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Theorem 2idlelb 21264
Description: Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21283. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.)
Hypotheses
Ref Expression
2idlel.i 𝐼 = (LIdeal‘𝑅)
2idlel.o 𝑂 = (oppr𝑅)
2idlel.j 𝐽 = (LIdeal‘𝑂)
2idlel.t 𝑇 = (2Ideal‘𝑅)
Assertion
Ref Expression
2idlelb (𝑈𝑇 ↔ (𝑈𝐼𝑈𝐽))

Proof of Theorem 2idlelb
StepHypRef Expression
1 2idlel.i . . 3 𝐼 = (LIdeal‘𝑅)
2 2idlel.o . . 3 𝑂 = (oppr𝑅)
3 2idlel.j . . 3 𝐽 = (LIdeal‘𝑂)
4 2idlel.t . . 3 𝑇 = (2Ideal‘𝑅)
51, 2, 3, 42idlval 21262 . 2 𝑇 = (𝐼𝐽)
65elin2 4202 1 (𝑈𝑇 ↔ (𝑈𝐼𝑈𝐽))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107  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:  df2idl2rng  21267  2idlelbas  21275  rng2idlsubgsubrng  21279  2idlcpblrng  21282  2idlcpbl  21283
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