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Theorem rngoidl 37530
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 4005 . 2 (𝑅 ∈ RingOps → 𝑋𝑋)
2 rngidl.1 . . 3 𝐺 = (1st𝑅)
3 rngidl.2 . . 3 𝑋 = ran 𝐺
4 eqid 2728 . . 3 (GId‘𝐺) = (GId‘𝐺)
52, 3, 4rngo0cl 37425 . 2 (𝑅 ∈ RingOps → (GId‘𝐺) ∈ 𝑋)
62, 3rngogcl 37418 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
763expa 1115 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
87ralrimiva 3143 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋)
9 eqid 2728 . . . . . . . . 9 (2nd𝑅) = (2nd𝑅)
102, 9, 3rngocl 37407 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧𝑋𝑥𝑋) → (𝑧(2nd𝑅)𝑥) ∈ 𝑋)
11103com23 1123 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → (𝑧(2nd𝑅)𝑥) ∈ 𝑋)
122, 9, 3rngocl 37407 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋𝑧𝑋) → (𝑥(2nd𝑅)𝑧) ∈ 𝑋)
1311, 12jca 510 . . . . . 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 3143 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋))
168, 15jca 510 . . 3 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋)))
1716ralrimiva 3143 . 2 (𝑅 ∈ RingOps → ∀𝑥𝑋 (∀𝑦𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧𝑋 ((𝑧(2nd𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝑋)))
182, 9, 3, 4isidl 37520 . 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 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  wss 3949  ran crn 5683  cfv 6553  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998  GIdcgi 30320  RingOpscrngo 37400  Idlcidl 37513
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-riota 7382  df-ov 7429  df-1st 7999  df-2nd 8000  df-grpo 30323  df-gid 30324  df-ablo 30375  df-rngo 37401  df-idl 37516
This theorem is referenced by:  divrngidl  37534  igenval  37567  igenidl  37569
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