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 | ||
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Hypotheses | ring1zr.b | |- B = ( Base ` R ) |
|
ring1zr.p | |- .+ = ( +g ` R ) |
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ring1zr.t | |- .* = ( .r ` R ) |
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ringen1zr.0 | |- Z = ( 0g ` R ) |
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Assertion | ringen1zr | |- ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. 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 | 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 >. } ) ) ) |