Theorem srg1zr 14291

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.) (Proof shortened by AV, 19-Jun-2026.)

Hypotheses

Ref	Expression
srg1zr.b	|- B = ( Base ` R )
srg1zr.p	|- .+ = ( +g ` R )
srg1zr.t	|- .* = ( .r ` R )

Assertion

Ref	Expression
srg1zr	|- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )

Proof of Theorem srg1zr

Step	Hyp	Ref	Expression
1		srgmnd 14271	. . . . . 6 |- ( R e. SRing -> R e. Mnd )
2		mndmgm 13735	. . . . . 6 |- ( R e. Mnd -> R e. Mgm )
3	1, 2	syl 14	. . . . 5 |- ( R e. SRing -> R e. Mgm )
4		eqid 2238	. . . . . . 7 |- (mulGrp ` R ) = (mulGrp ` R )
5	4	srgmgp 14272	. . . . . 6 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
6		mndmgm 13735	. . . . . 6 |- ( (mulGrp ` R ) e. Mnd -> (mulGrp ` R ) e. Mgm )
7	5, 6	syl 14	. . . . 5 |- ( R e. SRing -> (mulGrp ` R ) e. Mgm )
8	3, 7	jca 306	. . . 4 |- ( R e. SRing -> ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) )
9	8	3ad2ant1 1049	. . 3 |- ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) )
10	9	adantr 276	. 2 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) )
11		3simpc 1027	. . 3 |- ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) )
12	11	adantr 276	. 2 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) )
13		simpr 110	. 2 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> Z e. B )
14		srg1zr.b	. . 3 |- B = ( Base ` R )
15		srg1zr.p	. . 3 |- .+ = ( +g ` R )
16		srg1zr.t	. . 3 |- .* = ( .r ` R )
17	14, 15, 16	rng1zrlem 14258	. 2 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
18	10, 12, 13, 17	syl3anc 1278	1 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {csn 3709   <.cop 3712   X. cxp 4772   Fn wfn 5372   ` cfv 5377   Basecbs 13352   +g cplusg 13431   .rcmulr 13432  Mgmcmgm 13674   Mndcmnd 13729  mulGrpcmgp 14217  SRingcsrg 14267

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-0g 13612  df-plusf 13675  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-cmn 14089  df-mgp 14218  df-srg 14268

This theorem is used by:  srgen1zr0  14292  ring1zr  14621