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Theorem rngoidl 35407
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 ssidd 3976 . 2 (𝑅 ∈ RingOps → 𝑋𝑋)
2 rngidl.1 . . 3 𝐺 = (1st𝑅)
3 rngidl.2 . . 3 𝑋 = ran 𝐺
4 eqid 2824 . . 3 (GId‘𝐺) = (GId‘𝐺)
52, 3, 4rngo0cl 35302 . 2 (𝑅 ∈ RingOps → (GId‘𝐺) ∈ 𝑋)
62, 3rngogcl 35295 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
763expa 1115 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
87ralrimiva 3177 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋)
9 eqid 2824 . . . . . . . . 9 (2nd𝑅) = (2nd𝑅)
102, 9, 3rngocl 35284 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧𝑋𝑥𝑋) → (𝑧(2nd𝑅)𝑥) ∈ 𝑋)
11103com23 1123 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → (𝑧(2nd𝑅)𝑥) ∈ 𝑋)
122, 9, 3rngocl 35284 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → (𝑥(2nd𝑅)𝑧) ∈ 𝑋)
1311, 12jca 515 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋))
14133expa 1115 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ 𝑧𝑋) → ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋))
1514ralrimiva 3177 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋))
168, 15jca 515 . . 3 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋)))
1716ralrimiva 3177 . 2 (𝑅 ∈ RingOps → ∀𝑥𝑋 (∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋)))
182, 9, 3, 4isidl 35397 . 2 (𝑅 ∈ RingOps → (𝑋 ∈ (Idl‘𝑅) ↔ (𝑋𝑋 ∧ (GId‘𝐺) ∈ 𝑋 ∧ ∀𝑥𝑋 (∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋)))))
191, 5, 17, 18mpbir3and 1339 1 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  wss 3919  ran crn 5543  cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  GIdcgi 28276  RingOpscrngo 35277  Idlcidl 35390
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fo 6349  df-fv 6351  df-riota 7107  df-ov 7152  df-1st 7684  df-2nd 7685  df-grpo 28279  df-gid 28280  df-ablo 28331  df-rngo 35278  df-idl 35393
This theorem is referenced by:  divrngidl  35411  igenval  35444  igenidl  35446
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