Metamath Proof Explorer

Theorem cru

Description: The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of Gleason p. 130. (Contributed by NM, 9-May-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion cru 
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) <-> ( A = C /\ B = D ) ) )

Proof

Step Hyp Ref Expression
1 simplrl 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> C e. RR )
2 1 recnd 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> C e. CC )
3 simplll 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> A e. RR )
4 3 recnd 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> A e. CC )
5 simpr 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) )
6 ax-icn 
 |-  _i e. CC
7 6 a1i 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> _i e. CC )
8 simpllr 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> B e. RR )
9 8 recnd 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> B e. CC )
10 7 9 mulcld 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( _i x. B ) e. CC )
11 simplrr 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> D e. RR )
12 11 recnd 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> D e. CC )
13 7 12 mulcld 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( _i x. D ) e. CC )
14 4 10 2 13 addsubeq4d 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) <-> ( C - A ) = ( ( _i x. B ) - ( _i x. D ) ) ) )
15 5 14 mpbid 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( C - A ) = ( ( _i x. B ) - ( _i x. D ) ) )
16 8 11 resubcld 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( B - D ) e. RR )
17 7 9 12 subdid 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( _i x. ( B - D ) ) = ( ( _i x. B ) - ( _i x. D ) ) )
18 17 15 eqtr4d 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( _i x. ( B - D ) ) = ( C - A ) )
19 1 3 resubcld 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( C - A ) e. RR )
20 18 19 eqeltrd 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( _i x. ( B - D ) ) e. RR )
21 rimul 
 |-  ( ( ( B - D ) e. RR /\ ( _i x. ( B - D ) ) e. RR ) -> ( B - D ) = 0 )
22 16 20 21 syl2anc 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( B - D ) = 0 )
23 9 12 22 subeq0d 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> B = D )
24 23 oveq2d 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( _i x. B ) = ( _i x. D ) )
25 24 oveq1d 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( ( _i x. B ) - ( _i x. D ) ) = ( ( _i x. D ) - ( _i x. D ) ) )
26 13 subidd 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( ( _i x. D ) - ( _i x. D ) ) = 0 )
27 15 25 26 3eqtrd 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( C - A ) = 0 )
28 2 4 27 subeq0d 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> C = A )
29 28 eqcomd 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> A = C )
30 29 23 jca 
 |-  ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) ) -> ( A = C /\ B = D ) )
31 30 ex 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) -> ( A = C /\ B = D ) ) )
32 oveq2 
 |-  ( B = D -> ( _i x. B ) = ( _i x. D ) )
33 oveq12 
 |-  ( ( A = C /\ ( _i x. B ) = ( _i x. D ) ) -> ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) )
34 32 33 sylan2 
 |-  ( ( A = C /\ B = D ) -> ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) )
35 31 34 impbid1 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) <-> ( A = C /\ B = D ) ) )