Theorem ringadd2

Description: A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007) (Revised by Mario Carneiro, 22-Dec-2013) (Revised by AV, 24-Aug-2021)

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

Proof

Step	Hyp	Ref	Expression
1		ringadd2.b	|- B = ( Base ` R )
2		ringadd2.p	|- .+ = ( +g ` R )
3		ringadd2.t	|- .x. = ( .r ` R )
4	1 3	ringid	|- ( ( R e. Ring /\ X e. B ) -> E. x e. B ( ( x .x. X ) = X /\ ( X .x. x ) = X ) )
5		oveq12	|- ( ( ( x .x. X ) = X /\ ( x .x. X ) = X ) -> ( ( x .x. X ) .+ ( x .x. X ) ) = ( X .+ X ) )
6	5	anidms	|- ( ( x .x. X ) = X -> ( ( x .x. X ) .+ ( x .x. X ) ) = ( X .+ X ) )
7	6	eqcomd	|- ( ( x .x. X ) = X -> ( X .+ X ) = ( ( x .x. X ) .+ ( x .x. X ) ) )
8		simpll	|- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> R e. Ring )
9		simpr	|- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> x e. B )
10		simplr	|- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> X e. B )
11	1 2 3	ringdir	|- ( ( R e. Ring /\ ( x e. B /\ x e. B /\ X e. B ) ) -> ( ( x .+ x ) .x. X ) = ( ( x .x. X ) .+ ( x .x. X ) ) )
12	8 9 9 10 11	syl13anc	|- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( ( x .+ x ) .x. X ) = ( ( x .x. X ) .+ ( x .x. X ) ) )
13	12	eqeq2d	|- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( ( X .+ X ) = ( ( x .+ x ) .x. X ) <-> ( X .+ X ) = ( ( x .x. X ) .+ ( x .x. X ) ) ) )
14	7 13	syl5ibr	|- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( ( x .x. X ) = X -> ( X .+ X ) = ( ( x .+ x ) .x. X ) ) )
15	14	adantrd	|- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( ( ( x .x. X ) = X /\ ( X .x. x ) = X ) -> ( X .+ X ) = ( ( x .+ x ) .x. X ) ) )
16	15	reximdva	|- ( ( R e. Ring /\ X e. B ) -> ( E. x e. B ( ( x .x. X ) = X /\ ( X .x. x ) = X ) -> E. x e. B ( X .+ X ) = ( ( x .+ x ) .x. X ) ) )
17	4 16	mpd	|- ( ( R e. Ring /\ X e. B ) -> E. x e. B ( X .+ X ) = ( ( x .+ x ) .x. X ) )