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Theorem rngoidl 32993
Description: A ring 𝑅 is an 𝑅 ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
rngidl.1 𝐺 = (1st𝑅)
rngidl.2 𝑋 = ran 𝐺
Assertion
Ref Expression
rngoidl (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))

Proof of Theorem rngoidl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3587 . . 3 𝑋𝑋
21a1i 11 . 2 (𝑅 ∈ RingOps → 𝑋𝑋)
3 rngidl.1 . . 3 𝐺 = (1st𝑅)
4 rngidl.2 . . 3 𝑋 = ran 𝐺
5 eqid 2610 . . 3 (GId‘𝐺) = (GId‘𝐺)
63, 4, 5rngo0cl 32888 . 2 (𝑅 ∈ RingOps → (GId‘𝐺) ∈ 𝑋)
73, 4rngogcl 32881 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
873expa 1257 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
98ralrimiva 2949 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋)
10 eqid 2610 . . . . . . . . 9 (2nd𝑅) = (2nd𝑅)
113, 10, 4rngocl 32870 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧𝑋𝑥𝑋) → (𝑧(2nd𝑅)𝑥) ∈ 𝑋)
12113com23 1263 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → (𝑧(2nd𝑅)𝑥) ∈ 𝑋)
133, 10, 4rngocl 32870 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → (𝑥(2nd𝑅)𝑧) ∈ 𝑋)
1412, 13jca 553 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋))
15143expa 1257 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ 𝑧𝑋) → ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋))
1615ralrimiva 2949 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋))
179, 16jca 553 . . 3 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋)))
1817ralrimiva 2949 . 2 (𝑅 ∈ RingOps → ∀𝑥𝑋 (∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋)))
193, 10, 4, 5isidl 32983 . 2 (𝑅 ∈ RingOps → (𝑋 ∈ (Idl‘𝑅) ↔ (𝑋𝑋 ∧ (GId‘𝐺) ∈ 𝑋 ∧ ∀𝑥𝑋 (∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋)))))
202, 6, 18, 19mpbir3and 1238 1 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wss 3540  ran crn 5039  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  GIdcgi 26728  RingOpscrngo 32863  Idlcidl 32976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-riota 6511  df-ov 6552  df-1st 7059  df-2nd 7060  df-grpo 26731  df-gid 26732  df-ablo 26783  df-rngo 32864  df-idl 32979
This theorem is referenced by:  divrngidl  32997  igenval  33030  igenidl  33032
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