Theorem srgen1zr

Description: The only semiring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010) (Revised by AV, 25-Jan-2020)

	Ref	Expression
Hypotheses	srg1zr.b	|- B = ( Base ` R )
	srg1zr.p	|- .+ = ( +g ` R )
	srg1zr.t	|- .* = ( .r ` R )
	srgen1zr.p	|- Z = ( 0g ` R )
Assertion	srgen1zr	|- ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )

Proof

Step	Hyp	Ref	Expression
1		srg1zr.b	|- B = ( Base ` R )
2		srg1zr.p	|- .+ = ( +g ` R )
3		srg1zr.t	|- .* = ( .r ` R )
4		srgen1zr.p	|- Z = ( 0g ` R )
5	1 4	srg0cl	|- ( R e. SRing -> Z e. B )
6	5	3ad2ant1	|- ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> Z e. B )
7		en1eqsnbi	|- ( Z e. B -> ( B ~~ 1o <-> B = { Z } ) )
8	7	adantl	|- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B ~~ 1o <-> B = { Z } ) )
9	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 >. } ) ) )
10	8 9	bitrd	|- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
11	6 10	mpdan	|- ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )