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Theorem idlsubcl 36420
Description: An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
idlsubcl.1 𝐺 = (1st𝑅)
idlsubcl.2 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
idlsubcl (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼)

Proof of Theorem idlsubcl
StepHypRef Expression
1 idlsubcl.1 . . . . 5 𝐺 = (1st𝑅)
2 eqid 2737 . . . . 5 ran 𝐺 = ran 𝐺
31, 2idlcl 36414 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → 𝐴 ∈ ran 𝐺)
41, 2idlcl 36414 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐵𝐼) → 𝐵 ∈ ran 𝐺)
53, 4anim12dan 619 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴 ∈ ran 𝐺𝐵 ∈ ran 𝐺))
6 eqid 2737 . . . . . 6 (inv‘𝐺) = (inv‘𝐺)
7 idlsubcl.2 . . . . . 6 𝐷 = ( /𝑔𝐺)
81, 2, 6, 7rngosub 36327 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺𝐵 ∈ ran 𝐺) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
983expb 1120 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ ran 𝐺𝐵 ∈ ran 𝐺)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
109adantlr 713 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ ran 𝐺𝐵 ∈ ran 𝐺)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
115, 10syldan 591 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
12 simprl 769 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → 𝐴𝐼)
131, 6idlnegcl 36419 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐵𝐼) → ((inv‘𝐺)‘𝐵) ∈ 𝐼)
1413adantrl 714 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → ((inv‘𝐺)‘𝐵) ∈ 𝐼)
1512, 14jca 512 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐼 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝐼))
161idladdcl 36416 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝐼)) → (𝐴𝐺((inv‘𝐺)‘𝐵)) ∈ 𝐼)
1715, 16syldan 591 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐺((inv‘𝐺)‘𝐵)) ∈ 𝐼)
1811, 17eqeltrd 2838 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ran crn 5632  cfv 6493  (class class class)co 7351  1st c1st 7911  invcgn 29262   /𝑔 cgs 29263  RingOpscrngo 36291  Idlcidl 36404
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-grpo 29264  df-gid 29265  df-ginv 29266  df-gdiv 29267  df-ablo 29316  df-ass 36240  df-exid 36242  df-mgmOLD 36246  df-sgrOLD 36258  df-mndo 36264  df-rngo 36292  df-idl 36407
This theorem is referenced by: (None)
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