Theorem inidl 38767

Description: Obsolete theorem, use inlidl 33836 instead. The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
inidl	⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (Idl‘𝑅))

Proof of Theorem inidl

Step	Hyp	Ref	Expression
1		intprg 4944	. . 3 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ∩ {𝐼, 𝐽} = (𝐼 ∩ 𝐽))
2	1	3adant1 1148	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ∩ {𝐼, 𝐽} = (𝐼 ∩ 𝐽))
3		prnzg 4742	. . . . . 6 ⊢ (𝐼 ∈ (Idl‘𝑅) → {𝐼, 𝐽} ≠ ∅)
4	3	adantr 486	. . . . 5 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} ≠ ∅)
5		prssi 4785	. . . . 5 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} ⊆ (Idl‘𝑅))
6	4, 5	jca 521	. . . 4 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ({𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅)))
7		intidl 38766	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅)) → ∩ {𝐼, 𝐽} ∈ (Idl‘𝑅))
8	7	3expb 1138	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ({𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅))) → ∩ {𝐼, 𝐽} ∈ (Idl‘𝑅))
9	6, 8	sylan2 605	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅))) → ∩ {𝐼, 𝐽} ∈ (Idl‘𝑅))
10	9	3impb 1132	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ∩ {𝐼, 𝐽} ∈ (Idl‘𝑅))
11	2, 10	eqeltrrd 2863	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (Idl‘𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  {cpr 4589  ∩ cint 4910  ‘cfv 6537  RingOpscrngo 38631  Idlcidl 38744

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7419  df-idl 38747

This theorem is used by: (None)