Metamath Proof Explorer

Theorem mulid1

Description: The number 1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013)

Ref Expression
Assertion mulid1 
|- ( A e. CC -> ( A x. 1 ) = A )

Proof

Step Hyp Ref Expression
1 cnre 
 |-  ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
2 recn 
 |-  ( x e. RR -> x e. CC )
3 ax-icn 
 |-  _i e. CC
4 recn 
 |-  ( y e. RR -> y e. CC )
5 mulcl 
 |-  ( ( _i e. CC /\ y e. CC ) -> ( _i x. y ) e. CC )
6 3 4 5 sylancr 
 |-  ( y e. RR -> ( _i x. y ) e. CC )
7 ax-1cn 
 |-  1 e. CC
8 adddir 
 |-  ( ( x e. CC /\ ( _i x. y ) e. CC /\ 1 e. CC ) -> ( ( x + ( _i x. y ) ) x. 1 ) = ( ( x x. 1 ) + ( ( _i x. y ) x. 1 ) ) )
9 7 8 mp3an3 
 |-  ( ( x e. CC /\ ( _i x. y ) e. CC ) -> ( ( x + ( _i x. y ) ) x. 1 ) = ( ( x x. 1 ) + ( ( _i x. y ) x. 1 ) ) )
10 2 6 9 syl2an 
 |-  ( ( x e. RR /\ y e. RR ) -> ( ( x + ( _i x. y ) ) x. 1 ) = ( ( x x. 1 ) + ( ( _i x. y ) x. 1 ) ) )
11 ax-1rid 
 |-  ( x e. RR -> ( x x. 1 ) = x )
12 mulass 
 |-  ( ( _i e. CC /\ y e. CC /\ 1 e. CC ) -> ( ( _i x. y ) x. 1 ) = ( _i x. ( y x. 1 ) ) )
13 3 7 12 mp3an13 
 |-  ( y e. CC -> ( ( _i x. y ) x. 1 ) = ( _i x. ( y x. 1 ) ) )
14 4 13 syl 
 |-  ( y e. RR -> ( ( _i x. y ) x. 1 ) = ( _i x. ( y x. 1 ) ) )
15 ax-1rid 
 |-  ( y e. RR -> ( y x. 1 ) = y )
16 15 oveq2d 
 |-  ( y e. RR -> ( _i x. ( y x. 1 ) ) = ( _i x. y ) )
17 14 16 eqtrd 
 |-  ( y e. RR -> ( ( _i x. y ) x. 1 ) = ( _i x. y ) )
18 11 17 oveqan12d 
 |-  ( ( x e. RR /\ y e. RR ) -> ( ( x x. 1 ) + ( ( _i x. y ) x. 1 ) ) = ( x + ( _i x. y ) ) )
19 10 18 eqtrd 
 |-  ( ( x e. RR /\ y e. RR ) -> ( ( x + ( _i x. y ) ) x. 1 ) = ( x + ( _i x. y ) ) )
20 oveq1 
 |-  ( A = ( x + ( _i x. y ) ) -> ( A x. 1 ) = ( ( x + ( _i x. y ) ) x. 1 ) )
21 id 
 |-  ( A = ( x + ( _i x. y ) ) -> A = ( x + ( _i x. y ) ) )
22 20 21 eqeq12d 
 |-  ( A = ( x + ( _i x. y ) ) -> ( ( A x. 1 ) = A <-> ( ( x + ( _i x. y ) ) x. 1 ) = ( x + ( _i x. y ) ) ) )
23 19 22 syl5ibrcom 
 |-  ( ( x e. RR /\ y e. RR ) -> ( A = ( x + ( _i x. y ) ) -> ( A x. 1 ) = A ) )
24 23 rexlimivv 
 |-  ( E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) -> ( A x. 1 ) = A )
25 1 24 syl 
 |-  ( A e. CC -> ( A x. 1 ) = A )