MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isunit Structured version   Visualization version   GIF version

Theorem isunit 20087
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 π‘ˆ = (Unitβ€˜π‘…)
unit.2 1 = (1rβ€˜π‘…)
unit.3 βˆ₯ = (βˆ₯rβ€˜π‘…)
unit.4 𝑆 = (opprβ€˜π‘…)
unit.5 𝐸 = (βˆ₯rβ€˜π‘†)
Assertion
Ref Expression
isunit (𝑋 ∈ π‘ˆ ↔ (𝑋 βˆ₯ 1 ∧ 𝑋𝐸 1 ))

Proof of Theorem isunit
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6880 . . . 4 (𝑋 ∈ (Unitβ€˜π‘…) β†’ 𝑅 ∈ dom Unit)
2 unit.1 . . . 4 π‘ˆ = (Unitβ€˜π‘…)
31, 2eleq2s 2856 . . 3 (𝑋 ∈ π‘ˆ β†’ 𝑅 ∈ dom Unit)
43elexd 3466 . 2 (𝑋 ∈ π‘ˆ β†’ 𝑅 ∈ V)
5 df-br 5107 . . . 4 (𝑋 βˆ₯ 1 ↔ βŸ¨π‘‹, 1 ⟩ ∈ βˆ₯ )
6 elfvdm 6880 . . . . . 6 (βŸ¨π‘‹, 1 ⟩ ∈ (βˆ₯rβ€˜π‘…) β†’ 𝑅 ∈ dom βˆ₯r)
7 unit.3 . . . . . 6 βˆ₯ = (βˆ₯rβ€˜π‘…)
86, 7eleq2s 2856 . . . . 5 (βŸ¨π‘‹, 1 ⟩ ∈ βˆ₯ β†’ 𝑅 ∈ dom βˆ₯r)
98elexd 3466 . . . 4 (βŸ¨π‘‹, 1 ⟩ ∈ βˆ₯ β†’ 𝑅 ∈ V)
105, 9sylbi 216 . . 3 (𝑋 βˆ₯ 1 β†’ 𝑅 ∈ V)
1110adantr 482 . 2 ((𝑋 βˆ₯ 1 ∧ 𝑋𝐸 1 ) β†’ 𝑅 ∈ V)
12 fveq2 6843 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (βˆ₯rβ€˜π‘Ÿ) = (βˆ₯rβ€˜π‘…))
1312, 7eqtr4di 2795 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (βˆ₯rβ€˜π‘Ÿ) = βˆ₯ )
14 fveq2 6843 . . . . . . . . . . . 12 (π‘Ÿ = 𝑅 β†’ (opprβ€˜π‘Ÿ) = (opprβ€˜π‘…))
15 unit.4 . . . . . . . . . . . 12 𝑆 = (opprβ€˜π‘…)
1614, 15eqtr4di 2795 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (opprβ€˜π‘Ÿ) = 𝑆)
1716fveq2d 6847 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (βˆ₯rβ€˜(opprβ€˜π‘Ÿ)) = (βˆ₯rβ€˜π‘†))
18 unit.5 . . . . . . . . . 10 𝐸 = (βˆ₯rβ€˜π‘†)
1917, 18eqtr4di 2795 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (βˆ₯rβ€˜(opprβ€˜π‘Ÿ)) = 𝐸)
2013, 19ineq12d 4174 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ ((βˆ₯rβ€˜π‘Ÿ) ∩ (βˆ₯rβ€˜(opprβ€˜π‘Ÿ))) = ( βˆ₯ ∩ 𝐸))
2120cnveqd 5832 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ β—‘((βˆ₯rβ€˜π‘Ÿ) ∩ (βˆ₯rβ€˜(opprβ€˜π‘Ÿ))) = β—‘( βˆ₯ ∩ 𝐸))
22 fveq2 6843 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (1rβ€˜π‘Ÿ) = (1rβ€˜π‘…))
23 unit.2 . . . . . . . . 9 1 = (1rβ€˜π‘…)
2422, 23eqtr4di 2795 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (1rβ€˜π‘Ÿ) = 1 )
