Metamath Proof Explorer


Theorem diveq1d

Description: Equality in terms of unit ratio. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses div1d.1
|- ( ph -> A e. CC )
divcld.2
|- ( ph -> B e. CC )
divcld.3
|- ( ph -> B =/= 0 )
diveq1d.4
|- ( ph -> ( A / B ) = 1 )
Assertion diveq1d
|- ( ph -> A = B )

Proof

Step Hyp Ref Expression
1 div1d.1
 |-  ( ph -> A e. CC )
2 divcld.2
 |-  ( ph -> B e. CC )
3 divcld.3
 |-  ( ph -> B =/= 0 )
4 diveq1d.4
 |-  ( ph -> ( A / B ) = 1 )
5 diveq1
 |-  ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( A / B ) = 1 <-> A = B ) )
6 1 2 3 5 syl3anc
 |-  ( ph -> ( ( A / B ) = 1 <-> A = B ) )
7 4 6 mpbid
 |-  ( ph -> A = B )