Metamath Proof Explorer


Theorem ring1zr

Description: The only (unital) ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption R e. Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 13-Feb-2010) (Revised by AV, 25-Jan-2020) (Proof shortened by AV, 7-Feb-2020)

Ref Expression
Hypotheses ring1zr.b
|- B = ( Base ` R )
ring1zr.p
|- .+ = ( +g ` R )
ring1zr.t
|- .* = ( .r ` R )
Assertion ring1zr
|- ( ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )

Proof

Step Hyp Ref Expression
1 ring1zr.b
 |-  B = ( Base ` R )
2 ring1zr.p
 |-  .+ = ( +g ` R )
3 ring1zr.t
 |-  .* = ( .r ` R )
4 ringsrg
 |-  ( R e. Ring -> R e. SRing )
5 1 2 3 srg1zr
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
6 4 5 syl3anl1
 |-  ( ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )