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Theorem idlss 37489
Description: An ideal of 𝑅 is a subset of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idlss.1 𝐺 = (1st𝑅)
idlss.2 𝑋 = ran 𝐺
Assertion
Ref Expression
idlss ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼𝑋)

Proof of Theorem idlss
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idlss.1 . . . 4 𝐺 = (1st𝑅)
2 eqid 2728 . . . 4 (2nd𝑅) = (2nd𝑅)
3 idlss.2 . . . 4 𝑋 = ran 𝐺
4 eqid 2728 . . . 4 (GId‘𝐺) = (GId‘𝐺)
51, 2, 3, 4isidl 37487 . . 3 (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐼)))))
65biimpa 476 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐼))))
76simp1d 1140 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3058  wss 3947  ran crn 5679  cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992  GIdcgi 30299  RingOpscrngo 37367  Idlcidl 37480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-idl 37483
This theorem is referenced by:  idlcl  37490  idlnegcl  37495  1idl  37499  divrngidl  37501  intidl  37502  unichnidl  37504  ispridl2  37511  igenmin  37537  igenidl2  37538
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