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Theorem grposnOLD 32851
Description: The group operation for the singleton group. Obsolete, use grp1 17345. instead (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
grposnOLD.1 𝐴 ∈ V
Assertion
Ref Expression
grposnOLD {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp

Proof of Theorem grposnOLD
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4835 . 2 {𝐴} ∈ V
2 opex 4859 . . . . 5 𝐴, 𝐴⟩ ∈ V
3 grposnOLD.1 . . . . 5 𝐴 ∈ V
42, 3f1osn 6088 . . . 4 {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}–1-1-onto→{𝐴}
5 f1of 6050 . . . 4 ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}–1-1-onto→{𝐴} → {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}⟶{𝐴})
64, 5ax-mp 5 . . 3 {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}⟶{𝐴}
73, 3xpsn 6313 . . . 4 ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩}
87feq2i 5950 . . 3 ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:({𝐴} × {𝐴})⟶{𝐴} ↔ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}⟶{𝐴})
96, 8mpbir 220 . 2 {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:({𝐴} × {𝐴})⟶{𝐴}
10 velsn 4141 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11 velsn 4141 . . 3 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
12 velsn 4141 . . 3 (𝑧 ∈ {𝐴} ↔ 𝑧 = 𝐴)
13 oveq2 6557 . . . . . 6 (𝑧 = 𝐴 → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
14 oveq1 6556 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦))
15 oveq2 6557 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
16 df-ov 6552 . . . . . . . . . . 11 (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}‘⟨𝐴, 𝐴⟩)
172, 3fvsn 6351 . . . . . . . . . . 11 ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}‘⟨𝐴, 𝐴⟩) = 𝐴
1816, 17eqtri 2632 . . . . . . . . . 10 (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = 𝐴
1915, 18syl6eq 2660 . . . . . . . . 9 (𝑦 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = 𝐴)
2014, 19sylan9eq 2664 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = 𝐴)
2120oveq1d 6564 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
2221, 18syl6eq 2660 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = 𝐴)
2313, 22sylan9eqr 2666 . . . . 5 (((𝑥 = 𝐴𝑦 = 𝐴) ∧ 𝑧 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
24233impa 1251 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐴𝑧 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
25 oveq1 6556 . . . . . 6 (𝑥 = 𝐴 → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)))
26 oveq1 6556 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧))
27 oveq2 6557 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
2827, 18syl6eq 2660 . . . . . . . . 9 (𝑧 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
2926, 28sylan9eq 2664 . . . . . . . 8 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
3029oveq2d 6565 . . . . . . 7 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
3130, 18syl6eq 2660 . . . . . 6 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = 𝐴)
3225, 31sylan9eq 2664 . . . . 5 ((𝑥 = 𝐴 ∧ (𝑦 = 𝐴𝑧 = 𝐴)) → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = 𝐴)
33323impb 1252 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐴𝑧 = 𝐴) → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = 𝐴)
3424, 33eqtr4d 2647 . . 3 ((𝑥 = 𝐴𝑦 = 𝐴𝑧 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)))
3510, 11, 12, 34syl3anb 1361 . 2 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑧 ∈ {𝐴}) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)))
363snid 4155 . 2 𝐴 ∈ {𝐴}
37 oveq2 6557 . . . . 5 (𝑥 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
3837, 18syl6eq 2660 . . . 4 (𝑥 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝐴)
39 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
4038, 39eqtr4d 2647 . . 3 (𝑥 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝑥)
4110, 40sylbi 206 . 2 (𝑥 ∈ {𝐴} → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝑥)
4236a1i 11 . 2 (𝑥 ∈ {𝐴} → 𝐴 ∈ {𝐴})
4310, 38sylbi 206 . 2 (𝑥 ∈ {𝐴} → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝐴)
441, 9, 35, 36, 41, 42, 43isgrpoi 26736 1 {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp
Colors of variables: wff setvar class
Syntax hints:  wa 383  w3a 1031   = wceq 1475  wcel 1977  Vcvv 3173  {csn 4125  cop 4131   × cxp 5036  wf 5800  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  GrpOpcgr 26727
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-rep 4699  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-pw 4110  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-res 5050  df-ima 5051  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-grpo 26731
This theorem is referenced by:  gidsn  32921
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