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Theorem isunit 20373
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 6943 . . . 4 (𝑋 ∈ (Unit‘𝑅) → 𝑅 ∈ dom Unit)
2 unit.1 . . . 4 𝑈 = (Unit‘𝑅)
31, 2eleq2s 2859 . . 3 (𝑋𝑈𝑅 ∈ dom Unit)
43elexd 3504 . 2 (𝑋𝑈𝑅 ∈ V)
5 df-br 5144 . . . 4 (𝑋 1 ↔ ⟨𝑋, 1 ⟩ ∈ )
6 elfvdm 6943 . . . . . 6 (⟨𝑋, 1 ⟩ ∈ (∥r𝑅) → 𝑅 ∈ dom ∥r)
7 unit.3 . . . . . 6 = (∥r𝑅)
86, 7eleq2s 2859 . . . . 5 (⟨𝑋, 1 ⟩ ∈ 𝑅 ∈ dom ∥r)
98elexd 3504 . . . 4 (⟨𝑋, 1 ⟩ ∈ 𝑅 ∈ V)
105, 9sylbi 217 . . 3 (𝑋 1𝑅 ∈ V)
1110adantr 480 . 2 ((𝑋 1𝑋𝐸 1 ) → 𝑅 ∈ V)
12 fveq2 6906 . . . . . . . . . 10 (𝑟 = 𝑅 → (∥r𝑟) = (∥r𝑅))
1312, 7eqtr4di 2795 . . . . . . . . 9 (𝑟 = 𝑅 → (∥r𝑟) = )
14 fveq2 6906 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (oppr𝑟) = (oppr𝑅))
15 unit.4 . . . . . . . . . . . 12 𝑆 = (oppr𝑅)
1614, 15eqtr4di 2795 . . . . . . . . . . 11 (𝑟 = 𝑅 → (oppr𝑟) = 𝑆)
1716fveq2d 6910 . . . . . . . . . 10 (𝑟 = 𝑅 → (∥r‘(oppr𝑟)) = (∥r𝑆))
18 unit.5 . . . . . . . . . 10 𝐸 = (∥r𝑆)
1917, 18eqtr4di 2795 . . . . . . . . 9 (𝑟 = 𝑅 → (∥r‘(oppr𝑟)) = 𝐸)
2013, 19ineq12d 4221 . . . . . . . 8 (𝑟 = 𝑅 → ((∥r𝑟) ∩ (∥r‘(oppr𝑟))) = ( 𝐸))
2120cnveqd 5886 . . . . . . 7 (𝑟 = 𝑅((∥r𝑟) ∩ (∥r‘(oppr𝑟))) = ( 𝐸))
22 fveq2 6906 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
23 unit.2 . . . . . . . . 9 1 = (1r𝑅)
2422, 23eqtr4di 2795 . . . . . . . 8 (𝑟 = 𝑅 → (1r𝑟) = 1 )
2524sneqd 4638 . . . . . . 7 (𝑟 = 𝑅 → {(1r𝑟)} = { 1 })
2621, 25imaeq12d 6079 . . . . . 6 (𝑟 = 𝑅 → (((∥r𝑟) ∩ (∥r‘(oppr𝑟))) “ {(1r𝑟)}) = (( 𝐸) “ { 1 }))
27 df-unit 20358 . . . . . 6 Unit = (𝑟 ∈ V ↦ (((∥r𝑟) ∩ (∥r‘(oppr𝑟))) “ {(1r𝑟)}))
287fvexi 6920 . . . . . . . . 9 ∈ V
2928inex1 5317 . . . . . . . 8 ( 𝐸) ∈ V
3029cnvex 7947 . . . . . . 7 ( 𝐸) ∈ V
3130imaex 7936 . . . . . 6 (( 𝐸) “ { 1 }) ∈ V
3226, 27, 31fvmpt 7016 . . . . 5 (𝑅 ∈ V → (Unit‘𝑅) = (( 𝐸) “ { 1 }))
332, 32eqtrid 2789 . . . 4 (𝑅 ∈ V → 𝑈 = (( 𝐸) “ { 1 }))
3433eleq2d 2827 . . 3 (𝑅 ∈ V → (𝑋𝑈𝑋 ∈ (( 𝐸) “ { 1 })))
35 inss1 4237 . . . . . 6 ( 𝐸) ⊆
367reldvdsr 20360 . . . . . 6 Rel
37 relss 5791 . . . . . 6 (( 𝐸) ⊆ → (Rel → Rel ( 𝐸)))
3835, 36, 37mp2 9 . . . . 5 Rel ( 𝐸)
39 eliniseg2 6124 . . . . 5 (Rel ( 𝐸) → (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ 𝑋( 𝐸) 1 ))
4038, 39ax-mp 5 . . . 4 (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ 𝑋( 𝐸) 1 )
41 brin 5195 . . . 4 (𝑋( 𝐸) 1 ↔ (𝑋 1𝑋𝐸 1 ))
4240, 41bitri 275 . . 3 (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ (𝑋 1𝑋𝐸 1 ))
4334, 42bitrdi 287 . 2 (𝑅 ∈ V → (𝑋𝑈 ↔ (𝑋 1𝑋𝐸 1 )))
444, 11, 43pm5.21nii 378 1 (𝑋𝑈 ↔ (𝑋 1𝑋𝐸 1 ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  wss 3951  {csn 4626  cop 4632   class class class wbr 5143  ccnv 5684  dom cdm 5685  cima 5688  Rel wrel 5690  cfv 6561  1rcur 20178  opprcoppr 20333  rcdsr 20354  Unitcui 20355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-dvdsr 20357  df-unit 20358
This theorem is referenced by:  1unit  20374  unitcl  20375  opprunit  20377  crngunit  20378  unitmulcl  20380  unitgrp  20383  unitnegcl  20397  unitpropd  20417  elrhmunit  20510  subrguss  20587  subrgunit  20590  isdrng2  20743  fidomndrng  20774  invrvald  22682  isunit2  33244  isdrng4  33298
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