Theorem isunit 20399

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 6957	. . . 4 ⊢ (𝑋 ∈ (Unit‘𝑅) → 𝑅 ∈ dom Unit)
2		unit.1	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3	1, 2	eleq2s 2862	. . 3 ⊢ (𝑋 ∈ 𝑈 → 𝑅 ∈ dom Unit)
4	3	elexd 3512	. 2 ⊢ (𝑋 ∈ 𝑈 → 𝑅 ∈ V)
5		df-br 5167	. . . 4 ⊢ (𝑋 ∥ 1 ↔ ⟨𝑋, 1 ⟩ ∈ ∥ )
6		elfvdm 6957	. . . . . 6 ⊢ (⟨𝑋, 1 ⟩ ∈ (∥r‘𝑅) → 𝑅 ∈ dom ∥r)
7		unit.3	. . . . . 6 ⊢ ∥ = (∥r‘𝑅)
8	6, 7	eleq2s 2862	. . . . 5 ⊢ (⟨𝑋, 1 ⟩ ∈ ∥ → 𝑅 ∈ dom ∥r)
9	8	elexd 3512	. . . 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 6920	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (∥r‘𝑟) = (∥r‘𝑅))
13	12, 7	eqtr4di 2798	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (∥r‘𝑟) = ∥ )
14		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅))
15		unit.4	. . . . . . . . . . . 12 ⊢ 𝑆 = (oppr‘𝑅)
16	14, 15	eqtr4di 2798	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑆)
17	16	fveq2d 6924	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (∥r‘(oppr‘𝑟)) = (∥r‘𝑆))
18		unit.5	. . . . . . . . . 10 ⊢ 𝐸 = (∥r‘𝑆)
19	17, 18	eqtr4di 2798	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (∥r‘(oppr‘𝑟)) = 𝐸)
20	13, 19	ineq12d 4242	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) = ( ∥ ∩ 𝐸))
21	20	cnveqd 5900	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) = ◡( ∥ ∩ 𝐸))
22		fveq2 6920	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
23		unit.2	. . . . . . . . 9 ⊢ 1 = (1r‘𝑅)
24	22, 23	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
25	24	sneqd 4660	. . . . . . 7 ⊢ (𝑟 = 𝑅 → {(1r‘𝑟)} = { 1 })
26	21, 25	imaeq12d 6090	. . . . . 6 ⊢ (𝑟 = 𝑅 → (◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) “ {(1r‘𝑟)}) = (◡( ∥ ∩ 𝐸) “ { 1 }))
27		df-unit 20384	. . . . . 6 ⊢ Unit = (𝑟 ∈ V ↦ (◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) “ {(1r‘𝑟)}))
28	7	fvexi 6934	. . . . . . . . 9 ⊢ ∥ ∈ V
29	28	inex1 5335	. . . . . . . 8 ⊢ ( ∥ ∩ 𝐸) ∈ V
30	29	cnvex 7965	. . . . . . 7 ⊢ ◡( ∥ ∩ 𝐸) ∈ V
31	30	imaex 7954	. . . . . 6 ⊢ (◡( ∥ ∩ 𝐸) “ { 1 }) ∈ V
32	26, 27, 31	fvmpt 7029	. . . . 5 ⊢ (𝑅 ∈ V → (Unit‘𝑅) = (◡( ∥ ∩ 𝐸) “ { 1 }))
33	2, 32	eqtrid 2792	. . . 4 ⊢ (𝑅 ∈ V → 𝑈 = (◡( ∥ ∩ 𝐸) “ { 1 }))
34	33	eleq2d 2830	. . 3 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 })))
35		inss1 4258	. . . . . 6 ⊢ ( ∥ ∩ 𝐸) ⊆ ∥
36	7	reldvdsr 20386	. . . . . 6 ⊢ Rel ∥
37		relss 5805	. . . . . 6 ⊢ (( ∥ ∩ 𝐸) ⊆ ∥ → (Rel ∥ → Rel ( ∥ ∩ 𝐸)))
38	35, 36, 37	mp2 9	. . . . 5 ⊢ Rel ( ∥ ∩ 𝐸)
39		eliniseg2 6136	. . . . 5 ⊢ (Rel ( ∥ ∩ 𝐸) → (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ 𝑋( ∥ ∩ 𝐸) 1 ))
40	38, 39	ax-mp 5	. . . 4 ⊢ (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ 𝑋( ∥ ∩ 𝐸) 1 )
41		brin 5218	. . . 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 1537   ∈ wcel 2108  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  {csn 4648  ⟨cop 4654   class class class wbr 5166  ^◡ccnv 5699  dom cdm 5700   “ cima 5703  Rel wrel 5705  ‘cfv 6573  1_rcur 20208  opp_rcoppr 20359  ∥_rcdsr 20380  Unitcui 20381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-dvdsr 20383  df-unit 20384

This theorem is referenced by:  1unit  20400  unitcl  20401  opprunit  20403  crngunit  20404  unitmulcl  20406  unitgrp  20409  unitnegcl  20423  unitpropd  20443  elrhmunit  20536  subrguss  20615  subrgunit  20618  isdrng2  20765  fidomndrng  20796  invrvald  22703  isunit2  33220  isdrng4  33264