This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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) (Proof shortened by AV, 1-Feb-2025)

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 eqid 
 |-  ( 1r ` R ) = ( 1r ` R )
5 1 4 ringidcl 
 |-  ( R e. Ring -> ( 1r ` R ) e. B )
6 5 adantr 
 |-  ( ( R e. Ring /\ X e. B ) -> ( 1r ` R ) e. B )
7 simpr 
 |-  ( ( ( R e. Ring /\ X e. B ) /\ x = ( 1r ` R ) ) -> x = ( 1r ` R ) )
8 7 7 oveq12d 
 |-  ( ( ( R e. Ring /\ X e. B ) /\ x = ( 1r ` R ) ) -> ( x .+ x ) = ( ( 1r ` R ) .+ ( 1r ` R ) ) )
9 8 oveq1d 
 |-  ( ( ( R e. Ring /\ X e. B ) /\ x = ( 1r ` R ) ) -> ( ( x .+ x ) .x. X ) = ( ( ( 1r ` R ) .+ ( 1r ` R ) ) .x. X ) )
10 9 eqeq2d 
 |-  ( ( ( R e. Ring /\ X e. B ) /\ x = ( 1r ` R ) ) -> ( ( X .+ X ) = ( ( x .+ x ) .x. X ) <-> ( X .+ X ) = ( ( ( 1r ` R ) .+ ( 1r ` R ) ) .x. X ) ) )
11 1 2 3 4 ringo2times 
 |-  ( ( R e. Ring /\ X e. B ) -> ( X .+ X ) = ( ( ( 1r ` R ) .+ ( 1r ` R ) ) .x. X ) )
12 6 10 11 rspcedvd 
 |-  ( ( R e. Ring /\ X e. B ) -> E. x e. B ( X .+ X ) = ( ( x .+ x ) .x. X ) )