This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 >. } ) ) )