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 Description: An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypothesis
Ref Expression
Assertion
Ref Expression
idladdcl (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐺𝐵) ∈ 𝐼)

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idladdcl.1 . . . . . 6 𝐺 = (1st𝑅)
2 eqid 2770 . . . . . 6 (2nd𝑅) = (2nd𝑅)
3 eqid 2770 . . . . . 6 ran 𝐺 = ran 𝐺
4 eqid 2770 . . . . . 6 (GId‘𝐺) = (GId‘𝐺)
51, 2, 3, 4isidl 34138 . . . . 5 (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ ran 𝐺 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐼)))))
65biimpa 462 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 ⊆ ran 𝐺 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐼))))
76simp3d 1137 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐼)))
8 simpl 468 . . . 4 ((∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐼)) → ∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼)
98ralimi 3100 . . 3 (∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐼)) → ∀𝑥𝐼𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼)
107, 9syl 17 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥𝐼𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼)
11 oveq1 6799 . . . 4 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
1211eleq1d 2834 . . 3 (𝑥 = 𝐴 → ((𝑥𝐺𝑦) ∈ 𝐼 ↔ (𝐴𝐺𝑦) ∈ 𝐼))
13 oveq2 6800 . . . 4 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
1413eleq1d 2834 . . 3 (𝑦 = 𝐵 → ((𝐴𝐺𝑦) ∈ 𝐼 ↔ (𝐴𝐺𝐵) ∈ 𝐼))
1512, 14rspc2v 3470 . 2 ((𝐴𝐼𝐵𝐼) → (∀𝑥𝐼𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 → (𝐴𝐺𝐵) ∈ 𝐼))
1610, 15mpan9 490 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐺𝐵) ∈ 𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144  ∀wral 3060   ⊆ wss 3721  ran crn 5250  ‘cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  GIdcgi 27678  RingOpscrngo 34018  Idlcidl 34131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-idl 34134 This theorem is referenced by:  idlsubcl  34147  intidl  34153  unichnidl  34155
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