Theorem rngo2times

Description: A ring element plus itself is two times the element. "Two" in an arbitrary unital ring is the sum of the unit with itself. (Contributed by AV, 24-Aug-2021)

	Ref	Expression
Hypotheses	ringadd2.b	|- B = ( Base ` R )
	ringadd2.p	|- .+ = ( +g ` R )
	ringadd2.t	|- .x. = ( .r ` R )
	rngo2times.u	|- .1. = ( 1r ` R )
Assertion	rngo2times	|- ( ( R e. Ring /\ A e. B ) -> ( A .+ A ) = ( ( .1. .+ .1. ) .x. A ) )

Proof

Step	Hyp	Ref	Expression
1		ringadd2.b	|- B = ( Base ` R )
2		ringadd2.p	|- .+ = ( +g ` R )
3		ringadd2.t	|- .x. = ( .r ` R )
4		rngo2times.u	|- .1. = ( 1r ` R )
5	1 3 4	ringlidm	|- ( ( R e. Ring /\ A e. B ) -> ( .1. .x. A ) = A )
6	5	eqcomd	|- ( ( R e. Ring /\ A e. B ) -> A = ( .1. .x. A ) )
7	6 6	oveq12d	|- ( ( R e. Ring /\ A e. B ) -> ( A .+ A ) = ( ( .1. .x. A ) .+ ( .1. .x. A ) ) )
8		simpl	|- ( ( R e. Ring /\ A e. B ) -> R e. Ring )
9	1 4	ringidcl	|- ( R e. Ring -> .1. e. B )
10	9	adantr	|- ( ( R e. Ring /\ A e. B ) -> .1. e. B )
11		simpr	|- ( ( R e. Ring /\ A e. B ) -> A e. B )
12	1 2 3	ringdir	|- ( ( R e. Ring /\ ( .1. e. B /\ .1. e. B /\ A e. B ) ) -> ( ( .1. .+ .1. ) .x. A ) = ( ( .1. .x. A ) .+ ( .1. .x. A ) ) )
13	8 10 10 11 12	syl13anc	|- ( ( R e. Ring /\ A e. B ) -> ( ( .1. .+ .1. ) .x. A ) = ( ( .1. .x. A ) .+ ( .1. .x. A ) ) )
14	7 13	eqtr4d	|- ( ( R e. Ring /\ A e. B ) -> ( A .+ A ) = ( ( .1. .+ .1. ) .x. A ) )