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Theorem rngosn3 36429
Description: Obsolete as of 25-Jan-2020. Use ring1zr 20761 or srg1zr 19951 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 36415 . . . . . . . . 9 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 on1el3.2 . . . . . . . . . 10 𝑋 = ran 𝐺
43grpofo 29483 . . . . . . . . 9 (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto𝑋)
5 fof 6757 . . . . . . . . 9 (𝐺:(𝑋 × 𝑋)–onto𝑋𝐺:(𝑋 × 𝑋)⟶𝑋)
62, 4, 53syl 18 . . . . . . . 8 (𝑅 ∈ RingOps → 𝐺:(𝑋 × 𝑋)⟶𝑋)
76adantr 482 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
8 id 22 . . . . . . . . 9 (𝑋 = {𝐴} → 𝑋 = {𝐴})
98sqxpeqd 5666 . . . . . . . 8 (𝑋 = {𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴}))
109, 8feq23d 6664 . . . . . . 7 (𝑋 = {𝐴} → (𝐺:(𝑋 × 𝑋)⟶𝑋𝐺:({𝐴} × {𝐴})⟶{𝐴}))
117, 10syl5ibcom 244 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} → 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
127fdmd 6680 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → dom 𝐺 = (𝑋 × 𝑋))
1312eqcomd 2739 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 × 𝑋) = dom 𝐺)
14 fdm 6678 . . . . . . . . 9 (𝐺:({𝐴} × {𝐴})⟶{𝐴} → dom 𝐺 = ({𝐴} × {𝐴}))
1514eqeq2d 2744 . . . . . . . 8 (𝐺:({𝐴} × {𝐴})⟶{𝐴} → ((𝑋 × 𝑋) = dom 𝐺 ↔ (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
1613, 15syl5ibcom 244 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → (𝑋 × 𝑋) = ({𝐴} × {𝐴})))
17 xpid11 5888 . . . . . . 7 ((𝑋 × 𝑋) = ({𝐴} × {𝐴}) ↔ 𝑋 = {𝐴})
1816, 17syl6ib 251 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} → 𝑋 = {𝐴}))
1911, 18impbid 211 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝐺:({𝐴} × {𝐴})⟶{𝐴}))
20 simpr 486 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → 𝐴𝐵)
21 xpsng 7086 . . . . . . 7 ((𝐴𝐵𝐴𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
2220, 21sylancom 589 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
2322feq2d 6655 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:({𝐴} × {𝐴})⟶{𝐴} ↔ 𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴}))
24 opex 5422 . . . . . 6 𝐴, 𝐴⟩ ∈ V
25 fsng 7084 . . . . . 6 ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
2624, 20, 25sylancr 588 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝐺:{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
2719, 23, 263bitrd 305 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
281eqeq1i 2738 . . . 4 (𝐺 = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ↔ (1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
2927, 28bitrdi 287 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ (1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
3029anbi1d 631 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((𝑋 = {𝐴} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ↔ ((1st𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
31 eqid 2733 . . . . . . 7 (2nd𝑅) = (2nd𝑅)
321, 31, 3rngosm 36405 . . . . . 6 (𝑅 ∈ RingOps → (2nd𝑅):(𝑋 × 𝑋)⟶𝑋)
3332adantr 482 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (2nd𝑅):(𝑋 × 𝑋)⟶𝑋)
349, 8feq23d 6664 . . . . 5 (𝑋 = {𝐴} → ((2nd𝑅):(𝑋 × 𝑋)⟶𝑋 ↔ (2nd𝑅):({𝐴} × {𝐴})⟶{𝐴}))
3533, 34syl5ibcom 244 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} → (2nd𝑅):({𝐴} × {𝐴})⟶{𝐴}))
3622feq2d 6655 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((2nd𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴}))
37 fsng 7084 . . . . . 6 ((⟨𝐴, 𝐴⟩ ∈ V ∧ 𝐴𝐵) → ((2nd𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
3824, 20, 37sylancr 588 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((2nd𝑅):{⟨𝐴, 𝐴⟩}⟶{𝐴} ↔ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
3936, 38bitrd 279 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → ((2nd𝑅):({𝐴} × {𝐴})⟶{𝐴} ↔ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
4035, 39sylibd 238 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} → (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}))
4140pm4.71d 563 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ (𝑋 = {𝐴} ∧ (2nd𝑅) = {⟨⟨𝐴, 𝐴⟩, 𝐴⟩})))
42 relrngo 36401 . . . . . 6 Rel RingOps
43 df-rel 5641 . . . . . 6 (Rel RingOps ↔ RingOps ⊆ (V × V))
4442, 43mpbi 229 . . . . 5 RingOps ⊆ (V × V)
4544sseli 3941 . . . 4 (𝑅 ∈ RingOps → 𝑅 ∈ (V × V))
4645adantr 482 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → 𝑅 ∈ (V × V))
47 eqop 7964 . . 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 311 1 ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3444  wss 3911  {csn 4587  cop 4593   × cxp 5632  dom cdm 5634  ran crn 5635  Rel wrel 5639  wf 6493  ontowfo 6495  cfv 6497  1st c1st 7920  2nd c2nd 7921  GrpOpcgr 29473  RingOpscrngo 36399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-1st 7922  df-2nd 7923  df-grpo 29477  df-ablo 29529  df-rngo 36400
This theorem is referenced by:  rngosn4  36430
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