Metamath Proof Explorer

Theorem rec11d

Description: Reciprocal is one-to-one. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses div1d.1 
|- ( ph -> A e. CC )
divcld.2 
|- ( ph -> B e. CC )
divne0d.3 
|- ( ph -> A =/= 0 )
divne0d.4 
|- ( ph -> B =/= 0 )
rec11d.5 
|- ( ph -> ( 1 / A ) = ( 1 / B ) )
Assertion rec11d 
|- ( ph -> A = B )

Proof

Step Hyp Ref Expression
1 div1d.1 
 |-  ( ph -> A e. CC )
2 divcld.2 
 |-  ( ph -> B e. CC )
3 divne0d.3 
 |-  ( ph -> A =/= 0 )
4 divne0d.4 
 |-  ( ph -> B =/= 0 )
5 rec11d.5 
 |-  ( ph -> ( 1 / A ) = ( 1 / B ) )
6 rec11 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) )
7 1 3 2 4 6 syl22anc 
 |-  ( ph -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) )
8 5 7 mpbid 
 |-  ( ph -> A = B )