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Theorem 2idlelb 21152
Description: Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21171. (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 21150 . 2 𝑇 = (𝐼 ∩ 𝐽)
65elin2 4197 1 (π‘ˆ ∈ 𝑇 ↔ (π‘ˆ ∈ 𝐼 ∧ π‘ˆ ∈ 𝐽))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  β€˜cfv 6551  opprcoppr 20277  LIdealclidl 21107  2Idealc2idl 21148
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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-2idl 21149
This theorem is referenced by:  df2idl2rng  21155  2idlelbas  21163  rng2idlsubgsubrng  21167  2idlcpblrng  21170  2idlcpbl  21171
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