Theorem 2idllidld 21457

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 2760	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
3		eqid 2760	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
4		eqid 2760	. . . 4 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅))
5		eqid 2760	. . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
6	2, 3, 4, 5	2idlval 21454	. . 3 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅)))
7	1, 6	eleqtrdi 2870	. 2 ⊢ (𝜑 → 𝐼 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))))
8	7	elin1d 4150	1 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ∩ cin 3898  ‘cfv 6533  opp_rcoppr 20478  LIdealclidl 21394  2Idealc2idl 21452

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-2idl 21453

This theorem is used by:  df2idl2  21460  2idlss  21465  qusmul2idl  21482  rng2idl1cntr  21509  qsnzr  21547  opprqusmulr  33894  opprqus1r  33895  opprqusdrng  33896  qsdrnglem2  33899  qsdrng  33900  isidom3  49261