Theorem reim0

Description: The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005) (Revised by Mario Carneiro, 7-Nov-2013)

	Ref	Expression
Assertion	reim0	|- ( A e. RR -> ( Im ` A ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		recn	|- ( A e. RR -> A e. CC )
2		it0e0	|- ( _i x. 0 ) = 0
3	2	oveq2i	|- ( A + ( _i x. 0 ) ) = ( A + 0 )
4		addid1	|- ( A e. CC -> ( A + 0 ) = A )
5	3 4	eqtrid	|- ( A e. CC -> ( A + ( _i x. 0 ) ) = A )
6	1 5	syl	|- ( A e. RR -> ( A + ( _i x. 0 ) ) = A )
7	6	fveq2d	|- ( A e. RR -> ( Im ` ( A + ( _i x. 0 ) ) ) = ( Im ` A ) )
8		0re	|- 0 e. RR
9		crim	|- ( ( A e. RR /\ 0 e. RR ) -> ( Im ` ( A + ( _i x. 0 ) ) ) = 0 )
10	8 9	mpan2	|- ( A e. RR -> ( Im ` ( A + ( _i x. 0 ) ) ) = 0 )
11	7 10	eqtr3d	|- ( A e. RR -> ( Im ` A ) = 0 )