Metamath Proof Explorer

Theorem srg1zr

Description: The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010) (Revised by AV, 25-Jan-2020)

Ref Expression
Hypotheses srg1zr.b 
|- B = ( Base ` R )
srg1zr.p 
|- .+ = ( +g ` R )
srg1zr.t 
|- .* = ( .r ` R )
Assertion srg1zr 
|- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. 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 pm4.24 
 |-  ( B = { Z } <-> ( B = { Z } /\ B = { Z } ) )
5 srgmnd 
 |-  ( R e. SRing -> R e. Mnd )
6 5 3ad2ant1 
 |-  ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> R e. Mnd )
7 6 adantr 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> R e. Mnd )
8 mndmgm 
 |-  ( R e. Mnd -> R e. Mgm )
9 7 8 syl 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> R e. Mgm )
10 simpr 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> Z e. B )
11 simpl2 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> .+ Fn ( B X. B ) )
12 1 2 mgmb1mgm1 
 |-  ( ( R e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
13 9 10 11 12 syl3anc 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
14 simpl1 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> R e. SRing )
15 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
16 15 srgmgp 
 |-  ( R e. SRing -> ( mulGrp ` R ) e. Mnd )
17 mndmgm 
 |-  ( ( mulGrp ` R ) e. Mnd -> ( mulGrp ` R ) e. Mgm )
18 14 16 17 3syl 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( mulGrp ` R ) e. Mgm )
19 15 3 mgpplusg 
 |-  .* = ( +g ` ( mulGrp ` R ) )
20 19 fneq1i 
 |-  ( .* Fn ( B X. B ) <-> ( +g ` ( mulGrp ` R ) ) Fn ( B X. B ) )
21 20 biimpi 
 |-  ( .* Fn ( B X. B ) -> ( +g ` ( mulGrp ` R ) ) Fn ( B X. B ) )
22 21 3ad2ant3 
 |-  ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( +g ` ( mulGrp ` R ) ) Fn ( B X. B ) )
23 22 adantr 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( +g ` ( mulGrp ` R ) ) Fn ( B X. B ) )
24 15 1 mgpbas 
 |-  B = ( Base ` ( mulGrp ` R ) )
25 eqid 
 |-  ( +g ` ( mulGrp ` R ) ) = ( +g ` ( mulGrp ` R ) )
26 24 25 mgmb1mgm1 
 |-  ( ( ( mulGrp ` R ) e. Mgm /\ Z e. B /\ ( +g ` ( mulGrp ` R ) ) Fn ( B X. B ) ) -> ( B = { Z } <-> ( +g ` ( mulGrp ` R ) ) = { <. <. Z , Z >. , Z >. } ) )
27 18 10 23 26 syl3anc 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( +g ` ( mulGrp ` R ) ) = { <. <. Z , Z >. , Z >. } ) )
28 19 eqcomi 
 |-  ( +g ` ( mulGrp ` R ) ) = .*
29 28 a1i 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( +g ` ( mulGrp ` R ) ) = .* )
30 29 eqeq1d 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( ( +g ` ( mulGrp ` R ) ) = { <. <. Z , Z >. , Z >. } <-> .* = { <. <. Z , Z >. , Z >. } ) )
31 27 30 bitrd 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> .* = { <. <. Z , Z >. , Z >. } ) )
32 13 31 anbi12d 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( ( B = { Z } /\ B = { Z } ) <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
33 4 32 syl5bb 
 |-  ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )