Theorem inidl 38625

Description: Obsolete theorem, use inlidl 33695 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 4945	. . 3 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ∩ {𝐼, 𝐽} = (𝐼 ∩ 𝐽))
2	1	3adant1 1146	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ∩ {𝐼, 𝐽} = (𝐼 ∩ 𝐽))
3		prnzg 4743	. . . . . 6 ⊢ (𝐼 ∈ (Idl‘𝑅) → {𝐼, 𝐽} ≠ ∅)
4	3	adantr 485	. . . . 5 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} ≠ ∅)
5		prssi 4786	. . . . 5 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} ⊆ (Idl‘𝑅))
6	4, 5	jca 520	. . . 4 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ({𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅)))
7		intidl 38624	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅)) → ∩ {𝐼, 𝐽} ∈ (Idl‘𝑅))
8	7	3expb 1136	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ({𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅))) → ∩ {𝐼, 𝐽} ∈ (Idl‘𝑅))
9	6, 8	sylan2 604	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅))) → ∩ {𝐼, 𝐽} ∈ (Idl‘𝑅))
10	9	3impb 1130	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ∩ {𝐼, 𝐽} ∈ (Idl‘𝑅))
11	2, 10	eqeltrrd 2862	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (Idl‘𝑅))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ≠ wne 2956   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {cpr 4590  ∩ cint 4911  ‘cfv 6536  RingOpscrngo 38489  Idlcidl 38602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-idl 38605

This theorem is referenced by: (None)