Theorem rngosn3 38175

Description: Obsolete as of 25-Jan-2020. Use ring1zr 20721 or srg1zr 20162 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 38161	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		on1el3.2	. . . . . . . . . 10 ⊢ 𝑋 = ran 𝐺
4	3	grpofo 30587	. . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
5		fof 6754	. . . . . . . . 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 5664	. . . . . . . 8 ⊢ (𝑋 = {𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴}))
10	9, 8	feq23d 6665	. . . . . . 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 6680	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → dom 𝐺 = (𝑋 × 𝑋))
13	12	eqcomd 2743	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 × 𝑋) = dom 𝐺)
14		fdm 6679	. . . . . . . . 9 ⊢ (𝐺:({𝐴} × {𝐴})⟶{𝐴} → dom 𝐺 = ({𝐴} × {𝐴}))
15	14	eqeq2d 2748	. . . . . . . 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 5889	. . . . . . 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 7094	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
22	20, 21	sylancom 589	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
23	22	feq2d 6654	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} ↔ 𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴}))
24		opex 5419	. . . . . 6 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
25		fsng 7092	. . . . . 6 ⊢ ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴 ∈ 𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
26	24, 20, 25	sylancr 588	. . . . 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 2742	. . . 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 632	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ((𝑋 = {𝐴} ∧ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ↔ ((1st ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
31		eqid 2737	. . . . . . 7 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
32	1, 31, 3	rngosm 38151	. . . . . 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 6665	. . . . 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 6654	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → ((2nd ‘𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd ‘𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴}))
37		fsng 7092	. . . . . 6 ⊢ ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴 ∈ 𝐵) → ((2nd ‘𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd ‘𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
38	24, 20, 37	sylancr 588	. . . . 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 38147	. . . . . 6 ⊢ Rel RingOps
43		df-rel 5639	. . . . . 6 ⊢ (Rel RingOps ↔ RingOps ⊆ (V × V))
44	42, 43	mpbi 230	. . . . 5 ⊢ RingOps ⊆ (V × V)
45	44	sseli 3931	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑅 ∈ (V × V))
46	45	adantr 480	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → 𝑅 ∈ (V × V))
47		eqop 7985	. . 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 1542   ∈ wcel 2114  Vcvv 3442   ⊆ wss 3903  {csn 4582  ⟨cop 4588   × cxp 5630  dom cdm 5632  ran crn 5633  Rel wrel 5637  ⟶wf 6496  –onto→wfo 6498  ‘cfv 6500  1^st c1st 7941  2^nd c2nd 7942  GrpOpcgr 30577  RingOpscrngo 38145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-1st 7943  df-2nd 7944  df-grpo 30581  df-ablo 30633  df-rngo 38146

This theorem is referenced by:  rngosn4  38176