Theorem isunit 18480

Description: Property of being a unit of a ring. A unit is an element that left- and right-divides one. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 8-Dec-2015.)

Hypotheses

Ref	Expression
unit.1	⊢ 𝑈 = (Unit‘𝑅)
unit.2	⊢ 1 = (1r‘𝑅)
unit.3	⊢ ∥ = (∥r‘𝑅)
unit.4	⊢ 𝑆 = (oppr‘𝑅)
unit.5	⊢ 𝐸 = (∥r‘𝑆)

Assertion

Ref	Expression
isunit	⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 ))

Proof of Theorem isunit

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6130	. . . 4 ⊢ (𝑋 ∈ (Unit‘𝑅) → 𝑅 ∈ dom Unit)
2		unit.1	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3	1, 2	eleq2s 2706	. . 3 ⊢ (𝑋 ∈ 𝑈 → 𝑅 ∈ dom Unit)
4	3	elexd 3187	. 2 ⊢ (𝑋 ∈ 𝑈 → 𝑅 ∈ V)
5		df-br 4584	. . . 4 ⊢ (𝑋 ∥ 1 ↔ ⟨𝑋, 1 ⟩ ∈ ∥ )
6		elfvdm 6130	. . . . . 6 ⊢ (⟨𝑋, 1 ⟩ ∈ (∥r‘𝑅) → 𝑅 ∈ dom ∥r)
7		unit.3	. . . . . 6 ⊢ ∥ = (∥r‘𝑅)
8	6, 7	eleq2s 2706	. . . . 5 ⊢ (⟨𝑋, 1 ⟩ ∈ ∥ → 𝑅 ∈ dom ∥r)
9	8	elexd 3187	. . . 4 ⊢ (⟨𝑋, 1 ⟩ ∈ ∥ → 𝑅 ∈ V)
10	5, 9	sylbi 206	. . 3 ⊢ (𝑋 ∥ 1 → 𝑅 ∈ V)
11	10	adantr 480	. 2 ⊢ ((𝑋 ∥ 1 ∧ 𝑋𝐸 1 ) → 𝑅 ∈ V)
12		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (∥r‘𝑟) = (∥r‘𝑅))
13	12, 7	syl6eqr 2662	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (∥r‘𝑟) = ∥ )
14		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅))
15		unit.4	. . . . . . . . . . . 12 ⊢ 𝑆 = (oppr‘𝑅)
16	14, 15	syl6eqr 2662	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑆)
17	16	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (∥r‘(oppr‘𝑟)) = (∥r‘𝑆))
18		unit.5	. . . . . . . . . 10 ⊢ 𝐸 = (∥r‘𝑆)
19	17, 18	syl6eqr 2662	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (∥r‘(oppr‘𝑟)) = 𝐸)
20	13, 19	ineq12d 3777	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) = ( ∥ ∩ 𝐸))
21	20	cnveqd 5220	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) = ◡( ∥ ∩ 𝐸))
22		fveq2 6103	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
23		unit.2	. . . . . . . . 9 ⊢ 1 = (1r‘𝑅)
24	22, 23	syl6eqr 2662	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
25	24	sneqd 4137	. . . . . . 7 ⊢ (𝑟 = 𝑅 → {(1r‘𝑟)} = { 1 })
26	21, 25	imaeq12d 5386	. . . . . 6 ⊢ (𝑟 = 𝑅 → (◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) “ {(1r‘𝑟)}) = (◡( ∥ ∩ 𝐸) “ { 1 }))
27		df-unit 18465	. . . . . 6 ⊢ Unit = (𝑟 ∈ V ↦ (◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) “ {(1r‘𝑟)}))
28		fvex 6113	. . . . . . . . . 10 ⊢ (∥r‘𝑅) ∈ V
29	7, 28	eqeltri 2684	. . . . . . . . 9 ⊢ ∥ ∈ V
30	29	inex1 4727	. . . . . . . 8 ⊢ ( ∥ ∩ 𝐸) ∈ V
31	30	cnvex 7006	. . . . . . 7 ⊢ ◡( ∥ ∩ 𝐸) ∈ V
32	31	imaex 6996	. . . . . 6 ⊢ (◡( ∥ ∩ 𝐸) “ { 1 }) ∈ V
33	26, 27, 32	fvmpt 6191	. . . . 5 ⊢ (𝑅 ∈ V → (Unit‘𝑅) = (◡( ∥ ∩ 𝐸) “ { 1 }))
34	2, 33	syl5eq 2656	. . . 4 ⊢ (𝑅 ∈ V → 𝑈 = (◡( ∥ ∩ 𝐸) “ { 1 }))
35	34	eleq2d 2673	. . 3 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 })))
36		inss1 3795	. . . . . 6 ⊢ ( ∥ ∩ 𝐸) ⊆ ∥
37	7	reldvdsr 18467	. . . . . 6 ⊢ Rel ∥
38		relss 5129	. . . . . 6 ⊢ (( ∥ ∩ 𝐸) ⊆ ∥ → (Rel ∥ → Rel ( ∥ ∩ 𝐸)))
39	36, 37, 38	mp2 9	. . . . 5 ⊢ Rel ( ∥ ∩ 𝐸)
40		eliniseg2 5424	. . . . 5 ⊢ (Rel ( ∥ ∩ 𝐸) → (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ 𝑋( ∥ ∩ 𝐸) 1 ))
41	39, 40	ax-mp 5	. . . 4 ⊢ (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ 𝑋( ∥ ∩ 𝐸) 1 )
42		brin 4634	. . . 4 ⊢ (𝑋( ∥ ∩ 𝐸) 1 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 ))
43	41, 42	bitri 263	. . 3 ⊢ (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 ))
44	35, 43	syl6bb 275	. 2 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 )))
45	4, 11, 44	pm5.21nii 367	1 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 ))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ⟨cop 4131   class class class wbr 4583  ^◡ccnv 5037  dom cdm 5038   “ cima 5041  Rel wrel 5043  ‘cfv 5804  1_rcur 18324  opp_rcoppr 18445  ∥_rcdsr 18461  Unitcui 18462

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-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-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-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-fv 5812  df-dvdsr 18464  df-unit 18465

This theorem is referenced by:  1unit  18481  unitcl  18482  opprunit  18484  crngunit  18485  unitmulcl  18487  unitgrp  18490  unitnegcl  18504  unitpropd  18520  isdrng2  18580  subrguss  18618  subrgunit  18621  fidomndrng  19128  invrvald  20301  elrhmunit  29151