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Theorem inidl 36886
Description: The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
inidl ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼𝐽) ∈ (Idl‘𝑅))

Proof of Theorem inidl
StepHypRef Expression
1 intprg 4984 . . 3 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} = (𝐼𝐽))
213adant1 1130 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} = (𝐼𝐽))
3 prnzg 4781 . . . . . 6 (𝐼 ∈ (Idl‘𝑅) → {𝐼, 𝐽} ≠ ∅)
43adantr 481 . . . . 5 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} ≠ ∅)
5 prssi 4823 . . . . 5 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} ⊆ (Idl‘𝑅))
64, 5jca 512 . . . 4 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ({𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅)))
7 intidl 36885 . . . . 5 ((𝑅 ∈ RingOps ∧ {𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅)) → {𝐼, 𝐽} ∈ (Idl‘𝑅))
873expb 1120 . . . 4 ((𝑅 ∈ RingOps ∧ ({𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅))) → {𝐼, 𝐽} ∈ (Idl‘𝑅))
96, 8sylan2 593 . . 3 ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅))) → {𝐼, 𝐽} ∈ (Idl‘𝑅))
1093impb 1115 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} ∈ (Idl‘𝑅))
112, 10eqeltrrd 2834 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼𝐽) ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  cin 3946  wss 3947  c0 4321  {cpr 4629   cint 4949  cfv 6540  RingOpscrngo 36750  Idlcidl 36863
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-idl 36866
This theorem is referenced by: (None)
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