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Theorem 2idllidld 21265
Description: A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
Hypothesis
Ref Expression
2idllidld.1 (𝜑𝐼 ∈ (2Ideal‘𝑅))
Assertion
Ref Expression
2idllidld (𝜑𝐼 ∈ (LIdeal‘𝑅))

Proof of Theorem 2idllidld
StepHypRef Expression
1 2idllidld.1 . . 3 (𝜑𝐼 ∈ (2Ideal‘𝑅))
2 eqid 2736 . . . 4 (LIdeal‘𝑅) = (LIdeal‘𝑅)
3 eqid 2736 . . . 4 (oppr𝑅) = (oppr𝑅)
4 eqid 2736 . . . 4 (LIdeal‘(oppr𝑅)) = (LIdeal‘(oppr𝑅))
5 eqid 2736 . . . 4 (2Ideal‘𝑅) = (2Ideal‘𝑅)
62, 3, 4, 52idlval 21262 . . 3 (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr𝑅)))
71, 6eleqtrdi 2850 . 2 (𝜑𝐼 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr𝑅))))
87elin1d 4203 1 (𝜑𝐼 ∈ (LIdeal‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cin 3949  cfv 6560  opprcoppr 20334  LIdealclidl 21217  2Idealc2idl 21260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-2idl 21261
This theorem is referenced by:  df2idl2  21268  2idlss  21273  qusmul2idl  21290  rng2idl1cntr  21316  qsnzr  33484  opprqusmulr  33520  opprqus1r  33521  opprqusdrng  33522  qsdrnglem2  33525  qsdrng  33526
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