Theorem grposnOLD 37053

Description: The group operation for the singleton group. Obsolete, use grp1 18966. 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 5430	. 2 ⊢ {𝐴} ∈ V
2		opex 5463	. . . . 5 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
3		grposnOLD.1	. . . . 5 ⊢ 𝐴 ∈ V
4	2, 3	f1osn 6872	. . . 4 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}–1-1-onto→{𝐴}
5		f1of 6832	. . . 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 7140	. . . 4 ⊢ ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩}
8	7	feq2i 6708	. . 3 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:({𝐴} × {𝐴})⟶{𝐴} ↔ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:{⟨𝐴, 𝐴⟩}⟶{𝐴})
9	6, 8	mpbir 230	. 2 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}:({𝐴} × {𝐴})⟶{𝐴}
10		velsn 4643	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11		velsn 4643	. . 3 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
12		velsn 4643	. . 3 ⊢ (𝑧 ∈ {𝐴} ↔ 𝑧 = 𝐴)
13		oveq2 7419	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
14		oveq1 7418	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦))
15		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴))
16		df-ov 7414	. . . . . . . . . . 11 ⊢ (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝐴) = ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩}‘⟨𝐴, 𝐴⟩)
17	2, 3	fvsn 7180	. . . . . . . . . . 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 7426	. . . . . . 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 1108	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → ((𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑦){⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = 𝐴)
25		oveq1 7418	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩} (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧)))
26		oveq1 7418	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑦{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧) = (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑧))
27		oveq2 7419	. . . . . . . . . 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 7427	. . . . . . 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 1113	. . . 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 1159	. 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 4663	. 2 ⊢ 𝐴 ∈ {𝐴}
37		oveq2 7419	. . . . 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 216	. 2 ⊢ (𝑥 ∈ {𝐴} → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝑥)
42	36	a1i 11	. 2 ⊢ (𝑥 ∈ {𝐴} → 𝐴 ∈ {𝐴})
43	10, 38	sylbi 216	. 2 ⊢ (𝑥 ∈ {𝐴} → (𝐴{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}𝑥) = 𝐴)
44	1, 9, 35, 36, 41, 42, 43	isgrpoi 30018	1 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp

Syntax hints:   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  Vcvv 3472  {csn 4627  ⟨cop 4633   × cxp 5673  ⟶wf 6538  –1-1-onto→wf1o 6541  ‘cfv 6542  (class class class)co 7411  GrpOpcgr 30009

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-grpo 30013

This theorem is referenced by:  gidsn  37123