Metamath Proof Explorer

Theorem crim

Description: The real part of a complex number representation. Definition 10-3.1 of Gleason p. 132. (Contributed by NM, 12-May-2005) (Revised by Mario Carneiro, 7-Nov-2013)

Ref Expression
Assertion crim 
|- ( ( A e. RR /\ B e. RR ) -> ( Im ` ( A + ( _i x. B ) ) ) = B )

Proof

Step Hyp Ref Expression
1 recn 
 |-  ( A e. RR -> A e. CC )
2 ax-icn 
 |-  _i e. CC
3 recn 
 |-  ( B e. RR -> B e. CC )
4 mulcl 
 |-  ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
5 2 3 4 sylancr 
 |-  ( B e. RR -> ( _i x. B ) e. CC )
6 addcl 
 |-  ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A + ( _i x. B ) ) e. CC )
7 1 5 6 syl2an 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A + ( _i x. B ) ) e. CC )
8 imval 
 |-  ( ( A + ( _i x. B ) ) e. CC -> ( Im ` ( A + ( _i x. B ) ) ) = ( Re ` ( ( A + ( _i x. B ) ) / _i ) ) )
9 7 8 syl 
 |-  ( ( A e. RR /\ B e. RR ) -> ( Im ` ( A + ( _i x. B ) ) ) = ( Re ` ( ( A + ( _i x. B ) ) / _i ) ) )
10 2 4 mpan 
 |-  ( B e. CC -> ( _i x. B ) e. CC )
11 ine0 
 |-  _i =/= 0
12 divdir 
 |-  ( ( A e. CC /\ ( _i x. B ) e. CC /\ ( _i e. CC /\ _i =/= 0 ) ) -> ( ( A + ( _i x. B ) ) / _i ) = ( ( A / _i ) + ( ( _i x. B ) / _i ) ) )
13 12 3expa 
 |-  ( ( ( A e. CC /\ ( _i x. B ) e. CC ) /\ ( _i e. CC /\ _i =/= 0 ) ) -> ( ( A + ( _i x. B ) ) / _i ) = ( ( A / _i ) + ( ( _i x. B ) / _i ) ) )
14 2 11 13 mpanr12 
 |-  ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( ( A + ( _i x. B ) ) / _i ) = ( ( A / _i ) + ( ( _i x. B ) / _i ) ) )
15 10 14 sylan2 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( A + ( _i x. B ) ) / _i ) = ( ( A / _i ) + ( ( _i x. B ) / _i ) ) )
16 divrec2 
 |-  ( ( A e. CC /\ _i e. CC /\ _i =/= 0 ) -> ( A / _i ) = ( ( 1 / _i ) x. A ) )
17 2 11 16 mp3an23 
 |-  ( A e. CC -> ( A / _i ) = ( ( 1 / _i ) x. A ) )
18 irec 
 |-  ( 1 / _i ) = -u _i
19 18 oveq1i 
 |-  ( ( 1 / _i ) x. A ) = ( -u _i x. A )
20 19 a1i 
 |-  ( A e. CC -> ( ( 1 / _i ) x. A ) = ( -u _i x. A ) )
21 mulneg12 
 |-  ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. A ) = ( _i x. -u A ) )
22 2 21 mpan 
 |-  ( A e. CC -> ( -u _i x. A ) = ( _i x. -u A ) )
23 17 20 22 3eqtrd 
 |-  ( A e. CC -> ( A / _i ) = ( _i x. -u A ) )
24 divcan3 
 |-  ( ( B e. CC /\ _i e. CC /\ _i =/= 0 ) -> ( ( _i x. B ) / _i ) = B )
25 2 11 24 mp3an23 
 |-  ( B e. CC -> ( ( _i x. B ) / _i ) = B )
26 23 25 oveqan12d 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( A / _i ) + ( ( _i x. B ) / _i ) ) = ( ( _i x. -u A ) + B ) )
27 negcl 
 |-  ( A e. CC -> -u A e. CC )
28 mulcl 
 |-  ( ( _i e. CC /\ -u A e. CC ) -> ( _i x. -u A ) e. CC )
29 2 27 28 sylancr 
 |-  ( A e. CC -> ( _i x. -u A ) e. CC )
30 addcom 
 |-  ( ( ( _i x. -u A ) e. CC /\ B e. CC ) -> ( ( _i x. -u A ) + B ) = ( B + ( _i x. -u A ) ) )
31 29 30 sylan 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( _i x. -u A ) + B ) = ( B + ( _i x. -u A ) ) )
32 15 26 31 3eqtrrd 
 |-  ( ( A e. CC /\ B e. CC ) -> ( B + ( _i x. -u A ) ) = ( ( A + ( _i x. B ) ) / _i ) )
33 1 3 32 syl2an 
 |-  ( ( A e. RR /\ B e. RR ) -> ( B + ( _i x. -u A ) ) = ( ( A + ( _i x. B ) ) / _i ) )
34 33 fveq2d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( Re ` ( B + ( _i x. -u A ) ) ) = ( Re ` ( ( A + ( _i x. B ) ) / _i ) ) )
35 id 
 |-  ( B e. RR -> B e. RR )
36 renegcl 
 |-  ( A e. RR -> -u A e. RR )
37 crre 
 |-  ( ( B e. RR /\ -u A e. RR ) -> ( Re ` ( B + ( _i x. -u A ) ) ) = B )
38 35 36 37 syl2anr 
 |-  ( ( A e. RR /\ B e. RR ) -> ( Re ` ( B + ( _i x. -u A ) ) ) = B )
39 9 34 38 3eqtr2d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( Im ` ( A + ( _i x. B ) ) ) = B )