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 >. } ) ) ) |
| 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 >. } ) ) ) |