Theorem srg1zr 13358

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.)

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		pm4.24 395	. 2 |- ( B = { Z } <-> ( B = { Z } /\ B = { Z } ) )
2		srgmnd 13338	. . . . . . 7 |- ( R e. SRing -> R e. Mnd )
3	2	3ad2ant1 1020	. . . . . 6 |- ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> R e. Mnd )
4	3	adantr 276	. . . . 5 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> R e. Mnd )
5		mndmgm 12898	. . . . 5 |- ( R e. Mnd -> R e. Mgm )
6	4, 5	syl 14	. . . 4 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> R e. Mgm )
7		simpr 110	. . . 4 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> Z e. B )
8		simpl2 1003	. . . 4 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> .+ Fn ( B X. B ) )
9		srg1zr.b	. . . . 5 |- B = ( Base ` R )
10		srg1zr.p	. . . . 5 |- .+ = ( +g ` R )
11	9, 10	mgmb1mgm1 12847	. . . 4 |- ( ( R e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
12	6, 7, 8, 11	syl3anc 1249	. . 3 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
13		eqid 2189	. . . . . . . 8 |- (mulGrp ` R ) = (mulGrp ` R )
14	13, 9	mgpbasg 13297	. . . . . . 7 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
15	14	3ad2ant1 1020	. . . . . 6 |- ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> B = ( Base ` (mulGrp ` R ) ) )
16	15	adantr 276	. . . . 5 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> B = ( Base ` (mulGrp ` R ) ) )
17	16	eqeq1d 2198	. . . 4 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( Base ` (mulGrp ` R ) ) = { Z } ) )
18		simpl1 1002	. . . . . 6 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> R e. SRing )
19	13	srgmgp 13339	. . . . . 6 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
20		mndmgm 12898	. . . . . 6 |- ( (mulGrp ` R ) e. Mnd -> (mulGrp ` R ) e. Mgm )
21	18, 19, 20	3syl 17	. . . . 5 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> (mulGrp ` R ) e. Mgm )
22	7, 16	eleqtrd 2268	. . . . 5 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> Z e. ( Base ` (mulGrp ` R ) ) )
23		srg1zr.t	. . . . . . . . . . 11 |- .* = ( .r ` R )
24	13, 23	mgpplusgg 13295	. . . . . . . . . 10 |- ( R e. SRing -> .* = ( +g ` (mulGrp ` R ) ) )
25	24	fneq1d 5325	. . . . . . . . 9 |- ( R e. SRing -> ( .* Fn ( B X. B ) <-> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) ) )
26	25	biimpa 296	. . . . . . . 8 |- ( ( R e. SRing /\ .* Fn ( B X. B ) ) -> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) )
27	26	3adant2 1018	. . . . . . 7 |- ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) )
28	27	adantr 276	. . . . . 6 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) )
29	16	sqxpeqd 4670	. . . . . . 7 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B X. B ) = ( ( Base ` (mulGrp ` R ) ) X. ( Base ` (mulGrp ` R ) ) ) )
30	29	fneq2d 5326	. . . . . 6 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) <-> ( +g ` (mulGrp ` R ) ) Fn ( ( Base ` (mulGrp ` R ) ) X. ( Base ` (mulGrp ` R ) ) ) ) )
31	28, 30	mpbid 147	. . . . 5 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( +g ` (mulGrp ` R ) ) Fn ( ( Base ` (mulGrp ` R ) ) X. ( Base ` (mulGrp ` R ) ) ) )
32		eqid 2189	. . . . . 6 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
33		eqid 2189	. . . . . 6 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
34	32, 33	mgmb1mgm1 12847	. . . . 5 |- ( ( (mulGrp ` R ) e. Mgm /\ Z e. ( Base ` (mulGrp ` R ) ) /\ ( +g ` (mulGrp ` R ) ) Fn ( ( Base ` (mulGrp ` R ) ) X. ( Base ` (mulGrp ` R ) ) ) ) -> ( ( Base ` (mulGrp ` R ) ) = { Z } <-> ( +g ` (mulGrp ` R ) ) = { <. <. Z , Z >. , Z >. } ) )
35	21, 22, 31, 34	syl3anc 1249	. . . 4 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( ( Base ` (mulGrp ` R ) ) = { Z } <-> ( +g ` (mulGrp ` R ) ) = { <. <. Z , Z >. , Z >. } ) )
36	24	eqcomd 2195	. . . . . 6 |- ( R e. SRing -> ( +g ` (mulGrp ` R ) ) = .* )
37	18, 36	syl 14	. . . . 5 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( +g ` (mulGrp ` R ) ) = .* )
38	37	eqeq1d 2198	. . . 4 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( ( +g ` (mulGrp ` R ) ) = { <. <. Z , Z >. , Z >. } <-> .* = { <. <. Z , Z >. , Z >. } ) )
39	17, 35, 38	3bitrd 214	. . 3 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> .* = { <. <. Z , Z >. , Z >. } ) )
40	12, 39	anbi12d 473	. 2 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( ( B = { Z } /\ B = { Z } ) <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
41	1, 40	bitrid 192	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 980   = wceq 1364   e. wcel 2160   {csn 3607   <.cop 3610   X. cxp 4642   Fn wfn 5230   ` cfv 5235   Basecbs 12515   +g cplusg 12592   .rcmulr 12593  Mgmcmgm 12833   Mndcmnd 12892  mulGrpcmgp 13291  SRingcsrg 13334

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-pre-ltirr 7954  ax-pre-ltadd 7958

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-pnf 8025  df-mnf 8026  df-ltxr 8028  df-inn 8951  df-2 9009  df-3 9010  df-ndx 12518  df-slot 12519  df-base 12521  df-sets 12522  df-plusg 12605  df-mulr 12606  df-0g 12766  df-plusf 12834  df-mgm 12835  df-sgrp 12880  df-mnd 12893  df-cmn 13242  df-mgp 13292  df-srg 13335

This theorem is referenced by:  srgen1zr  13359