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Theorem isunit 19406
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 6701 . . . 4 (𝑋 ∈ (Unit‘𝑅) → 𝑅 ∈ dom Unit)
2 unit.1 . . . 4 𝑈 = (Unit‘𝑅)
31, 2eleq2s 2931 . . 3 (𝑋𝑈𝑅 ∈ dom Unit)
43elexd 3514 . 2 (𝑋𝑈𝑅 ∈ V)
5 df-br 5066 . . . 4 (𝑋 1 ↔ ⟨𝑋, 1 ⟩ ∈ )
6 elfvdm 6701 . . . . . 6 (⟨𝑋, 1 ⟩ ∈ (∥r𝑅) → 𝑅 ∈ dom ∥r)
7 unit.3 . . . . . 6 = (∥r𝑅)
86, 7eleq2s 2931 . . . . 5 (⟨𝑋, 1 ⟩ ∈ 𝑅 ∈ dom ∥r)
98elexd 3514 . . . 4 (⟨𝑋, 1 ⟩ ∈ 𝑅 ∈ V)
105, 9sylbi 219 . . 3 (𝑋 1𝑅 ∈ V)
1110adantr 483 . 2 ((𝑋 1𝑋𝐸 1 ) → 𝑅 ∈ V)
12 fveq2 6669 . . . . . . . . . 10 (𝑟 = 𝑅 → (∥r𝑟) = (∥r𝑅))
1312, 7syl6eqr 2874 . . . . . . . . 9 (𝑟 = 𝑅 → (∥r𝑟) = )
14 fveq2 6669 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (oppr𝑟) = (oppr𝑅))
15 unit.4 . . . . . . . . . . . 12 𝑆 = (oppr𝑅)
1614, 15syl6eqr 2874 . . . . . . . . . . 11 (𝑟 = 𝑅 → (oppr𝑟) = 𝑆)
1716fveq2d 6673 . . . . . . . . . 10 (𝑟 = 𝑅 → (∥r‘(oppr𝑟)) = (∥r𝑆))
18 unit.5 . . . . . . . . . 10 𝐸 = (∥r𝑆)
1917, 18syl6eqr 2874 . . . . . . . . 9 (𝑟 = 𝑅 → (∥r‘(oppr𝑟)) = 𝐸)
2013, 19ineq12d 4189 . . . . . . . 8 (𝑟 = 𝑅 → ((∥r𝑟) ∩ (∥r‘(oppr𝑟))) = ( 𝐸))
2120cnveqd 5745 . . . . . . 7 (𝑟 = 𝑅((∥r𝑟) ∩ (∥r‘(oppr𝑟))) = ( 𝐸))
22 fveq2 6669 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
23 unit.2 . . . . . . . . 9 1 = (1r𝑅)
2422, 23syl6eqr 2874 . . . . . . . 8 (𝑟 = 𝑅 → (1r𝑟) = 1 )
2524sneqd 4578 . . . . . . 7 (𝑟 = 𝑅 → {(1r𝑟)} = { 1 })
2621, 25imaeq12d 5929 . . . . . 6 (𝑟 = 𝑅 → (((∥r𝑟) ∩ (∥r‘(oppr𝑟))) “ {(1r𝑟)}) = (( 𝐸) “ { 1 }))
27 df-unit 19391 . . . . . 6 Unit = (𝑟 ∈ V ↦ (((∥r𝑟) ∩ (∥r‘(oppr𝑟))) “ {(1r𝑟)}))
287fvexi 6683 . . . . . . . . 9 ∈ V
2928inex1 5220 . . . . . . . 8 ( 𝐸) ∈ V
3029cnvex 7629 . . . . . . 7 ( 𝐸) ∈ V
3130imaex 7620 . . . . . 6 (( 𝐸) “ { 1 }) ∈ V
3226, 27, 31fvmpt 6767 . . . . 5 (𝑅 ∈ V → (Unit‘𝑅) = (( 𝐸) “ { 1 }))
332, 32syl5eq 2868 . . . 4 (𝑅 ∈ V → 𝑈 = (( 𝐸) “ { 1 }))
3433eleq2d 2898 . . 3 (𝑅 ∈ V → (𝑋𝑈𝑋 ∈ (( 𝐸) “ { 1 })))
35 inss1 4204 . . . . . 6 ( 𝐸) ⊆
367reldvdsr 19393 . . . . . 6 Rel
37 relss 5655 . . . . . 6 (( 𝐸) ⊆ → (Rel → Rel ( 𝐸)))
3835, 36, 37mp2 9 . . . . 5 Rel ( 𝐸)
39 eliniseg2 5968 . . . . 5 (Rel ( 𝐸) → (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ 𝑋( 𝐸) 1 ))
4038, 39ax-mp 5 . . . 4 (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ 𝑋( 𝐸) 1 )
41 brin 5117 . . . 4 (𝑋( 𝐸) 1 ↔ (𝑋 1𝑋𝐸 1 ))
4240, 41bitri 277 . . 3 (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ (𝑋 1𝑋𝐸 1 ))
4334, 42syl6bb 289 . 2 (𝑅 ∈ V → (𝑋𝑈 ↔ (𝑋 1𝑋𝐸 1 )))
444, 11, 43pm5.21nii 382 1 (𝑋𝑈 ↔ (𝑋 1𝑋𝐸 1 ))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1533  wcel 2110  Vcvv 3494  cin 3934  wss 3935  {csn 4566  cop 4572   class class class wbr 5065  ccnv 5553  dom cdm 5554  cima 5557  Rel wrel 5559  cfv 6354  1rcur 19250  opprcoppr 19371  rcdsr 19387  Unitcui 19388
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fv 6362  df-dvdsr 19390  df-unit 19391
This theorem is referenced by:  1unit  19407  unitcl  19408  opprunit  19410  crngunit  19411  unitmulcl  19413  unitgrp  19416  unitnegcl  19430  unitpropd  19446  isdrng2  19511  subrguss  19549  subrgunit  19552  fidomndrng  20079  invrvald  21284  elrhmunit  30893
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