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Theorem inidl 36835
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 4983 . . 3 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} = (𝐼𝐽))
213adant1 1131 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} = (𝐼𝐽))
3 prnzg 4780 . . . . . 6 (𝐼 ∈ (Idl‘𝑅) → {𝐼, 𝐽} ≠ ∅)
43adantr 482 . . . . 5 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} ≠ ∅)
5 prssi 4822 . . . . 5 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} ⊆ (Idl‘𝑅))
64, 5jca 513 . . . 4 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → ({𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅)))
7 intidl 36834 . . . . 5 ((𝑅 ∈ RingOps ∧ {𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅)) → {𝐼, 𝐽} ∈ (Idl‘𝑅))
873expb 1121 . . . 4 ((𝑅 ∈ RingOps ∧ ({𝐼, 𝐽} ≠ ∅ ∧ {𝐼, 𝐽} ⊆ (Idl‘𝑅))) → {𝐼, 𝐽} ∈ (Idl‘𝑅))
96, 8sylan2 594 . . 3 ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅))) → {𝐼, 𝐽} ∈ (Idl‘𝑅))
1093impb 1116 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → {𝐼, 𝐽} ∈ (Idl‘𝑅))
112, 10eqeltrrd 2835 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼𝐽) ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  cin 3945  wss 3946  c0 4320  {cpr 4628   cint 4948  cfv 6539  RingOpscrngo 36699  Idlcidl 36812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-int 4949  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6491  df-fun 6541  df-fv 6547  df-ov 7406  df-idl 36815
This theorem is referenced by: (None)
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