ) ) = ( x e. NN0 |-> if ( x = D , C , .0. ) )
11 iftrue 
 |-  ( x = D -> if ( x = D , C , .0. ) = C )
12 simp3 
 |-  ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> D e. NN0 )
13 simp2 
 |-  ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> C e. K )
14 10 11 12 13 fvmptd3 
 |-  ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( ( x e. NN0 |-> if ( x = D , C , .0. ) ) ` D ) = C )
15 9 14 eqtrd 
 |-  ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( ( coe1 ` ( C .x. ( D .^ X ) ) ) ` D ) = C )