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Metamath Proof Explorer

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 ) )