Metamath Proof Explorer


Theorem ringen1zr

Description: The only unital ring with 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, 15-Feb-2010) (Revised by AV, 25-Jan-2020)

Ref Expression
Hypotheses ring1zr.b
|- B = ( Base ` R )
ring1zr.p
|- .+ = ( +g ` R )
ring1zr.t
|- .* = ( .r ` R )
ringen1zr.0
|- Z = ( 0g ` R )
Assertion ringen1zr
|- ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. 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 ringen1zr.0
 |-  Z = ( 0g ` R )
5 1 4 ring0cl
 |-  ( R e. Ring -> Z e. B )
6 5 3ad2ant1
 |-  ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> Z e. B )
7 1 2 3 rngen1zr
 |-  ( ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
8 6 7 mpdan
 |-  ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )