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Theorem grposnOLD 37889
Description: The group operation for the singleton group. Obsolete, use grp1 19065. 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 5436 . 2 {𝐴} ∈ V
2 opex 5469 . . . . 5 𝐴, 𝐴⟩ ∈ V
3 grposnOLD.1 . . . . 5 𝐴 ∈ V
42, 3f1osn 6888 . . . 4 {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}–1-1-onto→{𝐴}
5 f1of 6848 . . . 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 7161 . . . 4 ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩}
87feq2i 6728 . . 3 ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:({𝐴} × {𝐴})⟶{𝐴} ↔ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}⟶{𝐴})
96, 8mpbir 231 . 2 {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:({𝐴} × {𝐴})⟶{𝐴}
10 velsn 4642 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11 velsn 4642 . . 3 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
12 velsn 4642 . . 3 (𝑧 ∈ {𝐴} ↔ 𝑧 = 𝐴)
13 oveq2 7439 . . . . . 6 (𝑧 = 𝐴 → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
14 oveq1 7438 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦))
15 oveq2 7439 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
16 df-ov 7434 . . . . . . . . . . 11 (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}‘⟨𝐴, 𝐴⟩)
172, 3fvsn 7201 . . . . . . . . . . 11 ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}‘⟨𝐴, 𝐴⟩) = 𝐴
1816, 17eqtri 2765 . . . . . . . . . 10 (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = 𝐴
1915, 18eqtrdi 2793 . . . . . . . . 9 (𝑦 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = 𝐴)
2014, 19sylan9eq 2797 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = 𝐴)
2120oveq1d 7446 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
2221, 18eqtrdi 2793 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = 𝐴)
2313, 22sylan9eqr 2799 . . . . 5 (((𝑥 = 𝐴𝑦 = 𝐴) ∧ 𝑧 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
24233impa 1110 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐴𝑧 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
25 oveq1 7438 . . . . . 6 (𝑥 = 𝐴 → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)))
26 oveq1 7438 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧))
27 oveq2 7439 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
2827, 18eqtrdi 2793 . . . . . . . . 9 (𝑧 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
2926, 28sylan9eq 2797 . . . . . . . 8 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
3029oveq2d 7447 . . . . . . 7 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
3130, 18eqtrdi 2793 . . . . . 6 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = 𝐴)
3225, 31sylan9eq 2797 . . . . 5 ((𝑥 = 𝐴 ∧ (𝑦 = 𝐴𝑧 = 𝐴)) → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = 𝐴)
33323impb 1115 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐴𝑧 = 𝐴) → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = 𝐴)
3424, 33eqtr4d 2780 . . 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 1162 . 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 4662 . 2 𝐴 ∈ {𝐴}
37 oveq2 7439 . . . . 5 (𝑥 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
3837, 18eqtrdi 2793 . . . 4 (𝑥 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝐴)
39 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
4038, 39eqtr4d 2780 . . 3 (𝑥 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝑥)
4110, 40sylbi 217 . 2 (𝑥 ∈ {𝐴} → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝑥)
4236a1i 11 . 2 (𝑥 ∈ {𝐴} → 𝐴 ∈ {𝐴})
4310, 38sylbi 217 . 2 (𝑥 ∈ {𝐴} → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝐴)
441, 9, 35, 36, 41, 42, 43isgrpoi 30517 1 {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632   × cxp 5683  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  GrpOpcgr 30508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-grpo 30512
This theorem is referenced by:  gidsn  37959
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