Theorem rngosn3 35204

Description: Obsolete as of 25-Jan-2020. Use ring1zr 20050 or srg1zr 19281 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 35190	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		on1el3.2	. . . . . . . . . 10 ⊢ 𝑋 = ran 𝐺
4	3	grpofo 28278	. . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
5		fof 6592	. . . . . . . . 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 483	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
8		id 22	. . . . . . . . 9 ⊢ (𝑋 = {𝐴} → 𝑋 = {𝐴})
9	8	sqxpeqd 5589	. . . . . . . 8 ⊢ (𝑋 = {𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴}))
10	9, 8	feq23d 6511	. . . . . . 7 ⊢ (𝑋 = {𝐴} → (𝐺:(𝑋 × 𝑋)⟶𝑋 ↔ 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
11	7, 10	syl5ibcom 247	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} → 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
12	7	fdmd 6525	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → dom 𝐺 = (𝑋 × 𝑋))
13	12	eqcomd 2829	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 × 𝑋) = dom 𝐺)
14		fdm 6524	. . . . . . . . 9 ⊢ (𝐺:({𝐴} × {𝐴})⟶{𝐴} → dom 𝐺 = ({𝐴} × {𝐴}))
15	14	eqeq2d 2834	. . . . . . . 8 ⊢ (𝐺:({𝐴} × {𝐴})⟶{𝐴} → ((𝑋 × 𝑋) = dom 𝐺 ↔ (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
16	13, 15	syl5ibcom 247	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
17		xpid11 5804	. . . . . . 7 ⊢ ((𝑋 × 𝑋) = ({𝐴} × {𝐴}) ↔ 𝑋 = {𝐴})
18	16, 17	syl6ib 253	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → 𝑋 = {𝐴}))
19	11, 18	impbid 214	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
20		simpr 487	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
21		xpsng 6903	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
22	20, 21	sylancom 590	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
23	22	feq2d 6502	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} ↔ 𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴}))
24		opex 5358	. . . . . 6 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
25		fsng 6901	. . . . . 6 ⊢ ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴 ∈ 𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
26	24, 20, 25	sylancr 589	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
27	19, 23, 26	3bitrd 307	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
28	1	eqeq1i 2828	. . . 4 ⊢ (𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↔ (1st ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
29	27, 28	syl6bb 289	. . 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 2823	. . . . . . 7 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
32	1, 31, 3	rngosm 35180	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (2nd ‘𝑅):(𝑋 × 𝑋)⟶𝑋)
33	32	adantr 483	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (2nd ‘𝑅):(𝑋 × 𝑋)⟶𝑋)
34	9, 8	feq23d 6511	. . . . 5 ⊢ (𝑋 = {𝐴} → ((2nd ‘𝑅):(𝑋 × 𝑋)⟶𝑋 ↔ (2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴}))
35	33, 34	syl5ibcom 247	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} → (2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴}))
36	22	feq2d 6502	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ((2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd ‘𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴}))
37		fsng 6901	. . . . . 6 ⊢ ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴 ∈ 𝐵) → ((2nd ‘𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
38	24, 20, 37	sylancr 589	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ((2nd ‘𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
39	36, 38	bitrd 281	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ((2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
40	35, 39	sylibd 241	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} → (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
41	40	pm4.71d 564	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ (𝑋 = {𝐴} ∧ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
42		relrngo 35176	. . . . . 6 ⊢ Rel RingOps
43		df-rel 5564	. . . . . 6 ⊢ (Rel RingOps ↔ RingOps ⊆ (V × V))
44	42, 43	mpbi 232	. . . . 5 ⊢ RingOps ⊆ (V × V)
45	44	sseli 3965	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑅 ∈ (V × V))
46	45	adantr 483	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → 𝑅 ∈ (V × V))
47		eqop 7733	. . 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 313	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ⊆ wss 3938  {csn 4569  ⟨cop 4575   × cxp 5555  dom cdm 5557  ran crn 5558  Rel wrel 5562  ⟶wf 6353  –onto→wfo 6355  ‘cfv 6357  1^st c1st 7689  2^nd c2nd 7690  GrpOpcgr 28268  RingOpscrngo 35174

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-1st 7691  df-2nd 7692  df-grpo 28272  df-ablo 28324  df-rngo 35175

This theorem is referenced by:  rngosn4  35205