2524sneqd 4599 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ {(1rβ€˜π‘Ÿ)} = { 1 })
2621, 25imaeq12d 6015 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (β—‘((βˆ₯rβ€˜π‘Ÿ) ∩ (βˆ₯rβ€˜(opprβ€˜π‘Ÿ))) β€œ {(1rβ€˜π‘Ÿ)}) = (β—‘( βˆ₯ ∩ 𝐸) β€œ { 1 }))
27 df-unit 20072 . . . . . 6 Unit = (π‘Ÿ ∈ V ↦ (β—‘((βˆ₯rβ€˜π‘Ÿ) ∩ (βˆ₯rβ€˜(opprβ€˜π‘Ÿ))) β€œ {(1rβ€˜π‘Ÿ)}))
287fvexi 6857 . . . . . . . . 9 βˆ₯ ∈ V
2928inex1 5275 . . . . . . . 8 ( βˆ₯ ∩ 𝐸) ∈ V
3029cnvex 7863 . . . . . . 7 β—‘( βˆ₯ ∩ 𝐸) ∈ V
3130imaex 7854 . . . . . 6 (β—‘( βˆ₯ ∩ 𝐸) β€œ { 1 }) ∈ V
3226, 27, 31fvmpt 6949 . . . . 5 (𝑅 ∈ V β†’ (Unitβ€˜π‘…) = (β—‘( βˆ₯ ∩ 𝐸) β€œ { 1 }))
332, 32eqtrid 2789 . . . 4 (𝑅 ∈ V β†’ π‘ˆ = (β—‘( βˆ₯ ∩ 𝐸) β€œ { 1 }))
3433eleq2d 2824 . . 3 (𝑅 ∈ V β†’ (𝑋 ∈ π‘ˆ ↔ 𝑋 ∈ (β—‘( βˆ₯ ∩ 𝐸) β€œ { 1 })))
35 inss1 4189 . . . . . 6 ( βˆ₯ ∩ 𝐸) βŠ† βˆ₯
367reldvdsr 20074 . . . . . 6 Rel βˆ₯
37 relss 5738 . . . . . 6 (( βˆ₯ ∩ 𝐸) βŠ† βˆ₯ β†’ (Rel βˆ₯ β†’ Rel ( βˆ₯ ∩ 𝐸)))
3835, 36, 37mp2 9 . . . . 5 Rel ( βˆ₯ ∩ 𝐸)
39 eliniseg2 6059 . . . . 5 (Rel ( βˆ₯ ∩ 𝐸) β†’ (𝑋 ∈ (β—‘( βˆ₯ ∩ 𝐸) β€œ { 1 }) ↔ 𝑋( βˆ₯ ∩ 𝐸) 1 ))
4038, 39ax-mp 5 . . . 4 (𝑋 ∈ (β—‘( βˆ₯ ∩ 𝐸) β€œ { 1 }) ↔ 𝑋( βˆ₯ ∩ 𝐸) 1 )
41 brin 5158 . . . 4 (𝑋( βˆ₯ ∩ 𝐸) 1 ↔ (𝑋 βˆ₯ 1 ∧ 𝑋𝐸 1 ))
4240, 41bitri 275 . . 3 (𝑋 ∈ (β—‘( βˆ₯ ∩ 𝐸) β€œ { 1 }) ↔ (𝑋 βˆ₯ 1 ∧ 𝑋𝐸 1 ))
4334, 42bitrdi 287 . 2 (𝑅 ∈ V β†’ (𝑋 ∈ π‘ˆ ↔ (𝑋 βˆ₯ 1 ∧ 𝑋𝐸 1 )))
444, 11, 43pm5.21nii 380 1 (𝑋 ∈ π‘ˆ ↔ (𝑋 βˆ₯ 1 ∧ 𝑋𝐸 1 ))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3446   ∩ cin 3910   βŠ† wss 3911  {csn 4587  βŸ¨cop 4593   class class class wbr 5106  β—‘ccnv 5633  dom cdm 5634   β€œ cima 5637  Rel wrel 5639  β€˜cfv 6497  1rcur 19914  opprcoppr 20049  βˆ₯rcdsr 20068  Unitcui 20069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-dvdsr 20071  df-unit 20072
This theorem is referenced by:  1unit  20088  unitcl  20089  opprunit  20091  crngunit  20092  unitmulcl  20094  unitgrp  20097  unitnegcl  20111  unitpropd  20127  elrhmunit  20184  isdrng2  20199  subrguss  20240  subrgunit  20243  fidomndrng  20781  invrvald  22028
  Copyright terms: Public domain W3C validator