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Theorem rngosn3 32893
Description: Obsolete as of 25-Jan-2020. Use ring1zr 19096 or srg1zr 18352 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 32879 . . . . . . . . 9 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 on1el3.2 . . . . . . . . . 10 𝑋 = ran 𝐺
43grpofo 26737 . . . . . . . . 9 (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto𝑋)
5 fof 6028 . . . . . . . . 9 (𝐺:(𝑋 × 𝑋)–onto𝑋𝐺:(𝑋 × 𝑋)⟶𝑋)
62, 4, 53syl 18 . . . . . . . 8 (𝑅 ∈ RingOps → 𝐺:(𝑋 × 𝑋)⟶𝑋)
76adantr 480 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
8 id 22 . . . . . . . . 9 (𝑋 = {𝐴} → 𝑋 = {𝐴})
98sqxpeqd 5065 . . . . . . . 8 (𝑋 = {𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴}))
109, 8feq23d 5953 . . . . . . 7 (𝑋 = {𝐴} → (𝐺:(𝑋 × 𝑋)⟶𝑋𝐺:({𝐴} × {𝐴})⟶{𝐴}))
117, 10syl5ibcom 234 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} → 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
12 fdm 5964 . . . . . . . . . 10 (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom 𝐺 = (𝑋 × 𝑋))
137, 12syl 17 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → dom 𝐺 = (𝑋 × 𝑋))
1413eqcomd 2616 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 × 𝑋) = dom 𝐺)
15 fdm 5964 . . . . . . . . 9 (𝐺:({𝐴} × {𝐴})⟶{𝐴} → dom 𝐺 = ({𝐴} × {𝐴}))
1615eqeq2d 2620 . . . . . . . 8 (𝐺:({𝐴} × {𝐴})⟶{𝐴} → ((𝑋 × 𝑋) = dom 𝐺 ↔ (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
1714, 16syl5ibcom 234 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
18 xpid11 5268 . . . . . . 7 ((𝑋 × 𝑋) = ({𝐴} × {𝐴}) ↔ 𝑋 = {𝐴})
1917, 18syl6ib 240 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → 𝑋 = {𝐴}))
2011, 19impbid 201 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
21 simpr 476 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → 𝐴𝐵)
22 xpsng 6312 . . . . . . 7 ((𝐴𝐵𝐴𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
2321, 22sylancom 698 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
2423feq2d 5944 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} ↔ 𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴}))
25 opex 4859 . . . . . 6 𝐴, 𝐴⟩ ∈ V
26 fsng 6310 . . . . . 6 ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
2725, 21, 26sylancr 694 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
2820, 24, 273bitrd 293 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
291eqeq1i 2615 . . . 4 (𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↔ (1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
3028, 29syl6bb 275 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ (1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
3130anbi1d 737 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((𝑋 = {𝐴} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ↔ ((1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
32 eqid 2610 . . . . . . 7 (2nd𝑅) = (2nd𝑅)
331, 32, 3rngosm 32869 . . . . . 6 (𝑅 ∈ RingOps → (2nd𝑅):(𝑋 × 𝑋)⟶𝑋)
3433adantr 480 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (2nd𝑅):(𝑋 × 𝑋)⟶𝑋)
359, 8feq23d 5953 . . . . 5 (𝑋 = {𝐴} → ((2nd𝑅):(𝑋 × 𝑋)⟶𝑋 ↔ (2nd𝑅):({𝐴} × {𝐴})⟶{𝐴}))
3634, 35syl5ibcom 234 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} → (2nd𝑅):({𝐴} × {𝐴})⟶{𝐴}))
3723feq2d 5944 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((2nd𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴}))
38 fsng 6310 . . . . . 6 ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴𝐵) → ((2nd𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
3925, 21, 38sylancr 694 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((2nd𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
4037, 39bitrd 267 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((2nd𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
4136, 40sylibd 228 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} → (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
4241pm4.71d 664 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ (𝑋 = {𝐴} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
43 relrngo 32865 . . . . . 6 Rel RingOps
44 df-rel 5045 . . . . . 6 (Rel RingOps ↔ RingOps ⊆ (V × V))
4543, 44mpbi 219 . . . . 5 RingOps ⊆ (V × V)
4645sseli 3564 . . . 4 (𝑅 ∈ RingOps → 𝑅 ∈ (V × V))
4746adantr 480 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → 𝑅 ∈ (V × V))
48 eqop 7099 . . 3 (𝑅 ∈ (V × V) → (𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ↔ ((1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
4947, 48syl 17 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ↔ ((1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
5031, 42, 493bitr4d 299 1 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  wss 3540  {csn 4125  cop 4131   × cxp 5036  dom cdm 5038  ran crn 5039  Rel wrel 5043  wf 5800  ontowfo 5802  cfv 5804  1st c1st 7057  2nd c2nd 7058  GrpOpcgr 26727  RingOpscrngo 32863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-1st 7059  df-2nd 7060  df-grpo 26731  df-ablo 26783  df-rngo 32864
This theorem is referenced by:  rngosn4  32894
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