Theorem 2idllidld 21207

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 2734	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
3		eqid 2734	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
4		eqid 2734	. . . 4 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅))
5		eqid 2734	. . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
6	2, 3, 4, 5	2idlval 21204	. . 3 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅)))
7	1, 6	eleqtrdi 2844	. 2 ⊢ (𝜑 → 𝐼 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))))
8	7	elin1d 4154	1 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))

Syntax hints:   → wi 4   ∈ wcel 2113   ∩ cin 3898  ‘cfv 6490  opp_rcoppr 20270  LIdealclidl 21159  2Idealc2idl 21202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-2idl 21203

This theorem is referenced by:  df2idl2  21210  2idlss  21215  qusmul2idl  21232  rng2idl1cntr  21258  qsnzr  33485  opprqusmulr  33521  opprqus1r  33522  opprqusdrng  33523  qsdrnglem2  33526  qsdrng  33527