Theorem grposnOLD 38256

Description: The group operation for the singleton group. Obsolete, use grp1 19021. 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 5375	. 2 ⊢ {𝐴} ∈ V
2		opex 5410	. . . . 5 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
3		grposnOLD.1	. . . . 5 ⊢ 𝐴 ∈ V
4	2, 3	f1osn 6815	. . . 4 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}–1-1-onto→{𝐴}
5		f1of 6774	. . . 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 7090	. . . 4 ⊢ ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩}
8	7	feq2i 6654	. . 3 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:({𝐴} × {𝐴})⟶{𝐴} ↔ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}⟶{𝐴})
9	6, 8	mpbir 232	. 2 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:({𝐴} × {𝐴})⟶{𝐴}
10		velsn 4578	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11		velsn 4578	. . 3 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
12		velsn 4578	. . 3 ⊢ (𝑧 ∈ {𝐴} ↔ 𝑧 = 𝐴)
13		oveq2 7371	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
14		oveq1 7370	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦))
15		oveq2 7371	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
16		df-ov 7366	. . . . . . . . . . 11 ⊢ (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}‘⟨𝐴, 𝐴⟩)
17	2, 3	fvsn 7132	. . . . . . . . . . 11 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}‘⟨𝐴, 𝐴⟩) = 𝐴
18	16, 17	eqtri 2763	. . . . . . . . . 10 ⊢ (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = 𝐴
19	15, 18	eqtrdi 2791	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = 𝐴)
20	14, 19	sylan9eq 2795	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = 𝐴)
21	20	oveq1d 7378	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
22	21, 18	eqtrdi 2791	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = 𝐴)
23	13, 22	sylan9eqr 2797	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) ∧ 𝑧 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
24	23	3impa 1115	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
25		oveq1 7370	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)))
26		oveq1 7370	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧))
27		oveq2 7371	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
28	27, 18	eqtrdi 2791	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
29	26, 28	sylan9eq 2795	. . . . . . . 8 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
30	29	oveq2d 7379	. . . . . . 7 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
31	30, 18	eqtrdi 2791	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = 𝐴)
32	25, 31	sylan9eq 2795	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ (𝑦 = 𝐴 ∧ 𝑧 = 𝐴)) → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = 𝐴)
33	32	3impb 1120	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = 𝐴)
34	24, 33	eqtr4d 2778	. . 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 1167	. 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 4601	. 2 ⊢ 𝐴 ∈ {𝐴}
37		oveq2 7371	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
38	37, 18	eqtrdi 2791	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝐴)
39		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
40	38, 39	eqtr4d 2778	. . 3 ⊢ (𝑥 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝑥)
41	10, 40	sylbi 218	. 2 ⊢ (𝑥 ∈ {𝐴} → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝑥)
42	36	a1i 11	. 2 ⊢ (𝑥 ∈ {𝐴} → 𝐴 ∈ {𝐴})
43	10, 38	sylbi 218	. 2 ⊢ (𝑥 ∈ {𝐴} → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝐴)
44	1, 9, 35, 36, 41, 42, 43	isgrpoi 30594	1 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp

Syntax hints:   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  Vcvv 3432  {csn 4562  ⟨cop 4568   × cxp 5623  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7363  GrpOpcgr 30585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-grpo 30589

This theorem is referenced by:  gidsn  38326