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Theorem idlss 34302
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 2799 . . . 4 (2nd𝑅) = (2nd𝑅)
3 idlss.2 . . . 4 𝑋 = ran 𝐺
4 eqid 2799 . . . 4 (GId‘𝐺) = (GId‘𝐺)
51, 2, 3, 4isidl 34300 . . 3 (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐼)))))
65biimpa 469 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐼))))
76simp1d 1173 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  wss 3769  ran crn 5313  cfv 6101  (class class class)co 6878  1st c1st 7399  2nd c2nd 7400  GIdcgi 27870  RingOpscrngo 34180  Idlcidl 34293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-idl 34296
This theorem is referenced by:  idlcl  34303  idlnegcl  34308  1idl  34312  divrngidl  34314  intidl  34315  unichnidl  34317  ispridl2  34324  igenmin  34350  igenidl2  34351
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