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Theorem rngosn3 35819
Description: Obsolete as of 25-Jan-2020. Use ring1zr 20313 or srg1zr 19544 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 = (1st𝑅)
on1el3.2 𝑋 = ran 𝐺
Assertion
Ref Expression
rngosn3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))

Proof of Theorem rngosn3
StepHypRef Expression
1 on1el3.1 . . . . . . . . . 10 𝐺 = (1st𝑅)
21rngogrpo 35805 . . . . . . . . 9 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 on1el3.2 . . . . . . . . . 10 𝑋 = ran 𝐺
43grpofo 28580 . . . . . . . . 9 (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto𝑋)
5 fof 6633 . . . . . . . . 9 (𝐺:(𝑋 × 𝑋)–onto𝑋𝐺:(𝑋 × 𝑋)⟶𝑋)
62, 4, 53syl 18 . . . . . . . 8 (𝑅 ∈ RingOps → 𝐺:(𝑋 × 𝑋)⟶𝑋)
76adantr 484 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
8 id 22 . . . . . . . . 9 (𝑋 = {𝐴} → 𝑋 = {𝐴})
98sqxpeqd 5583 . . . . . . . 8 (𝑋 = {𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴}))
109, 8feq23d 6540 . . . . . . 7 (𝑋 = {𝐴} → (𝐺:(𝑋 × 𝑋)⟶𝑋𝐺:({𝐴} × {𝐴})⟶{𝐴}))
117, 10syl5ibcom 248 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} → 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
127fdmd 6556 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → dom 𝐺 = (𝑋 × 𝑋))
1312eqcomd 2743 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 × 𝑋) = dom 𝐺)
14 fdm 6554 . . . . . . . . 9 (𝐺:({𝐴} × {𝐴})⟶{𝐴} → dom 𝐺 = ({𝐴} × {𝐴}))
1514eqeq2d 2748 . . . . . . . 8 (𝐺:({𝐴} × {𝐴})⟶{𝐴} → ((𝑋 × 𝑋) = dom 𝐺 ↔ (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
1613, 15syl5ibcom 248 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
17 xpid11 5801 . . . . . . 7 ((𝑋 × 𝑋) = ({𝐴} × {𝐴}) ↔ 𝑋 = {𝐴})
1816, 17syl6ib 254 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → 𝑋 = {𝐴}))
1911, 18impbid 215 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
20 simpr 488 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → 𝐴𝐵)
21 xpsng 6954 . . . . . . 7 ((𝐴𝐵𝐴𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
2220, 21sylancom 591 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
2322feq2d 6531 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} ↔ 𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴}))
24 opex 5348 . . . . . 6 𝐴, 𝐴⟩ ∈ V
25 fsng 6952 . . . . . 6 ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
2624, 20, 25sylancr 590 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
2719, 23, 263bitrd 308 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
281eqeq1i 2742 . . . 4 (𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↔ (1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
2927, 28bitrdi 290 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ (1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
3029anbi1d 633 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((𝑋 = {𝐴} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ↔ ((1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
31 eqid 2737 . . . . . . 7 (2nd𝑅) = (2nd𝑅)
321, 31, 3rngosm 35795 . . . . . 6 (𝑅 ∈ RingOps → (2nd𝑅):(𝑋 × 𝑋)⟶𝑋)
3332adantr 484 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (2nd𝑅):(𝑋 × 𝑋)⟶𝑋)
349, 8feq23d 6540 . . . . 5 (𝑋 = {𝐴} → ((2nd𝑅):(𝑋 × 𝑋)⟶𝑋 ↔ (2nd𝑅):({𝐴} × {𝐴})⟶{𝐴}))
3533, 34syl5ibcom 248 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} → (2nd𝑅):({𝐴} × {𝐴})⟶{𝐴}))
3622feq2d 6531 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((2nd𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴}))
37 fsng 6952 . . . . . 6 ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴𝐵) → ((2nd𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
3824, 20, 37sylancr 590 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((2nd𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
3936, 38bitrd 282 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((2nd𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
4035, 39sylibd 242 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} → (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
4140pm4.71d 565 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ (𝑋 = {𝐴} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
42 relrngo 35791 . . . . . 6 Rel RingOps
43 df-rel 5558 . . . . . 6 (Rel RingOps ↔ RingOps ⊆ (V × V))
4442, 43mpbi 233 . . . . 5 RingOps ⊆ (V × V)
4544sseli 3896 . . . 4 (𝑅 ∈ RingOps → 𝑅 ∈ (V × V))
4645adantr 484 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → 𝑅 ∈ (V × V))
47 eqop 7803 . . 3 (𝑅 ∈ (V × V) → (𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ↔ ((1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
4846, 47syl 17 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ↔ ((1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
4930, 41, 483bitr4d 314 1 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  Vcvv 3408  wss 3866  {csn 4541  cop 4547   × cxp 5549  dom cdm 5551  ran crn 5552  Rel wrel 5556  wf 6376  ontowfo 6378  cfv 6380  1st c1st 7759  2nd c2nd 7760  GrpOpcgr 28570  RingOpscrngo 35789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-1st 7761  df-2nd 7762  df-grpo 28574  df-ablo 28626  df-rngo 35790
This theorem is referenced by:  rngosn4  35820
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