Theorem srg1zr 14230

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 14210	. . . . . 6 |- ( R e. SRing -> R e. Mnd )
2		mndmgm 13683	. . . . . 6 |- ( R e. Mnd -> R e. Mgm )
3	1, 2	syl 14	. . . . 5 |- ( R e. SRing -> R e. Mgm )
4		eqid 2234	. . . . . . 7 |- (mulGrp ` R ) = (mulGrp ` R )
5	4	srgmgp 14211	. . . . . 6 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
6		mndmgm 13683	. . . . . 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 1045	. . 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 1023	. . 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 14198	. 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 1274	1 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   {csn 3694   <.cop 3697   X. cxp 4752   Fn wfn 5352   ` cfv 5357   Basecbs 13296   +g cplusg 13374   .rcmulr 13375  Mgmcmgm 13617   Mndcmnd 13677  mulGrpcmgp 14159  SRingcsrg 14206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-plusf 13618  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-cmn 14039  df-mgp 14160  df-srg 14207

This theorem is referenced by:  srgen1zr0  14231  ring1zr  14559