Theorem srg1zr 13483

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 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 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 13463	. . . . . . 7 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
3	2	3ad2ant1 1020	. . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → 𝑅 ∈ Mnd)
4	3	adantr 276	. . . . 5 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑅 ∈ Mnd)
5		mndmgm 13003	. . . . 5 ⊢ (𝑅 ∈ Mnd → 𝑅 ∈ Mgm)
6	4, 5	syl 14	. . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑅 ∈ Mgm)
7		simpr 110	. . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
8		simpl2 1003	. . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → + Fn (𝐵 × 𝐵))
9		srg1zr.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
10		srg1zr.p	. . . . 5 ⊢ + = (+g‘𝑅)
11	9, 10	mgmb1mgm1 12951	. . . 4 ⊢ ((𝑅 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
12	6, 7, 8, 11	syl3anc 1249	. . 3 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
13		eqid 2193	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
14	13, 9	mgpbasg 13422	. . . . . . 7 ⊢ (𝑅 ∈ SRing → 𝐵 = (Base‘(mulGrp‘𝑅)))
15	14	3ad2ant1 1020	. . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → 𝐵 = (Base‘(mulGrp‘𝑅)))
16	15	adantr 276	. . . . 5 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝐵 = (Base‘(mulGrp‘𝑅)))
17	16	eqeq1d 2202	. . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ (Base‘(mulGrp‘𝑅)) = {𝑍}))
18		simpl1 1002	. . . . . 6 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑅 ∈ SRing)
19	13	srgmgp 13464	. . . . . 6 ⊢ (𝑅 ∈ SRing → (mulGrp‘𝑅) ∈ Mnd)
20		mndmgm 13003	. . . . . 6 ⊢ ((mulGrp‘𝑅) ∈ Mnd → (mulGrp‘𝑅) ∈ Mgm)
21	18, 19, 20	3syl 17	. . . . 5 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (mulGrp‘𝑅) ∈ Mgm)
22	7, 16	eleqtrd 2272	. . . . 5 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ (Base‘(mulGrp‘𝑅)))
23		srg1zr.t	. . . . . . . . . . 11 ⊢ ∗ = (.r‘𝑅)
24	13, 23	mgpplusgg 13420	. . . . . . . . . 10 ⊢ (𝑅 ∈ SRing → ∗ = (+g‘(mulGrp‘𝑅)))
25	24	fneq1d 5344	. . . . . . . . 9 ⊢ (𝑅 ∈ SRing → ( ∗ Fn (𝐵 × 𝐵) ↔ (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)))
26	25	biimpa 296	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ ∗ Fn (𝐵 × 𝐵)) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵))
27	26	3adant2 1018	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵))
28	27	adantr 276	. . . . . 6 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵))
29	16	sqxpeqd 4685	. . . . . . 7 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 × 𝐵) = ((Base‘(mulGrp‘𝑅)) × (Base‘(mulGrp‘𝑅))))
30	29	fneq2d 5345	. . . . . 6 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵) ↔ (+g‘(mulGrp‘𝑅)) Fn ((Base‘(mulGrp‘𝑅)) × (Base‘(mulGrp‘𝑅)))))
31	28, 30	mpbid 147	. . . . 5 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (+g‘(mulGrp‘𝑅)) Fn ((Base‘(mulGrp‘𝑅)) × (Base‘(mulGrp‘𝑅))))
32		eqid 2193	. . . . . 6 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
33		eqid 2193	. . . . . 6 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅))
34	32, 33	mgmb1mgm1 12951	. . . . 5 ⊢ (((mulGrp‘𝑅) ∈ Mgm ∧ 𝑍 ∈ (Base‘(mulGrp‘𝑅)) ∧ (+g‘(mulGrp‘𝑅)) Fn ((Base‘(mulGrp‘𝑅)) × (Base‘(mulGrp‘𝑅)))) → ((Base‘(mulGrp‘𝑅)) = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
35	21, 22, 31, 34	syl3anc 1249	. . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((Base‘(mulGrp‘𝑅)) = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
36	24	eqcomd 2199	. . . . . 6 ⊢ (𝑅 ∈ SRing → (+g‘(mulGrp‘𝑅)) = ∗ )
37	18, 36	syl 14	. . . . 5 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (+g‘(mulGrp‘𝑅)) = ∗ )
38	37	eqeq1d 2202	. . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ↔ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
39	17, 35, 38	3bitrd 214	. . 3 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
40	12, 39	anbi12d 473	. 2 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((𝐵 = {𝑍} ∧ 𝐵 = {𝑍}) ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
41	1, 40	bitrid 192	1 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  {csn 3618  ⟨cop 3621   × cxp 4657   Fn wfn 5249  ‘cfv 5254  Basecbs 12618  +_gcplusg 12695  ._rcmulr 12696  Mgmcmgm 12937  Mndcmnd 12997  mulGrpcmgp 13416  SRingcsrg 13459

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-0g 12869  df-plusf 12938  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-cmn 13356  df-mgp 13417  df-srg 13460

This theorem is referenced by:  srgen1zr  13484