Theorem srg1zr 12963

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 12943	. . . . . . 7 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
3	2	3ad2ant1 1018	. . . . . 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 12688	. . . . 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 1001	. . . 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 12652	. . . 4 ⊢ ((𝑅 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
12	6, 7, 8, 11	syl3anc 1238	. . 3 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
13		eqid 2175	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
14	13, 9	mgpbasg 12930	. . . . . . 7 ⊢ (𝑅 ∈ SRing → 𝐵 = (Base‘(mulGrp‘𝑅)))
15	14	3ad2ant1 1018	. . . . . 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 2184	. . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ (Base‘(mulGrp‘𝑅)) = {𝑍}))
18		simpl1 1000	. . . . . 6 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑅 ∈ SRing)
19	13	srgmgp 12944	. . . . . 6 ⊢ (𝑅 ∈ SRing → (mulGrp‘𝑅) ∈ Mnd)
20		mndmgm 12688	. . . . . 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 2254	. . . . 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 12929	. . . . . . . . . 10 ⊢ (𝑅 ∈ SRing → ∗ = (+g‘(mulGrp‘𝑅)))
25	24	fneq1d 5298	. . . . . . . . 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 1016	. . . . . . 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 4646	. . . . . . 7 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 × 𝐵) = ((Base‘(mulGrp‘𝑅)) × (Base‘(mulGrp‘𝑅))))
30	29	fneq2d 5299	. . . . . 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 2175	. . . . . 6 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
33		eqid 2175	. . . . . 6 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅))
34	32, 33	mgmb1mgm1 12652	. . . . 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 1238	. . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((Base‘(mulGrp‘𝑅)) = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
36	24	eqcomd 2181	. . . . . 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 2184	. . . 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 978   = wceq 1353   ∈ wcel 2146  {csn 3589  ⟨cop 3592   × cxp 4618   Fn wfn 5203  ‘cfv 5208  Basecbs 12428  +_gcplusg 12492  ._rcmulr 12493  Mgmcmgm 12638  Mndcmnd 12682  mulGrpcmgp 12925  SRingcsrg 12939

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-pre-ltirr 7898  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-inn 8891  df-2 8949  df-3 8950  df-ndx 12431  df-slot 12432  df-base 12434  df-sets 12435  df-plusg 12505  df-mulr 12506  df-0g 12628  df-plusf 12639  df-mgm 12640  df-sgrp 12673  df-mnd 12683  df-cmn 12886  df-mgp 12926  df-srg 12940

This theorem is referenced by:  srgen1zr  12964