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Theorem 2idlelb 21108
Description: Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21127. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.)
Hypotheses
Ref Expression
2idlel.i 𝐼 = (LIdealβ€˜π‘…)
2idlel.o 𝑂 = (opprβ€˜π‘…)
2idlel.j 𝐽 = (LIdealβ€˜π‘‚)
2idlel.t 𝑇 = (2Idealβ€˜π‘…)
Assertion
Ref Expression
2idlelb (π‘ˆ ∈ 𝑇 ↔ (π‘ˆ ∈ 𝐼 ∧ π‘ˆ ∈ 𝐽))

Proof of Theorem 2idlelb
StepHypRef Expression
1 2idlel.i . . 3 𝐼 = (LIdealβ€˜π‘…)
2 2idlel.o . . 3 𝑂 = (opprβ€˜π‘…)
3 2idlel.j . . 3 𝐽 = (LIdealβ€˜π‘‚)
4 2idlel.t . . 3 𝑇 = (2Idealβ€˜π‘…)
51, 2, 3, 42idlval 21106 . 2 𝑇 = (𝐼 ∩ 𝐽)
65elin2 4192 1 (π‘ˆ ∈ 𝑇 ↔ (π‘ˆ ∈ 𝐼 ∧ π‘ˆ ∈ 𝐽))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  β€˜cfv 6536  opprcoppr 20233  LIdealclidl 21063  2Idealc2idl 21104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-2idl 21105
This theorem is referenced by:  df2idl2rng  21111  2idlelbas  21119  rng2idlsubgsubrng  21123  2idlcpblrng  21126  2idlcpbl  21127
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