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Theorem isunit 18877
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 = (1r𝑅)
unit.3 = (∥r𝑅)
unit.4 𝑆 = (oppr𝑅)
unit.5 𝐸 = (∥r𝑆)
Assertion
Ref Expression
isunit (𝑋𝑈 ↔ (𝑋 1𝑋𝐸 1 ))

Proof of Theorem isunit
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6382 . . . 4 (𝑋 ∈ (Unit‘𝑅) → 𝑅 ∈ dom Unit)
2 unit.1 . . . 4 𝑈 = (Unit‘𝑅)
31, 2eleq2s 2857 . . 3 (𝑋𝑈𝑅 ∈ dom Unit)
43elexd 3354 . 2 (𝑋𝑈𝑅 ∈ V)
5 df-br 4805 . . . 4 (𝑋 1 ↔ ⟨𝑋, 1 ⟩ ∈ )
6 elfvdm 6382 . . . . . 6 (⟨𝑋, 1 ⟩ ∈ (∥r𝑅) → 𝑅 ∈ dom ∥r)
7 unit.3 . . . . . 6 = (∥r𝑅)
86, 7eleq2s 2857 . . . . 5 (⟨𝑋, 1 ⟩ ∈ 𝑅 ∈ dom ∥r)
98elexd 3354 . . . 4 (⟨𝑋, 1 ⟩ ∈ 𝑅 ∈ V)
105, 9sylbi 207 . . 3 (𝑋 1𝑅 ∈ V)
1110adantr 472 . 2 ((𝑋 1𝑋𝐸 1 ) → 𝑅 ∈ V)
12 fveq2 6353 . . . . . . . . . 10 (𝑟 = 𝑅 → (∥r𝑟) = (∥r𝑅))
1312, 7syl6eqr 2812 . . . . . . . . 9 (𝑟 = 𝑅 → (∥r𝑟) = )
14 fveq2 6353 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (oppr𝑟) = (oppr𝑅))
15 unit.4 . . . . . . . . . . . 12 𝑆 = (oppr𝑅)
1614, 15syl6eqr 2812 . . . . . . . . . . 11 (𝑟 = 𝑅 → (oppr𝑟) = 𝑆)
1716fveq2d 6357 . . . . . . . . . 10 (𝑟 = 𝑅 → (∥r‘(oppr𝑟)) = (∥r𝑆))
18 unit.5 . . . . . . . . . 10 𝐸 = (∥r𝑆)
1917, 18syl6eqr 2812 . . . . . . . . 9 (𝑟 = 𝑅 → (∥r‘(oppr𝑟)) = 𝐸)
2013, 19ineq12d 3958 . . . . . . . 8 (𝑟 = 𝑅 → ((∥r𝑟) ∩ (∥r‘(oppr𝑟))) = ( 𝐸))
2120cnveqd 5453 . . . . . . 7 (𝑟 = 𝑅((∥r𝑟) ∩ (∥r‘(oppr𝑟))) = ( 𝐸))
22 fveq2 6353 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
23 unit.2 . . . . . . . . 9 1 = (1r𝑅)
2422, 23syl6eqr 2812 . . . . . . . 8 (𝑟 = 𝑅 → (1r𝑟) = 1 )
2524sneqd 4333 . . . . . . 7 (𝑟 = 𝑅 → {(1r𝑟)} = { 1 })
2621, 25imaeq12d 5625 . . . . . 6 (𝑟 = 𝑅 → (((∥r𝑟) ∩ (∥r‘(oppr𝑟))) “ {(1r𝑟)}) = (( 𝐸) “ { 1 }))
27 df-unit 18862 . . . . . 6 Unit = (𝑟 ∈ V ↦ (((∥r𝑟) ∩ (∥r‘(oppr𝑟))) “ {(1r𝑟)}))
28 fvex 6363 . . . . . . . . . 10 (∥r𝑅) ∈ V
297, 28eqeltri 2835 . . . . . . . . 9 ∈ V
3029inex1 4951 . . . . . . . 8 ( 𝐸) ∈ V
3130cnvex 7279 . . . . . . 7 ( 𝐸) ∈ V
3231imaex 7270 . . . . . 6 (( 𝐸) “ { 1 }) ∈ V
3326, 27, 32fvmpt 6445 . . . . 5 (𝑅 ∈ V → (Unit‘𝑅) = (( 𝐸) “ { 1 }))
342, 33syl5eq 2806 . . . 4 (𝑅 ∈ V → 𝑈 = (( 𝐸) “ { 1 }))
3534eleq2d 2825 . . 3 (𝑅 ∈ V → (𝑋𝑈𝑋 ∈ (( 𝐸) “ { 1 })))
36 inss1 3976 . . . . . 6 ( 𝐸) ⊆
377reldvdsr 18864 . . . . . 6 Rel
38 relss 5363 . . . . . 6 (( 𝐸) ⊆ → (Rel → Rel ( 𝐸)))
3936, 37, 38mp2 9 . . . . 5 Rel ( 𝐸)
40 eliniseg2 5663 . . . . 5 (Rel ( 𝐸) → (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ 𝑋( 𝐸) 1 ))
4139, 40ax-mp 5 . . . 4 (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ 𝑋( 𝐸) 1 )
42 brin 4856 . . . 4 (𝑋( 𝐸) 1 ↔ (𝑋 1𝑋𝐸 1 ))
4341, 42bitri 264 . . 3 (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ (𝑋 1𝑋𝐸 1 ))
4435, 43syl6bb 276 . 2 (𝑅 ∈ V → (𝑋𝑈 ↔ (𝑋 1𝑋𝐸 1 )))
454, 11, 44pm5.21nii 367 1 (𝑋𝑈 ↔ (𝑋 1𝑋𝐸 1 ))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  wss 3715  {csn 4321  cop 4327   class class class wbr 4804  ccnv 5265  dom cdm 5266  cima 5269  Rel wrel 5271  cfv 6049  1rcur 18721  opprcoppr 18842  rcdsr 18858  Unitcui 18859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-dvdsr 18861  df-unit 18862
This theorem is referenced by:  1unit  18878  unitcl  18879  opprunit  18881  crngunit  18882  unitmulcl  18884  unitgrp  18887  unitnegcl  18901  unitpropd  18917  isdrng2  18979  subrguss  19017  subrgunit  19020  fidomndrng  19529  invrvald  20704  elrhmunit  30150
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