Metamath Proof Explorer


Theorem rngmgp

Description: A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020)

Ref Expression
Hypothesis rngmgp.g G = mulGrp R
Assertion rngmgp R Rng G Smgrp

Proof

Step Hyp Ref Expression
1 rngmgp.g G = mulGrp R
2 eqid Base R = Base R
3 eqid + R = + R
4 eqid R = R
5 2 1 3 4 isrng R Rng R Abel G Smgrp x Base R y Base R z Base R x R y + R z = x R y + R x R z x + R y R z = x R z + R y R z
6 5 simp2bi R Rng G Smgrp