Theorem grposnOLD 37906

Description: The group operation for the singleton group. Obsolete, use grp1 19030. 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.

Step	Hyp	Ref	Expression
1		snex 5406	. 2 ⊢ {𝐴} ∈ V
2		opex 5439	. . . . 5 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
3		grposnOLD.1	. . . . 5 ⊢ 𝐴 ∈ V
4	2, 3	f1osn 6858	. . . 4 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}–1-1-onto→{𝐴}
5		f1of 6818	. . . 4 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}–1-1-onto→{𝐴} → {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}⟶{𝐴})
6	4, 5	ax-mp 5	. . 3 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}⟶{𝐴}
7	3, 3	xpsn 7131	. . . 4 ⊢ ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩}
8	7	feq2i 6698	. . 3 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:({𝐴} × {𝐴})⟶{𝐴} ↔ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}⟶{𝐴})
9	6, 8	mpbir 231	. 2 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:({𝐴} × {𝐴})⟶{𝐴}
10		velsn 4617	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11		velsn 4617	. . 3 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
12		velsn 4617	. . 3 ⊢ (𝑧 ∈ {𝐴} ↔ 𝑧 = 𝐴)
13		oveq2 7413	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
14		oveq1 7412	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦))
15		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
16		df-ov 7408	. . . . . . . . . . 11 ⊢ (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}‘⟨𝐴, 𝐴⟩)
17	2, 3	fvsn 7173	. . . . . . . . . . 11 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}‘⟨𝐴, 𝐴⟩) = 𝐴
18	16, 17	eqtri 2758	. . . . . . . . . 10 ⊢ (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = 𝐴
19	15, 18	eqtrdi 2786	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = 𝐴)
20	14, 19	sylan9eq 2790	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = 𝐴)
21	20	oveq1d 7420	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
22	21, 18	eqtrdi 2786	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = 𝐴)
23	13, 22	sylan9eqr 2792	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) ∧ 𝑧 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
24	23	3impa 1109	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
25		oveq1 7412	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)))
26		oveq1 7412	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧))
27		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
28	27, 18	eqtrdi 2786	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
29	26, 28	sylan9eq 2790	. . . . . . . 8 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
30	29	oveq2d 7421	. . . . . . 7 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
31	30, 18	eqtrdi 2786	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = 𝐴)
32	25, 31	sylan9eq 2790	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ (𝑦 = 𝐴 ∧ 𝑧 = 𝐴)) → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = 𝐴)
33	32	3impb 1114	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = 𝐴)
34	24, 33	eqtr4d 2773	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)))
35	10, 11, 12, 34	syl3anb 1161	. 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)))
36	3	snid 4638	. 2 ⊢ 𝐴 ∈ {𝐴}
37		oveq2 7413	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
38	37, 18	eqtrdi 2786	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝐴)
39		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
40	38, 39	eqtr4d 2773	. . 3 ⊢ (𝑥 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝑥)
41	10, 40	sylbi 217	. 2 ⊢ (𝑥 ∈ {𝐴} → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝑥)
42	36	a1i 11	. 2 ⊢ (𝑥 ∈ {𝐴} → 𝐴 ∈ {𝐴})
43	10, 38	sylbi 217	. 2 ⊢ (𝑥 ∈ {𝐴} → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝐴)
44	1, 9, 35, 36, 41, 42, 43	isgrpoi 30479	1 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  Vcvv 3459  {csn 4601  ⟨cop 4607   × cxp 5652  ⟶wf 6527  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7405  GrpOpcgr 30470

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-grpo 30474

This theorem is referenced by:  gidsn  37976