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Metamath Proof Explorer

Theorem mul02

Description: Multiplication by 0 . Theorem I.6 of Apostol p. 18. Based on ideas by Eric Schmidt. (Contributed by NM, 10-Aug-1999) (Revised by Scott Fenton, 3-Jan-2013)

Ref Expression
Assertion mul02 
|- ( A e. CC -> ( 0 x. A ) = 0 )

Proof

Step Hyp Ref Expression
1 cnre 
 |-  ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
2 0cn 
 |-  0 e. CC
3 recn 
 |-  ( x e. RR -> x e. CC )
4 ax-icn 
 |-  _i e. CC
5 recn 
 |-  ( y e. RR -> y e. CC )
6 mulcl 
 |-  ( ( _i e. CC /\ y e. CC ) -> ( _i x. y ) e. CC )
7 4 5 6 sylancr 
 |-  ( y e. RR -> ( _i x. y ) e. CC )
8 adddi 
 |-  ( ( 0 e. CC /\ x e. CC /\ ( _i x. y ) e. CC ) -> ( 0 x. ( x + ( _i x. y ) ) ) = ( ( 0 x. x ) + ( 0 x. ( _i x. y ) ) ) )
9 2 3 7 8 mp3an3an 
 |-  ( ( x e. RR /\ y e. RR ) -> ( 0 x. ( x + ( _i x. y ) ) ) = ( ( 0 x. x ) + ( 0 x. ( _i x. y ) ) ) )
10 mul02lem2 
 |-  ( x e. RR -> ( 0 x. x ) = 0 )
11 mul12 
 |-  ( ( 0 e. CC /\ _i e. CC /\ y e. CC ) -> ( 0 x. ( _i x. y ) ) = ( _i x. ( 0 x. y ) ) )
12 2 4 5 11 mp3an12i 
 |-  ( y e. RR -> ( 0 x. ( _i x. y ) ) = ( _i x. ( 0 x. y ) ) )
13 mul02lem2 
 |-  ( y e. RR -> ( 0 x. y ) = 0 )
14 13 oveq2d 
 |-  ( y e. RR -> ( _i x. ( 0 x. y ) ) = ( _i x. 0 ) )
15 12 14 eqtrd 
 |-  ( y e. RR -> ( 0 x. ( _i x. y ) ) = ( _i x. 0 ) )
16 10 15 oveqan12d 
 |-  ( ( x e. RR /\ y e. RR ) -> ( ( 0 x. x ) + ( 0 x. ( _i x. y ) ) ) = ( 0 + ( _i x. 0 ) ) )
17 9 16 eqtrd 
 |-  ( ( x e. RR /\ y e. RR ) -> ( 0 x. ( x + ( _i x. y ) ) ) = ( 0 + ( _i x. 0 ) ) )
18 cnre 
 |-  ( 0 e. CC -> E. x e. RR E. y e. RR 0 = ( x + ( _i x. y ) ) )
19 2 18 ax-mp 
 |-  E. x e. RR E. y e. RR 0 = ( x + ( _i x. y ) )
20 oveq2 
 |-  ( 0 = ( x + ( _i x. y ) ) -> ( 0 x. 0 ) = ( 0 x. ( x + ( _i x. y ) ) ) )
21 20 eqeq1d 
 |-  ( 0 = ( x + ( _i x. y ) ) -> ( ( 0 x. 0 ) = ( 0 + ( _i x. 0 ) ) <-> ( 0 x. ( x + ( _i x. y ) ) ) = ( 0 + ( _i x. 0 ) ) ) )
22 17 21 syl5ibrcom 
 |-  ( ( x e. RR /\ y e. RR ) -> ( 0 = ( x + ( _i x. y ) ) -> ( 0 x. 0 ) = ( 0 + ( _i x. 0 ) ) ) )
23 22 rexlimivv 
 |-  ( E. x e. RR E. y e. RR 0 = ( x + ( _i x. y ) ) -> ( 0 x. 0 ) = ( 0 + ( _i x. 0 ) ) )
24 19 23 ax-mp 
 |-  ( 0 x. 0 ) = ( 0 + ( _i x. 0 ) )
25 0re 
 |-  0 e. RR
26 mul02lem2 
 |-  ( 0 e. RR -> ( 0 x. 0 ) = 0 )
27 25 26 ax-mp 
 |-  ( 0 x. 0 ) = 0
28 24 27 eqtr3i 
 |-  ( 0 + ( _i x. 0 ) ) = 0
29 17 28 eqtrdi 
 |-  ( ( x e. RR /\ y e. RR ) -> ( 0 x. ( x + ( _i x. y ) ) ) = 0 )
30 oveq2 
 |-  ( A = ( x + ( _i x. y ) ) -> ( 0 x. A ) = ( 0 x. ( x + ( _i x. y ) ) ) )
31 30 eqeq1d 
 |-  ( A = ( x + ( _i x. y ) ) -> ( ( 0 x. A ) = 0 <-> ( 0 x. ( x + ( _i x. y ) ) ) = 0 ) )
32 29 31 syl5ibrcom 
 |-  ( ( x e. RR /\ y e. RR ) -> ( A = ( x + ( _i x. y ) ) -> ( 0 x. A ) = 0 ) )
33 32 rexlimivv 
 |-  ( E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) -> ( 0 x. A ) = 0 )
34 1 33 syl 
 |-  ( A e. CC -> ( 0 x. A ) = 0 )