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Theorem 2idlridld 20808
Description: A two-sided ideal is a right ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
Hypotheses
Ref Expression
2idllidld.1 (𝜑𝐼 ∈ (2Ideal‘𝑅))
2idlridld.o 𝑂 = (oppr𝑅)
Assertion
Ref Expression
2idlridld (𝜑𝐼 ∈ (LIdeal‘𝑂))

Proof of Theorem 2idlridld
StepHypRef Expression
1 2idllidld.1 . . 3 (𝜑𝐼 ∈ (2Ideal‘𝑅))
2 eqid 2732 . . . 4 (LIdeal‘𝑅) = (LIdeal‘𝑅)
3 2idlridld.o . . . 4 𝑂 = (oppr𝑅)
4 eqid 2732 . . . 4 (LIdeal‘𝑂) = (LIdeal‘𝑂)
5 eqid 2732 . . . 4 (2Ideal‘𝑅) = (2Ideal‘𝑅)
62, 3, 4, 52idlval 20806 . . 3 (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂))
71, 6eleqtrdi 2843 . 2 (𝜑𝐼 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂)))
87elin2d 4196 1 (𝜑𝐼 ∈ (LIdeal‘𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cin 3944  cfv 6533  opprcoppr 20103  LIdealclidl 20734  2Idealc2idl 20804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-iota 6485  df-fun 6535  df-fv 6541  df-2idl 20805
This theorem is referenced by:  qsdrng  32521
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