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 >. } ) ) )