Theorem isunit 20390

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 6944	. . . 4 ⊢ (𝑋 ∈ (Unit‘𝑅) → 𝑅 ∈ dom Unit)
2		unit.1	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3	1, 2	eleq2s 2857	. . 3 ⊢ (𝑋 ∈ 𝑈 → 𝑅 ∈ dom Unit)
4	3	elexd 3502	. 2 ⊢ (𝑋 ∈ 𝑈 → 𝑅 ∈ V)
5		df-br 5149	. . . 4 ⊢ (𝑋 ∥ 1 ↔ ⟨𝑋, 1 ⟩ ∈ ∥ )
6		elfvdm 6944	. . . . . 6 ⊢ (⟨𝑋, 1 ⟩ ∈ (∥r‘𝑅) → 𝑅 ∈ dom ∥r)
7		unit.3	. . . . . 6 ⊢ ∥ = (∥r‘𝑅)
8	6, 7	eleq2s 2857	. . . . 5 ⊢ (⟨𝑋, 1 ⟩ ∈ ∥ → 𝑅 ∈ dom ∥r)
9	8	elexd 3502	. . . 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 6907	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (∥r‘𝑟) = (∥r‘𝑅))
13	12, 7	eqtr4di 2793	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (∥r‘𝑟) = ∥ )
14		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅))
15		unit.4	. . . . . . . . . . . 12 ⊢ 𝑆 = (oppr‘𝑅)
16	14, 15	eqtr4di 2793	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑆)
17	16	fveq2d 6911	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (∥r‘(oppr‘𝑟)) = (∥r‘𝑆))
18		unit.5	. . . . . . . . . 10 ⊢ 𝐸 = (∥r‘𝑆)
19	17, 18	eqtr4di 2793	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (∥r‘(oppr‘𝑟)) = 𝐸)
20	13, 19	ineq12d 4229	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) = ( ∥ ∩ 𝐸))
21	20	cnveqd 5889	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) = ◡( ∥ ∩ 𝐸))
22		fveq2 6907	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
23		unit.2	. . . . . . . . 9 ⊢ 1 = (1r‘𝑅)
24	22, 23	eqtr4di 2793	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
25	24	sneqd 4643	. . . . . . 7 ⊢ (𝑟 = 𝑅 → {(1r‘𝑟)} = { 1 })
26	21, 25	imaeq12d 6081	. . . . . 6 ⊢ (𝑟 = 𝑅 → (◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) “ {(1r‘𝑟)}) = (◡( ∥ ∩ 𝐸) “ { 1 }))
27		df-unit 20375	. . . . . 6 ⊢ Unit = (𝑟 ∈ V ↦ (◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) “ {(1r‘𝑟)}))
28	7	fvexi 6921	. . . . . . . . 9 ⊢ ∥ ∈ V
29	28	inex1 5323	. . . . . . . 8 ⊢ ( ∥ ∩ 𝐸) ∈ V
30	29	cnvex 7948	. . . . . . 7 ⊢ ◡( ∥ ∩ 𝐸) ∈ V
31	30	imaex 7937	. . . . . 6 ⊢ (◡( ∥ ∩ 𝐸) “ { 1 }) ∈ V
32	26, 27, 31	fvmpt 7016	. . . . 5 ⊢ (𝑅 ∈ V → (Unit‘𝑅) = (◡( ∥ ∩ 𝐸) “ { 1 }))
33	2, 32	eqtrid 2787	. . . 4 ⊢ (𝑅 ∈ V → 𝑈 = (◡( ∥ ∩ 𝐸) “ { 1 }))
34	33	eleq2d 2825	. . 3 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 })))
35		inss1 4245	. . . . . 6 ⊢ ( ∥ ∩ 𝐸) ⊆ ∥
36	7	reldvdsr 20377	. . . . . 6 ⊢ Rel ∥
37		relss 5794	. . . . . 6 ⊢ (( ∥ ∩ 𝐸) ⊆ ∥ → (Rel ∥ → Rel ( ∥ ∩ 𝐸)))
38	35, 36, 37	mp2 9	. . . . 5 ⊢ Rel ( ∥ ∩ 𝐸)
39		eliniseg2 6127	. . . . 5 ⊢ (Rel ( ∥ ∩ 𝐸) → (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ 𝑋( ∥ ∩ 𝐸) 1 ))
40	38, 39	ax-mp 5	. . . 4 ⊢ (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ 𝑋( ∥ ∩ 𝐸) 1 )
41		brin 5200	. . . 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 2106  Vcvv 3478   ∩ cin 3962   ⊆ wss 3963  {csn 4631  ⟨cop 4637   class class class wbr 5148  ^◡ccnv 5688  dom cdm 5689   “ cima 5692  Rel wrel 5694  ‘cfv 6563  1_rcur 20199  opp_rcoppr 20350  ∥_rcdsr 20371  Unitcui 20372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-dvdsr 20374  df-unit 20375

This theorem is referenced by:  1unit  20391  unitcl  20392  opprunit  20394  crngunit  20395  unitmulcl  20397  unitgrp  20400  unitnegcl  20414  unitpropd  20434  elrhmunit  20527  subrguss  20604  subrgunit  20607  isdrng2  20760  fidomndrng  20791  invrvald  22698  isunit2  33230  isdrng4  33279