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Theorem idl0cl 36409
Description: An ideal contains 0. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idl0cl.1 𝐺 = (1st𝑅)
idl0cl.2 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
idl0cl ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑍𝐼)

Proof of Theorem idl0cl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idl0cl.1 . . . 4 𝐺 = (1st𝑅)
2 eqid 2736 . . . 4 (2nd𝑅) = (2nd𝑅)
3 eqid 2736 . . . 4 ran 𝐺 = ran 𝐺
4 idl0cl.2 . . . 4 𝑍 = (GId‘𝐺)
51, 2, 3, 4isidl 36405 . . 3 (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ ran 𝐺𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐼)))))
65biimpa 477 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 ⊆ ran 𝐺𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐼))))
76simp2d 1143 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑍𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3062  wss 3908  ran crn 5632  cfv 6493  (class class class)co 7351  1st c1st 7911  2nd c2nd 7912  GIdcgi 29277  RingOpscrngo 36285  Idlcidl 36398
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 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-idl 36401
This theorem is referenced by:  divrngidl  36419  intidl  36420  unichnidl  36422  maxidln0  36436
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