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Theorem rngoidl 37404
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 4000 . 2 (𝑅 ∈ RingOps → 𝑋𝑋)
2 rngidl.1 . . 3 𝐺 = (1st𝑅)
3 rngidl.2 . . 3 𝑋 = ran 𝐺
4 eqid 2726 . . 3 (GId‘𝐺) = (GId‘𝐺)
52, 3, 4rngo0cl 37299 . 2 (𝑅 ∈ RingOps → (GId‘𝐺) ∈ 𝑋)
62, 3rngogcl 37292 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
763expa 1115 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
87ralrimiva 3140 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋)
9 eqid 2726 . . . . . . . . 9 (2nd𝑅) = (2nd𝑅)
102, 9, 3rngocl 37281 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧𝑋𝑥𝑋) → (𝑧(2nd𝑅)𝑥) ∈ 𝑋)
11103com23 1123 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → (𝑧(2nd𝑅)𝑥) ∈ 𝑋)
122, 9, 3rngocl 37281 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → (𝑥(2nd𝑅)𝑧) ∈ 𝑋)
1311, 12jca 511 . . . . . 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 3140 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋))
168, 15jca 511 . . 3 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋)))
1716ralrimiva 3140 . 2 (𝑅 ∈ RingOps → ∀𝑥𝑋 (∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋)))
182, 9, 3, 4isidl 37394 . 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 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  wss 3943  ran crn 5670  cfv 6536  (class class class)co 7404  1st c1st 7969  2nd c2nd 7970  GIdcgi 30247  RingOpscrngo 37274  Idlcidl 37387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-riota 7360  df-ov 7407  df-1st 7971  df-2nd 7972  df-grpo 30250  df-gid 30251  df-ablo 30302  df-rngo 37275  df-idl 37390
This theorem is referenced by:  divrngidl  37408  igenval  37441  igenidl  37443
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