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Theorem isunit 20187
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 6929 . . . 4 (𝑋 ∈ (Unitβ€˜π‘…) β†’ 𝑅 ∈ dom Unit)
2 unit.1 . . . 4 π‘ˆ = (Unitβ€˜π‘…)
31, 2eleq2s 2852 . . 3 (𝑋 ∈ π‘ˆ β†’ 𝑅 ∈ dom Unit)
43elexd 3495 . 2 (𝑋 ∈ π‘ˆ β†’ 𝑅 ∈ V)
5 df-br 5150 . . . 4 (𝑋 βˆ₯ 1 ↔ βŸ¨π‘‹, 1 ⟩ ∈ βˆ₯ )
6 elfvdm 6929 . . . . . 6 (βŸ¨π‘‹, 1 ⟩ ∈ (βˆ₯rβ€˜π‘…) β†’ 𝑅 ∈ dom βˆ₯r)
7 unit.3 . . . . . 6 βˆ₯ = (βˆ₯rβ€˜π‘…)
86, 7eleq2s 2852 . . . . 5 (βŸ¨π‘‹, 1 ⟩ ∈ βˆ₯ β†’ 𝑅 ∈ dom βˆ₯r)
98elexd 3495 . . . 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 6892 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (βˆ₯rβ€˜π‘Ÿ) = (βˆ₯rβ€˜π‘…))
1312, 7eqtr4di 2791 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (βˆ₯rβ€˜π‘Ÿ) = βˆ₯ )
14 fveq2 6892 . . . . . . . . . . . 12 (π‘Ÿ = 𝑅 β†’ (opprβ€˜π‘Ÿ) = (opprβ€˜π‘…))
15 unit.4 . . . . . . . . . . . 12 𝑆 = (opprβ€˜π‘…)
1614, 15eqtr4di 2791 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (opprβ€˜π‘Ÿ) = 𝑆)
1716fveq2d 6896 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (βˆ₯rβ€˜(opprβ€˜π‘Ÿ)) = (βˆ₯rβ€˜π‘†))
18 unit.5 . . . . . . . . . 10 𝐸 = (βˆ₯rβ€˜π‘†)
1917, 18eqtr4di 2791 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (βˆ₯rβ€˜(opprβ€˜π‘Ÿ)) = 𝐸)
2013, 19ineq12d 4214 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ ((βˆ₯rβ€˜π‘Ÿ) ∩ (βˆ₯rβ€˜(opprβ€˜π‘Ÿ))) = ( βˆ₯ ∩ 𝐸))
2120cnveqd 5876 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ β—‘((βˆ₯rβ€˜π‘Ÿ) ∩ (βˆ₯rβ€˜(opprβ€˜π‘Ÿ))) = β—‘( βˆ₯ ∩ 𝐸))
22 fveq2 6892 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (1rβ€˜π‘Ÿ) = (1rβ€˜π‘…))
23 unit.2 . . . . . . . . 9 1 = (1rβ€˜π‘…)
2422, 23eqtr4di 2791 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (1rβ€˜π‘Ÿ) = 1 )
2524sneqd 4641 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ {(1rβ€˜π‘Ÿ)} = { 1 })
2621, 25imaeq12d 6061 . . . . . 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 20172 . . . . . 6 Unit = (π‘Ÿ ∈ V ↦ (β—‘((βˆ₯rβ€˜π‘Ÿ) ∩ (βˆ₯rβ€˜(opprβ€˜π‘Ÿ))) β€œ {(1rβ€˜π‘Ÿ)}))
287fvexi 6906 . . . . . . . . 9 βˆ₯ ∈ V
2928inex1 5318 . . . . . . . 8 ( βˆ₯ ∩ 𝐸) ∈ V
3029cnvex 7916 . . . . . . 7 β—‘( βˆ₯ ∩ 𝐸) ∈ V
3130imaex 7907 . . . . . 6 (β—‘( βˆ₯ ∩ 𝐸) β€œ { 1 }) ∈ V
3226, 27, 31fvmpt 6999 . . . . 5 (𝑅 ∈ V β†’ (Unitβ€˜π‘…) = (β—‘( βˆ₯ ∩ 𝐸) β€œ { 1 }))
332, 32eqtrid 2785 . . . 4 (𝑅 ∈ V β†’ π‘ˆ = (β—‘( βˆ₯ ∩ 𝐸) β€œ { 1 }))
3433eleq2d 2820 . . 3 (𝑅 ∈ V β†’ (𝑋 ∈ π‘ˆ ↔ 𝑋 ∈ (β—‘( βˆ₯ ∩ 𝐸) β€œ { 1 })))
35 inss1 4229 . . . . . 6 ( βˆ₯ ∩ 𝐸) βŠ† βˆ₯
367reldvdsr 20174 . . . . . 6 Rel βˆ₯
37 relss 5782 . . . . . 6 (( βˆ₯ ∩ 𝐸) βŠ† βˆ₯ β†’ (Rel βˆ₯ β†’ Rel ( βˆ₯ ∩ 𝐸)))
3835, 36, 37mp2 9 . . . . 5 Rel ( βˆ₯ ∩ 𝐸)
39 eliniseg2 6106 . . . . 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 5201 . . . 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 3475   ∩ cin 3948   βŠ† wss 3949  {csn 4629  βŸ¨cop 4635   class class class wbr 5149  β—‘ccnv 5676  dom cdm 5677   β€œ cima 5680  Rel wrel 5682  β€˜cfv 6544  1rcur 20004  opprcoppr 20149  βˆ₯rcdsr 20168  Unitcui 20169
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 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-dvdsr 20171  df-unit 20172
This theorem is referenced by:  1unit  20188  unitcl  20189  opprunit  20191  crngunit  20192  unitmulcl  20194  unitgrp  20197  unitnegcl  20211  unitpropd  20231  elrhmunit  20289  subrguss  20334  subrgunit  20337  isdrng2  20371  fidomndrng  20926  invrvald  22178  isdrng4  32426
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