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Theorem idlnegcl 37723
Description: An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idlnegcl.1 𝐺 = (1st𝑅)
idlnegcl.2 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
idlnegcl (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁𝐴) ∈ 𝐼)

Proof of Theorem idlnegcl
StepHypRef Expression
1 idlnegcl.1 . . . 4 𝐺 = (1st𝑅)
2 eqid 2726 . . . 4 ran 𝐺 = ran 𝐺
31, 2idlss 37717 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran 𝐺)
4 ssel2 3974 . . . . 5 ((𝐼 ⊆ ran 𝐺𝐴𝐼) → 𝐴 ∈ ran 𝐺)
5 eqid 2726 . . . . . 6 (2nd𝑅) = (2nd𝑅)
6 idlnegcl.2 . . . . . 6 𝑁 = (inv‘𝐺)
7 eqid 2726 . . . . . 6 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
81, 5, 2, 6, 7rngonegmn1l 37642 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
94, 8sylan2 591 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran 𝐺𝐴𝐼)) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
109anassrs 466 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran 𝐺) ∧ 𝐴𝐼) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
113, 10syldanl 600 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
121rneqi 5943 . . . . . 6 ran 𝐺 = ran (1st𝑅)
1312, 5, 7rngo1cl 37640 . . . . 5 (𝑅 ∈ RingOps → (GId‘(2nd𝑅)) ∈ ran 𝐺)
141, 2, 6rngonegcl 37628 . . . . 5 ((𝑅 ∈ RingOps ∧ (GId‘(2nd𝑅)) ∈ ran 𝐺) → (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)
1513, 14mpdan 685 . . . 4 (𝑅 ∈ RingOps → (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)
1615ad2antrr 724 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)
171, 5, 2idllmulcl 37721 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼 ∧ (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)) → ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴) ∈ 𝐼)
1817anassrs 466 . . 3 ((((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) ∧ (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺) → ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴) ∈ 𝐼)
1916, 18mpdan 685 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴) ∈ 𝐼)
2011, 19eqeltrd 2826 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁𝐴) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wss 3947  ran crn 5683  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  GIdcgi 30423  invcgn 30424  RingOpscrngo 37595  Idlcidl 37708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  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 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-1st 8003  df-2nd 8004  df-grpo 30426  df-gid 30427  df-ginv 30428  df-ablo 30478  df-ass 37544  df-exid 37546  df-mgmOLD 37550  df-sgrOLD 37562  df-mndo 37568  df-rngo 37596  df-idl 37711
This theorem is referenced by:  idlsubcl  37724
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