Theorem rng1zr 14234

Description: The only ring 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, 18-Jun-2026.)

Hypotheses

Ref	Expression
rng1zr.b	|- B = ( Base ` R )
rng1zr.p	|- .+ = ( +g ` R )
rng1zr.t	|- .* = ( .r ` R )

Assertion

Ref	Expression
rng1zr	|- ( ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )

Proof of Theorem rng1zr

Step	Hyp	Ref	Expression
1		rnggrp 14212	. . . . . 6 |- ( R e. Rng -> R e. Grp )
2	1	grpmgmd 13808	. . . . 5 |- ( R e. Rng -> R e. Mgm )
3		eqid 2238	. . . . . . 7 |- (mulGrp ` R ) = (mulGrp ` R )
4	3	rngmgp 14210	. . . . . 6 |- ( R e. Rng -> (mulGrp ` R ) e. Smgrp )
5		sgrpmgm 13699	. . . . . 6 |- ( (mulGrp ` R ) e. Smgrp -> (mulGrp ` R ) e. Mgm )
6	4, 5	syl 14	. . . . 5 |- ( R e. Rng -> (mulGrp ` R ) e. Mgm )
7	2, 6	jca 306	. . . 4 |- ( R e. Rng -> ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) )
8	7	3ad2ant1 1049	. . 3 |- ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) )
9	8	adantr 276	. 2 |- ( ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) )
10		3simpc 1027	. . 3 |- ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) )
11	10	adantr 276	. 2 |- ( ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) )
12		simpr 110	. 2 |- ( ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> Z e. B )
13		rng1zr.b	. . 3 |- B = ( Base ` R )
14		rng1zr.p	. . 3 |- .+ = ( +g ` R )
15		rng1zr.t	. . 3 |- .* = ( .r ` R )
16	13, 14, 15	rng1zrlem 14233	. 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 >. } ) ) )
17	9, 11, 12, 16	syl3anc 1278	1 |- ( ( ( R e. Rng /\ .+ 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 1009   = wceq 1402   e. wcel 2209   {csn 3705   <.cop 3708   X. cxp 4767   Fn wfn 5367   ` cfv 5372   Basecbs 13330   +g cplusg 13408   .rcmulr 13409  Mgmcmgm 13651  Smgrpcsgrp 13693  mulGrpcmgp 14194  Rngcrng 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 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 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-plusf 13652  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-abl 14067  df-mgp 14195  df-rng 14207

This theorem is referenced by:  rngen1zr  14235