Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngosn3 Structured version   Visualization version   GIF version

Theorem rngosn3 33853
Description: Obsolete as of 25-Jan-2020. Use ring1zr 19323 or srg1zr 18575 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 33839 . . . . . . . . 9 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 on1el3.2 . . . . . . . . . 10 𝑋 = ran 𝐺
43grpofo 27481 . . . . . . . . 9 (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto𝑋)
5 fof 6153 . . . . . . . . 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 5175 . . . . . . . 8 (𝑋 = {𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴}))
109, 8feq23d 6078 . . . . . . 7 (𝑋 = {𝐴} → (𝐺:(𝑋 × 𝑋)⟶𝑋𝐺:({𝐴} × {𝐴})⟶{𝐴}))
117, 10syl5ibcom 235 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} → 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
12 fdm 6089 . . . . . . . . . 10 (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom 𝐺 = (𝑋 × 𝑋))
137, 12syl 17 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → dom 𝐺 = (𝑋 × 𝑋))
1413eqcomd 2657 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 × 𝑋) = dom 𝐺)
15 fdm 6089 . . . . . . . . 9 (𝐺:({𝐴} × {𝐴})⟶{𝐴} → dom 𝐺 = ({𝐴} × {𝐴}))
1615eqeq2d 2661 . . . . . . . 8 (𝐺:({𝐴} × {𝐴})⟶{𝐴} → ((𝑋 × 𝑋) = dom 𝐺 ↔ (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
1714, 16syl5ibcom 235 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
18 xpid11 5379 . . . . . . 7 ((𝑋 × 𝑋) = ({𝐴} × {𝐴}) ↔ 𝑋 = {𝐴})
1917, 18syl6ib 241 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → 𝑋 = {𝐴}))
2011, 19impbid 202 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
21 simpr 476 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → 𝐴𝐵)
22 xpsng 6446 . . . . . . 7 ((𝐴𝐵𝐴𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
2321, 22sylancom 702 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
2423feq2d 6069 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} ↔ 𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴}))
25 opex 4962 . . . . . 6 𝐴, 𝐴⟩ ∈ V
26 fsng 6444 . . . . . 6 ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
2725, 21, 26sylancr 696 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
2820, 24, 273bitrd 294 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
291eqeq1i 2656 . . . 4 (𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↔ (1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
3028, 29syl6bb 276 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ (1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
3130anbi1d 741 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((𝑋 = {𝐴} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ↔ ((1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
32 eqid 2651 . . . . . . 7 (2nd𝑅) = (2nd𝑅)
331, 32, 3rngosm 33829 . . . . . 6 (𝑅 ∈ RingOps → (2nd𝑅):(𝑋 × 𝑋)⟶𝑋)
3433adantr 480 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (2nd𝑅):(𝑋 × 𝑋)⟶𝑋)
359, 8feq23d 6078 . . . . 5 (𝑋 = {𝐴} → ((2nd𝑅):(𝑋 × 𝑋)⟶𝑋 ↔ (2nd𝑅):({𝐴} × {𝐴})⟶{𝐴}))
3634, 35syl5ibcom 235 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} → (2nd𝑅):({𝐴} × {𝐴})⟶{𝐴}))
3723feq2d 6069 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((2nd𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴}))
38 fsng 6444 . . . . . 6 ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴𝐵) → ((2nd𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
3925, 21, 38sylancr 696 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((2nd𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
4037, 39bitrd 268 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((2nd𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
4136, 40sylibd 229 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} → (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
4241pm4.71d 667 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ (𝑋 = {𝐴} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
43 relrngo 33825 . . . . . 6 Rel RingOps
44 df-rel 5150 . . . . . 6 (Rel RingOps ↔ RingOps ⊆ (V × V))
4543, 44mpbi 220 . . . . 5 RingOps ⊆ (V × V)
4645sseli 3632 . . . 4 (𝑅 ∈ RingOps → 𝑅 ∈ (V × V))
4746adantr 480 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → 𝑅 ∈ (V × V))
48 eqop 7252 . . 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 300 1 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  {csn 4210  cop 4216   × cxp 5141  dom cdm 5143  ran crn 5144  Rel wrel 5148  wf 5922  ontowfo 5924  cfv 5926  1st c1st 7208  2nd c2nd 7209  GrpOpcgr 27471  RingOpscrngo 33823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-1st 7210  df-2nd 7211  df-grpo 27475  df-ablo 27527  df-rngo 33824
This theorem is referenced by:  rngosn4  33854
  Copyright terms: Public domain W3C validator