Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idl0cl Structured version   Visualization version   GIF version

Theorem idl0cl 35799
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 2738 . . . 4 (2nd𝑅) = (2nd𝑅)
3 eqid 2738 . . . 4 ran 𝐺 = ran 𝐺
4 idl0cl.2 . . . 4 𝑍 = (GId‘𝐺)
51, 2, 3, 4isidl 35795 . . 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 480 . 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 1144 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑍𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053  wss 3843  ran crn 5526  cfv 6339  (class class class)co 7170  1st c1st 7712  2nd c2nd 7713  GIdcgi 28425  RingOpscrngo 35675  Idlcidl 35788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173  df-idl 35791
This theorem is referenced by:  divrngidl  35809  intidl  35810  unichnidl  35812  maxidln0  35826
  Copyright terms: Public domain W3C validator