Metamath Proof Explorer


Theorem imre

Description: The imaginary part of a complex number in terms of the real part function. (Contributed by NM, 12-May-2005) (Revised by Mario Carneiro, 6-Nov-2013)

Ref Expression
Assertion imre A A = i A

Proof

Step Hyp Ref Expression
1 imval A A = A i
2 ax-icn i
3 ine0 i 0
4 divrec2 A i i 0 A i = 1 i A
5 2 3 4 mp3an23 A A i = 1 i A
6 irec 1 i = i
7 6 oveq1i 1 i A = i A
8 5 7 eqtrdi A A i = i A
9 8 fveq2d A A i = i A
10 1 9 eqtrd A A = i A