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Theorem rngoidl 34364
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 3848 . 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 34259 . 2 (𝑅 ∈ RingOps → (GId‘𝐺) ∈ 𝑋)
62, 3rngogcl 34252 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
763expa 1153 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
87ralrimiva 3174 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋)
9 eqid 2824 . . . . . . . . 9 (2nd𝑅) = (2nd𝑅)
102, 9, 3rngocl 34241 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧𝑋𝑥𝑋) → (𝑧(2nd𝑅)𝑥) ∈ 𝑋)
11103com23 1162 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → (𝑧(2nd𝑅)𝑥) ∈ 𝑋)
122, 9, 3rngocl 34241 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → (𝑥(2nd𝑅)𝑧) ∈ 𝑋)
1311, 12jca 509 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋))
14133expa 1153 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ 𝑧𝑋) → ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋))
1514ralrimiva 3174 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋))
168, 15jca 509 . . 3 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋)))
1716ralrimiva 3174 . 2 (𝑅 ∈ RingOps → ∀𝑥𝑋 (∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋)))
182, 9, 3, 4isidl 34354 . 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 1448 1 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3116  wss 3797  ran crn 5342  cfv 6122  (class class class)co 6904  1st c1st 7425  2nd c2nd 7426  GIdcgi 27899  RingOpscrngo 34234  Idlcidl 34347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-fo 6128  df-fv 6130  df-riota 6865  df-ov 6907  df-1st 7427  df-2nd 7428  df-grpo 27902  df-gid 27903  df-ablo 27954  df-rngo 34235  df-idl 34350
This theorem is referenced by:  divrngidl  34368  igenval  34401  igenidl  34403
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