Metamath Proof Explorer

Theorem mpaaroot

Description: The minimal polynomial of an algebraic number has the number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014)

Ref Expression
Assertion mpaaroot 
|- ( A e. AA -> ( ( minPolyAA ` A ) ` A ) = 0 )

Proof

Step Hyp Ref Expression
1 mpaalem 
 |-  ( A e. AA -> ( ( minPolyAA ` A ) e. ( Poly ` QQ ) /\ ( ( deg ` ( minPolyAA ` A ) ) = ( degAA ` A ) /\ ( ( minPolyAA ` A ) ` A ) = 0 /\ ( ( coeff ` ( minPolyAA ` A ) ) ` ( degAA ` A ) ) = 1 ) ) )
2 simpr2 
 |-  ( ( ( minPolyAA ` A ) e. ( Poly ` QQ ) /\ ( ( deg ` ( minPolyAA ` A ) ) = ( degAA ` A ) /\ ( ( minPolyAA ` A ) ` A ) = 0 /\ ( ( coeff ` ( minPolyAA ` A ) ) ` ( degAA ` A ) ) = 1 ) ) -> ( ( minPolyAA ` A ) ` A ) = 0 )
3 1 2 syl 
 |-  ( A e. AA -> ( ( minPolyAA ` A ) ` A ) = 0 )