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Theorem isunit 18480
 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 6130 . . . 4 (𝑋 ∈ (Unit‘𝑅) → 𝑅 ∈ dom Unit)
2 unit.1 . . . 4 𝑈 = (Unit‘𝑅)
31, 2eleq2s 2706 . . 3 (𝑋𝑈𝑅 ∈ dom Unit)
43elexd 3187 . 2 (𝑋𝑈𝑅 ∈ V)
5 df-br 4584 . . . 4 (𝑋 1 ↔ ⟨𝑋, 1 ⟩ ∈ )
6 elfvdm 6130 . . . . . 6 (⟨𝑋, 1 ⟩ ∈ (∥r𝑅) → 𝑅 ∈ dom ∥r)
7 unit.3 . . . . . 6 = (∥r𝑅)
86, 7eleq2s 2706 . . . . 5 (⟨𝑋, 1 ⟩ ∈ 𝑅 ∈ dom ∥r)
98elexd 3187 . . . 4 (⟨𝑋, 1 ⟩ ∈ 𝑅 ∈ V)
105, 9sylbi 206 . . 3 (𝑋 1𝑅 ∈ V)
1110adantr 480 . 2 ((𝑋 1𝑋𝐸 1 ) → 𝑅 ∈ V)
12 fveq2 6103 . . . . . . . . . 10 (𝑟 = 𝑅 → (∥r𝑟) = (∥r𝑅))
1312, 7syl6eqr 2662 . . . . . . . . 9 (𝑟 = 𝑅 → (∥r𝑟) = )
14 fveq2 6103 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (oppr𝑟) = (oppr𝑅))
15 unit.4 . . . . . . . . . . . 12 𝑆 = (oppr𝑅)
1614, 15syl6eqr 2662 . . . . . . . . . . 11 (𝑟 = 𝑅 → (oppr𝑟) = 𝑆)
1716fveq2d 6107 . . . . . . . . . 10 (𝑟 = 𝑅 → (∥r‘(oppr𝑟)) = (∥r𝑆))
18 unit.5 . . . . . . . . . 10 𝐸 = (∥r𝑆)
1917, 18syl6eqr 2662 . . . . . . . . 9 (𝑟 = 𝑅 → (∥r‘(oppr𝑟)) = 𝐸)
2013, 19ineq12d 3777 . . . . . . . 8 (𝑟 = 𝑅 → ((∥r𝑟) ∩ (∥r‘(oppr𝑟))) = ( 𝐸))
2120cnveqd 5220 . . . . . . 7 (𝑟 = 𝑅((∥r𝑟) ∩ (∥r‘(oppr𝑟))) = ( 𝐸))
22 fveq2 6103 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
23 unit.2 . . . . . . . . 9 1 = (1r𝑅)
2422, 23syl6eqr 2662 . . . . . . . 8 (𝑟 = 𝑅 → (1r𝑟) = 1 )
2524sneqd 4137 . . . . . . 7 (𝑟 = 𝑅 → {(1r𝑟)} = { 1 })
2621, 25imaeq12d 5386 . . . . . 6 (𝑟 = 𝑅 → (((∥r𝑟) ∩ (∥r‘(oppr𝑟))) “ {(1r𝑟)}) = (( 𝐸) “ { 1 }))
27 df-unit 18465 . . . . . 6 Unit = (𝑟 ∈ V ↦ (((∥r𝑟) ∩ (∥r‘(oppr𝑟))) “ {(1r𝑟)}))
28 fvex 6113 . . . . . . . . . 10 (∥r𝑅) ∈ V
297, 28eqeltri 2684 . . . . . . . . 9 ∈ V
3029inex1 4727 . . . . . . . 8 ( 𝐸) ∈ V
3130cnvex 7006 . . . . . . 7 ( 𝐸) ∈ V
3231imaex 6996 . . . . . 6 (( 𝐸) “ { 1 }) ∈ V
3326, 27, 32fvmpt 6191 . . . . 5 (𝑅 ∈ V → (Unit‘𝑅) = (( 𝐸) “ { 1 }))
342, 33syl5eq 2656 . . . 4 (𝑅 ∈ V → 𝑈 = (( 𝐸) “ { 1 }))
3534eleq2d 2673 . . 3 (𝑅 ∈ V → (𝑋𝑈𝑋 ∈ (( 𝐸) “ { 1 })))
36 inss1 3795 . . . . . 6 ( 𝐸) ⊆
377reldvdsr 18467 . . . . . 6 Rel
38 relss 5129 . . . . . 6 (( 𝐸) ⊆ → (Rel → Rel ( 𝐸)))
3936, 37, 38mp2 9 . . . . 5 Rel ( 𝐸)
40 eliniseg2 5424 . . . . 5 (Rel ( 𝐸) → (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ 𝑋( 𝐸) 1 ))
4139, 40ax-mp 5 . . . 4 (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ 𝑋( 𝐸) 1 )
42 brin 4634 . . . 4 (𝑋( 𝐸) 1 ↔ (𝑋 1𝑋𝐸 1 ))
4341, 42bitri 263 . . 3 (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ (𝑋 1𝑋𝐸 1 ))
4435, 43syl6bb 275 . 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ⟨cop 4131   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038   “ cima 5041  Rel wrel 5043  ‘cfv 5804  1rcur 18324  opprcoppr 18445  ∥rcdsr 18461  Unitcui 18462 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-dvdsr 18464  df-unit 18465 This theorem is referenced by:  1unit  18481  unitcl  18482  opprunit  18484  crngunit  18485  unitmulcl  18487  unitgrp  18490  unitnegcl  18504  unitpropd  18520  isdrng2  18580  subrguss  18618  subrgunit  18621  fidomndrng  19128  invrvald  20301  elrhmunit  29151
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