Theorem rngabl

Description: A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020)

	Ref	Expression
Assertion	rngabl	$$⊢R\in \mathrm{Rng}\to R\in \mathrm{Abel}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_R$$
2		eqid	$$⊢{\mathrm{mulGrp}}_R={\mathrm{mulGrp}}_R$$
3		eqid	$$⊢{+}_R={+}_R$$
4		eqid	$$⊢{\cdot }_R={\cdot }_R$$
5	1 2 3 4	isrng	Could not format ( R e. Rng <-> ( R e. Abel /\ ( mulGrp ` R ) e. Smgrp /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) ) : No typesetting found for |- ( R e. Rng <-> ( R e. Abel /\ ( mulGrp ` R ) e. Smgrp /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) ) with typecode |-
6	5	simp1bi	$$⊢R\in \mathrm{Rng}\to R\in \mathrm{Abel}$$