Theorem isunit 20351

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 6868	. . . 4 ⊢ (𝑋 ∈ (Unit‘𝑅) → 𝑅 ∈ dom Unit)
2		unit.1	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3	1, 2	eleq2s 2858	. . 3 ⊢ (𝑋 ∈ 𝑈 → 𝑅 ∈ dom Unit)
4	3	elexd 3456	. 2 ⊢ (𝑋 ∈ 𝑈 → 𝑅 ∈ V)
5		df-br 5080	. . . 4 ⊢ (𝑋 ∥ 1 ↔ ⟨𝑋, 1 ⟩ ∈ ∥ )
6		elfvdm 6868	. . . . . 6 ⊢ (⟨𝑋, 1 ⟩ ∈ (∥r‘𝑅) → 𝑅 ∈ dom ∥r)
7		unit.3	. . . . . 6 ⊢ ∥ = (∥r‘𝑅)
8	6, 7	eleq2s 2858	. . . . 5 ⊢ (⟨𝑋, 1 ⟩ ∈ ∥ → 𝑅 ∈ dom ∥r)
9	8	elexd 3456	. . . 4 ⊢ (⟨𝑋, 1 ⟩ ∈ ∥ → 𝑅 ∈ V)
10	5, 9	sylbi 218	. . 3 ⊢ (𝑋 ∥ 1 → 𝑅 ∈ V)
11	10	adantr 481	. 2 ⊢ ((𝑋 ∥ 1 ∧ 𝑋𝐸 1 ) → 𝑅 ∈ V)
12		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (∥r‘𝑟) = (∥r‘𝑅))
13	12, 7	eqtr4di 2793	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (∥r‘𝑟) = ∥ )
14		fveq2 6834	. . . . . . . . . . . 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 6838	. . . . . . . . . 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 4157	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) = ( ∥ ∩ 𝐸))
21	20	cnveqd 5824	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) = ◡( ∥ ∩ 𝐸))
22		fveq2 6834	. . . . . . . . 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 4574	. . . . . . 7 ⊢ (𝑟 = 𝑅 → {(1r‘𝑟)} = { 1 })
26	21, 25	imaeq12d 6020	. . . . . 6 ⊢ (𝑟 = 𝑅 → (◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) “ {(1r‘𝑟)}) = (◡( ∥ ∩ 𝐸) “ { 1 }))
27		df-unit 20336	. . . . . 6 ⊢ Unit = (𝑟 ∈ V ↦ (◡((∥r‘𝑟) ∩ (∥r‘(oppr‘𝑟))) “ {(1r‘𝑟)}))
28	7	fvexi 6848	. . . . . . . . 9 ⊢ ∥ ∈ V
29	28	inex1 5252	. . . . . . . 8 ⊢ ( ∥ ∩ 𝐸) ∈ V
30	29	cnvex 7872	. . . . . . 7 ⊢ ◡( ∥ ∩ 𝐸) ∈ V
31	30	imaex 7861	. . . . . 6 ⊢ (◡( ∥ ∩ 𝐸) “ { 1 }) ∈ V
32	26, 27, 31	fvmpt 6942	. . . . 5 ⊢ (𝑅 ∈ V → (Unit‘𝑅) = (◡( ∥ ∩ 𝐸) “ { 1 }))
33	2, 32	eqtrid 2787	. . . 4 ⊢ (𝑅 ∈ V → 𝑈 = (◡( ∥ ∩ 𝐸) “ { 1 }))
34	33	eleq2d 2826	. . 3 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 })))
35		inss1 4172	. . . . . 6 ⊢ ( ∥ ∩ 𝐸) ⊆ ∥
36	7	reldvdsr 20338	. . . . . 6 ⊢ Rel ∥
37		relss 5732	. . . . . 6 ⊢ (( ∥ ∩ 𝐸) ⊆ ∥ → (Rel ∥ → Rel ( ∥ ∩ 𝐸)))
38	35, 36, 37	mp2 9	. . . . 5 ⊢ Rel ( ∥ ∩ 𝐸)
39		eliniseg2 6065	. . . . 5 ⊢ (Rel ( ∥ ∩ 𝐸) → (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ 𝑋( ∥ ∩ 𝐸) 1 ))
40	38, 39	ax-mp 5	. . . 4 ⊢ (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ 𝑋( ∥ ∩ 𝐸) 1 )
41		brin 5131	. . . 4 ⊢ (𝑋( ∥ ∩ 𝐸) 1 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 ))
42	40, 41	bitri 276	. . 3 ⊢ (𝑋 ∈ (◡( ∥ ∩ 𝐸) “ { 1 }) ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 ))
43	34, 42	bitrdi 288	. 2 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 )))
44	4, 11, 43	pm5.21nii 379	1 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 ))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432   ∩ cin 3889   ⊆ wss 3890  {csn 4562  ⟨cop 4568   class class class wbr 5079  ^◡ccnv 5624  dom cdm 5625   “ cima 5628  Rel wrel 5630  ‘cfv 6492  1_rcur 20160  opp_rcoppr 20314  ∥_rcdsr 20332  Unitcui 20333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fv 6500  df-dvdsr 20335  df-unit 20336

This theorem is referenced by:  1unit  20352  unitcl  20353  opprunit  20355  crngunit  20356  unitmulcl  20358  unitgrp  20361  unitnegcl  20375  unitpropd  20395  elrhmunit  20489  subrguss  20566  subrgunit  20569  isdrng2  20722  fidomndrng  20752  invrvald  22666  isunit2  33328  isdrng4  33386