Theorem 2idllidld 21140

Description: A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)

Hypothesis

Ref	Expression
2idllidld.1	⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))

Assertion

Ref	Expression
2idllidld	⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))

Proof of Theorem 2idllidld

Step	Hyp	Ref	Expression
1		2idllidld.1	. . 3 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
2		eqid 2729	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
3		eqid 2729	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
4		eqid 2729	. . . 4 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅))
5		eqid 2729	. . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
6	2, 3, 4, 5	2idlval 21137	. . 3 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅)))
7	1, 6	eleqtrdi 2838	. 2 ⊢ (𝜑 → 𝐼 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))))
8	7	elin1d 4163	1 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))

Syntax hints:   → wi 4   ∈ wcel 2109   ∩ cin 3910  ‘cfv 6499  opp_rcoppr 20221  LIdealclidl 21092  2Idealc2idl 21135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-2idl 21136

This theorem is referenced by:  df2idl2  21143  2idlss  21148  qusmul2idl  21165  rng2idl1cntr  21191  qsnzr  33399  opprqusmulr  33435  opprqus1r  33436  opprqusdrng  33437  qsdrnglem2  33440  qsdrng  33441