Theorem 2idllidld 21252

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 2736	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
3		eqid 2736	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
4		eqid 2736	. . . 4 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅))
5		eqid 2736	. . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
6	2, 3, 4, 5	2idlval 21249	. . 3 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅)))
7	1, 6	eleqtrdi 2846	. 2 ⊢ (𝜑 → 𝐼 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))))
8	7	elin1d 4144	1 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))

Syntax hints:   → wi 4   ∈ wcel 2114   ∩ cin 3888  ‘cfv 6498  opp_rcoppr 20316  LIdealclidl 21204  2Idealc2idl 21247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-2idl 21248

This theorem is referenced by:  df2idl2  21255  2idlss  21260  qusmul2idl  21277  rng2idl1cntr  21303  qsnzr  33515  opprqusmulr  33551  opprqus1r  33552  opprqusdrng  33553  qsdrnglem2  33556  qsdrng  33557