Theorem rngosn3 34680

Description: Obsolete as of 25-Jan-2020. Use ring1zr 19725 or srg1zr 18957 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 34666	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		on1el3.2	. . . . . . . . . 10 ⊢ 𝑋 = ran 𝐺
4	3	grpofo 27955	. . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
5		fof 6450	. . . . . . . . 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 481	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
8		id 22	. . . . . . . . 9 ⊢ (𝑋 = {𝐴} → 𝑋 = {𝐴})
9	8	sqxpeqd 5467	. . . . . . . 8 ⊢ (𝑋 = {𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴}))
10	9, 8	feq23d 6369	. . . . . . 7 ⊢ (𝑋 = {𝐴} → (𝐺:(𝑋 × 𝑋)⟶𝑋 ↔ 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
11	7, 10	syl5ibcom 246	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} → 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
12	7	fdmd 6383	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → dom 𝐺 = (𝑋 × 𝑋))
13	12	eqcomd 2799	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 × 𝑋) = dom 𝐺)
14		fdm 6382	. . . . . . . . 9 ⊢ (𝐺:({𝐴} × {𝐴})⟶{𝐴} → dom 𝐺 = ({𝐴} × {𝐴}))
15	14	eqeq2d 2803	. . . . . . . 8 ⊢ (𝐺:({𝐴} × {𝐴})⟶{𝐴} → ((𝑋 × 𝑋) = dom 𝐺 ↔ (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
16	13, 15	syl5ibcom 246	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
17		xpid11 5676	. . . . . . 7 ⊢ ((𝑋 × 𝑋) = ({𝐴} × {𝐴}) ↔ 𝑋 = {𝐴})
18	16, 17	syl6ib 252	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → 𝑋 = {𝐴}))
19	11, 18	impbid 213	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
20		simpr 485	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
21		xpsng 6755	. . . . . . 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 6360	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} ↔ 𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴}))
24		opex 5241	. . . . . 6 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
25		fsng 6753	. . . . . 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 306	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
28	1	eqeq1i 2798	. . . 4 ⊢ (𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↔ (1st ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
29	27, 28	syl6bb 288	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ (1st ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
30	29	anbi1d 629	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ((𝑋 = {𝐴} ∧ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ↔ ((1st ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
31		eqid 2793	. . . . . . 7 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
32	1, 31, 3	rngosm 34656	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (2nd ‘𝑅):(𝑋 × 𝑋)⟶𝑋)
33	32	adantr 481	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (2nd ‘𝑅):(𝑋 × 𝑋)⟶𝑋)
34	9, 8	feq23d 6369	. . . . 5 ⊢ (𝑋 = {𝐴} → ((2nd ‘𝑅):(𝑋 × 𝑋)⟶𝑋 ↔ (2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴}))
35	33, 34	syl5ibcom 246	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} → (2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴}))
36	22	feq2d 6360	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ((2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd ‘𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴}))
37		fsng 6753	. . . . . 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 280	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ((2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
40	35, 39	sylibd 240	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} → (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
41	40	pm4.71d 562	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ (𝑋 = {𝐴} ∧ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
42		relrngo 34652	. . . . . 6 ⊢ Rel RingOps
43		df-rel 5442	. . . . . 6 ⊢ (Rel RingOps ↔ RingOps ⊆ (V × V))
44	42, 43	mpbi 231	. . . . 5 ⊢ RingOps ⊆ (V × V)
45	44	sseli 3880	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑅 ∈ (V × V))
46	45	adantr 481	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → 𝑅 ∈ (V × V))
47		eqop 7578	. . 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 312	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1520   ∈ wcel 2079  Vcvv 3432   ⊆ wss 3854  {csn 4466  ⟨cop 4472   × cxp 5433  dom cdm 5435  ran crn 5436  Rel wrel 5440  ⟶wf 6213  –onto→wfo 6215  ‘cfv 6217  1^st c1st 7534  2^nd c2nd 7535  GrpOpcgr 27945  RingOpscrngo 34650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-1st 7536  df-2nd 7537  df-grpo 27949  df-ablo 28001  df-rngo 34651

This theorem is referenced by:  rngosn4  34681