Theorem 2idllidld 21598

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 2763	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
3		eqid 2763	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
4		eqid 2763	. . . 4 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅))
5		eqid 2763	. . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
6	2, 3, 4, 5	2idlval 21595	. . 3 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅)))
7	1, 6	eleqtrdi 2874	. 2 ⊢ (𝜑 → 𝐼 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))))
8	7	elin1d 4104	1 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ∩ cin 4092  ‘cfv 6566  opp_rcoppr 20617  LIdealclidl 21535  2Idealc2idl 21593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-2idl 21594

This theorem is used by:  2idl1el  21600  df2idl2  21602  2idlss  21607  qusmul2idl  21625  rng2idl1cntr  21652  qsnzr  21690  rsp2idlid  34035  opprqusmulr  34119  opprqus1r  34120  opprqusdrng  34121  qsdrnglem2  34124  qsdrng  34125  isidom3  49773