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Metamath Proof Explorer

Theorem srgo2times

Description: A semiring element plus itself is two times the element. "Two" in an arbitrary (unital) semiring is the sum of the unity element with itself. (Contributed by AV, 24-Aug-2021) Variant of o2timesd for semirings. (Revised by AV, 1-Feb-2025)

Ref Expression
Hypotheses srgo2times.b 
|- B = ( Base ` R )
srgo2times.p 
|- .+ = ( +g ` R )
srgo2times.t 
|- .x. = ( .r ` R )
srgo2times.u 
|- .1. = ( 1r ` R )
Assertion srgo2times 
|- ( ( R e. SRing /\ A e. B ) -> ( A .+ A ) = ( ( .1. .+ .1. ) .x. A ) )

Proof

Step Hyp Ref Expression
1 srgo2times.b 
 |-  B = ( Base ` R )
2 srgo2times.p 
 |-  .+ = ( +g ` R )
3 srgo2times.t 
 |-  .x. = ( .r ` R )
4 srgo2times.u 
 |-  .1. = ( 1r ` R )
5 1 2 3 srgdir 
 |-  ( ( R e. SRing /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) )
6 5 ralrimivvva 
 |-  ( R e. SRing -> 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. SRing /\ 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 srgidcl 
 |-  ( R e. SRing -> .1. e. B )
9 8 adantr 
 |-  ( ( R e. SRing /\ A e. B ) -> .1. e. B )
10 1 3 4 srglidm 
 |-  ( ( R e. SRing /\ x e. B ) -> ( .1. .x. x ) = x )
11 10 ralrimiva 
 |-  ( R e. SRing -> A. x e. B ( .1. .x. x ) = x )
12 11 adantr 
 |-  ( ( R e. SRing /\ A e. B ) -> A. x e. B ( .1. .x. x ) = x )
13 simpr 
 |-  ( ( R e. SRing /\ A e. B ) -> A e. B )
14 7 9 12 13 o2timesd 
 |-  ( ( R e. SRing /\ A e. B ) -> ( A .+ A ) = ( ( .1. .+ .1. ) .x. A ) )