Theorem ringo2times

Description: A ring element plus itself is two times the element. "Two" in an arbitrary unital ring is the sum of the unity element with itself. (Contributed by AV, 24-Aug-2021) Variant of o2timesd for rings. (Revised by AV, 5-Feb-2025)

	Ref	Expression
Hypotheses	ringo2times.b	|- B = ( Base ` R )
	ringo2times.p	|- .+ = ( +g ` R )
	ringo2times.t	|- .x. = ( .r ` R )
	ringo2times.u	|- .1. = ( 1r ` R )
Assertion	ringo2times	|- ( ( R e. Ring /\ A e. B ) -> ( A .+ A ) = ( ( .1. .+ .1. ) .x. A ) )

Proof

Step	Hyp	Ref	Expression
1		ringo2times.b	|- B = ( Base ` R )
2		ringo2times.p	|- .+ = ( +g ` R )
3		ringo2times.t	|- .x. = ( .r ` R )
4		ringo2times.u	|- .1. = ( 1r ` R )
5	1 2 3	ringdir	|- ( ( R e. Ring /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) )
6	5	ralrimivvva	|- ( R e. Ring -> A. x e. B A. y e. B A. z e. B ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) )
7	6	adantr	|- ( ( R e. Ring /\ A e. B ) -> A. x e. B A. y e. B A. z e. B ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) )
8	1 4	ringidcl	|- ( R e. Ring -> .1. e. B )
9	8	adantr	|- ( ( R e. Ring /\ A e. B ) -> .1. e. B )
10	1 3 4	ringlidm	|- ( ( R e. Ring /\ x e. B ) -> ( .1. .x. x ) = x )
11	10	ralrimiva	|- ( R e. Ring -> A. x e. B ( .1. .x. x ) = x )
12	11	adantr	|- ( ( R e. Ring /\ A e. B ) -> A. x e. B ( .1. .x. x ) = x )
13		simpr	|- ( ( R e. Ring /\ A e. B ) -> A e. B )
14	7 9 12 13	o2timesd	|- ( ( R e. Ring /\ A e. B ) -> ( A .+ A ) = ( ( .1. .+ .1. ) .x. A ) )