Theorem isunit 20289

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 6856	. . . 4 ⊢ (𝑋 ∈ (Unit‘𝑅) → 𝑅 ∈ dom Unit)
2		unit.1	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3	1, 2	eleq2s 2849	. . 3 ⊢ (𝑋 ∈ 𝑈 → 𝑅 ∈ dom Unit)
4	3	elexd 3460	. 2 ⊢ (𝑋 ∈ 𝑈 → 𝑅 ∈ V)
5		df-br 5092	. . . 4 ⊢ (𝑋 ∥ 1 ↔ ⟨𝑋, 1 ⟩ ∈ ∥ )
6		elfvdm 6856	. . . . . 6 ⊢ (⟨𝑋, 1 ⟩ ∈ (∥r‘𝑅) → 𝑅 ∈ dom ∥r)
7		unit.3	. . . . . 6 ⊢ ∥ = (∥r‘𝑅)
8	6, 7	eleq2s 2849	. . . . 5 ⊢ (⟨𝑋, 1 ⟩ ∈ ∥ → 𝑅 ∈ dom ∥r)
9	8	elexd 3460	. . . 4 ⊢ (⟨𝑋, 1 ⟩ ∈ ∥ → 𝑅 ∈ V)
10	5, 9	sylbi 217	. . 3 ⊢ (𝑋 ∥ 1 → 𝑅 ∈ V)
11	10	adantr 480	. 2 ⊢ ((𝑋 ∥ 1 ∧ 𝑋𝐸 1 ) → 𝑅 ∈ V)
12		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (∥r‘𝑟) = (∥r‘𝑅))
13	12, 7	eqtr4di 2784	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (∥r‘𝑟) = ∥ )
14		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅))
15		unit.4	. . . . . . . . . . . 12 ⊢ 𝑆 = (oppr‘𝑅)
16	14, 15	eqtr4di 2784	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑆)
17	16	fveq2d 6826	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (∥r‘(oppr‘𝑟)) = (∥r‘𝑆))
18		unit.5	. . . . . . . . . 10 ⊢ 𝐸 = (∥r‘𝑆)
19	17, 18	eqtr4di 2784	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (∥r‘(oppr‘𝑟)) = 𝐸)
20	13, 19	ineq12d 4171	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) = ( ∥ ∩ 𝐸))
21	20	cnveqd 5815	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) = ◡( ∥ ∩ 𝐸))
22		fveq2 6822	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
23		unit.2	. . . . . . . . 9 ⊢ 1 = (1r‘𝑅)
24	22, 23	eqtr4di 2784	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
25	24	sneqd 4588	. . . . . . 7 ⊢ (𝑟 = 𝑅 → {(1r‘𝑟)} = { 1 })
26	21, 25	imaeq12d 6010	. . . . . 6 ⊢ (𝑟 = 𝑅 → (◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) “ {(1r‘𝑟)}) = (◡( ∥ ∩ 𝐸) “ { 1 }))
27		df-unit 20274	. . . . . 6 ⊢ Unit = (𝑟 ∈ V ↦ (◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) “ {(1r‘𝑟)}))
28	7	fvexi 6836	. . . . . . . . 9 ⊢ ∥ ∈ V
29	28	inex1 5255	. . . . . . . 8 ⊢ ( ∥ ∩ 𝐸) ∈ V
30	29	cnvex 7855	. . . . . . 7 ⊢ ◡( ∥ ∩ 𝐸) ∈ V
31	30	imaex 7844	. . . . . 6 ⊢ (◡( ∥ ∩ 𝐸) “ { 1 }) ∈ V
32	26, 27, 31	fvmpt 6929	. . . . 5 ⊢ (𝑅 ∈ V → (Unit‘𝑅) = (◡( ∥ ∩ 𝐸) “ { 1 }))
33	2, 32	eqtrid 2778	. . . 4 ⊢ (𝑅 ∈ V → 𝑈 = (◡( ∥ ∩ 𝐸) “ { 1 }))
34	33	eleq2d 2817	. . 3 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 })))
35		inss1 4187	. . . . . 6 ⊢ ( ∥ ∩ 𝐸) ⊆ ∥
36	7	reldvdsr 20276	. . . . . 6 ⊢ Rel ∥
37		relss 5722	. . . . . 6 ⊢ (( ∥ ∩ 𝐸) ⊆ ∥ → (Rel ∥ → Rel ( ∥ ∩ 𝐸)))
38	35, 36, 37	mp2 9	. . . . 5 ⊢ Rel ( ∥ ∩ 𝐸)
39		eliniseg2 6055	. . . . 5 ⊢ (Rel ( ∥ ∩ 𝐸) → (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ 𝑋( ∥ ∩ 𝐸) 1 ))
40	38, 39	ax-mp 5	. . . 4 ⊢ (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ 𝑋( ∥ ∩ 𝐸) 1 )
41		brin 5143	. . . 4 ⊢ (𝑋( ∥ ∩ 𝐸) 1 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 ))
42	40, 41	bitri 275	. . 3 ⊢ (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 ))
43	34, 42	bitrdi 287	. 2 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 )))
44	4, 11, 43	pm5.21nii 378	1 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∩ cin 3901   ⊆ wss 3902  {csn 4576  ⟨cop 4582   class class class wbr 5091  ^◡ccnv 5615  dom cdm 5616   “ cima 5619  Rel wrel 5621  ‘cfv 6481  1_rcur 20097  opp_rcoppr 20252  ∥_rcdsr 20270  Unitcui 20271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fv 6489  df-dvdsr 20273  df-unit 20274

This theorem is referenced by:  1unit  20290  unitcl  20291  opprunit  20293  crngunit  20294  unitmulcl  20296  unitgrp  20299  unitnegcl  20313  unitpropd  20333  elrhmunit  20423  subrguss  20500  subrgunit  20503  isdrng2  20656  fidomndrng  20686  invrvald  22589  isunit2  33202  isdrng4  33256