Metamath Proof Explorer


Theorem rec11d

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

Ref Expression
Hypotheses div1d.1 φA
divcld.2 φB
divne0d.3 φA0
divne0d.4 φB0
rec11d.5 φ1A=1B
Assertion rec11d φA=B

Proof

Step Hyp Ref Expression
1 div1d.1 φA
2 divcld.2 φB
3 divne0d.3 φA0
4 divne0d.4 φB0
5 rec11d.5 φ1A=1B
6 rec11 AA0BB01A=1BA=B
7 1 3 2 4 6 syl22anc φ1A=1BA=B
8 5 7 mpbid φA=B