Theorem rngosn3 38125

Description: Obsolete as of 25-Jan-2020. Use ring1zr 20709 or srg1zr 20150 instead. The only unital ring with a base set consisting in one element is the zero ring. (Contributed by FL, 13-Feb-2010.) (Proof shortened by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
on1el3.1	⊢ 𝐺 = (1st ‘𝑅)
on1el3.2	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
rngosn3	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))

Proof of Theorem rngosn3

Step	Hyp	Ref	Expression
1		on1el3.1	. . . . . . . . . 10 ⊢ 𝐺 = (1st ‘𝑅)
2	1	rngogrpo 38111	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		on1el3.2	. . . . . . . . . 10 ⊢ 𝑋 = ran 𝐺
4	3	grpofo 30574	. . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
5		fof 6746	. . . . . . . . 9 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → 𝐺:(𝑋 × 𝑋)⟶𝑋)
6	2, 4, 5	3syl 18	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → 𝐺:(𝑋 × 𝑋)⟶𝑋)
7	6	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
8		id 22	. . . . . . . . 9 ⊢ (𝑋 = {𝐴} → 𝑋 = {𝐴})
9	8	sqxpeqd 5656	. . . . . . . 8 ⊢ (𝑋 = {𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴}))
10	9, 8	feq23d 6657	. . . . . . 7 ⊢ (𝑋 = {𝐴} → (𝐺:(𝑋 × 𝑋)⟶𝑋 ↔ 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
11	7, 10	syl5ibcom 245	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} → 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
12	7	fdmd 6672	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → dom 𝐺 = (𝑋 × 𝑋))
13	12	eqcomd 2742	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 × 𝑋) = dom 𝐺)
14		fdm 6671	. . . . . . . . 9 ⊢ (𝐺:({𝐴} × {𝐴})⟶{𝐴} → dom 𝐺 = ({𝐴} × {𝐴}))
15	14	eqeq2d 2747	. . . . . . . 8 ⊢ (𝐺:({𝐴} × {𝐴})⟶{𝐴} → ((𝑋 × 𝑋) = dom 𝐺 ↔ (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
16	13, 15	syl5ibcom 245	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
17		xpid11 5881	. . . . . . 7 ⊢ ((𝑋 × 𝑋) = ({𝐴} × {𝐴}) ↔ 𝑋 = {𝐴})
18	16, 17	imbitrdi 251	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → 𝑋 = {𝐴}))
19	11, 18	impbid 212	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
20		simpr 484	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
21		xpsng 7084	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
22	20, 21	sylancom 588	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
23	22	feq2d 6646	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} ↔ 𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴}))
24		opex 5412	. . . . . 6 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
25		fsng 7082	. . . . . 6 ⊢ ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴 ∈ 𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
26	24, 20, 25	sylancr 587	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
27	19, 23, 26	3bitrd 305	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
28	1	eqeq1i 2741	. . . 4 ⊢ (𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↔ (1st ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
29	27, 28	bitrdi 287	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ (1st ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
30	29	anbi1d 631	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ((𝑋 = {𝐴} ∧ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ↔ ((1st ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
31		eqid 2736	. . . . . . 7 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
32	1, 31, 3	rngosm 38101	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (2nd ‘𝑅):(𝑋 × 𝑋)⟶𝑋)
33	32	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (2nd ‘𝑅):(𝑋 × 𝑋)⟶𝑋)
34	9, 8	feq23d 6657	. . . . 5 ⊢ (𝑋 = {𝐴} → ((2nd ‘𝑅):(𝑋 × 𝑋)⟶𝑋 ↔ (2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴}))
35	33, 34	syl5ibcom 245	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} → (2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴}))
36	22	feq2d 6646	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ((2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd ‘𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴}))
37		fsng 7082	. . . . . 6 ⊢ ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴 ∈ 𝐵) → ((2nd ‘𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
38	24, 20, 37	sylancr 587	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ((2nd ‘𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
39	36, 38	bitrd 279	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ((2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
40	35, 39	sylibd 239	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} → (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
41	40	pm4.71d 561	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ (𝑋 = {𝐴} ∧ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
42		relrngo 38097	. . . . . 6 ⊢ Rel RingOps
43		df-rel 5631	. . . . . 6 ⊢ (Rel RingOps ↔ RingOps ⊆ (V × V))
44	42, 43	mpbi 230	. . . . 5 ⊢ RingOps ⊆ (V × V)
45	44	sseli 3929	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑅 ∈ (V × V))
46	45	adantr 480	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → 𝑅 ∈ (V × V))
47		eqop 7975	. . 3 ⊢ (𝑅 ∈ (V × V) → (𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ↔ ((1st ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
48	46, 47	syl 17	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ↔ ((1st ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
49	30, 41, 48	3bitr4d 311	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ⊆ wss 3901  {csn 4580  ⟨cop 4586   × cxp 5622  dom cdm 5624  ran crn 5625  Rel wrel 5629  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  1^st c1st 7931  2^nd c2nd 7932  GrpOpcgr 30564  RingOpscrngo 38095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-1st 7933  df-2nd 7934  df-grpo 30568  df-ablo 30620  df-rngo 38096

This theorem is referenced by:  rngosn4  38126