 Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idlnegcl Structured version   Visualization version   GIF version

Theorem idlnegcl 33450
 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 2621 . . . 4 ran 𝐺 = ran 𝐺
31, 2idlss 33444 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran 𝐺)
4 ssel2 3578 . . . . 5 ((𝐼 ⊆ ran 𝐺𝐴𝐼) → 𝐴 ∈ ran 𝐺)
5 eqid 2621 . . . . . 6 (2nd𝑅) = (2nd𝑅)
6 idlnegcl.2 . . . . . 6 𝑁 = (inv‘𝐺)
7 eqid 2621 . . . . . 6 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
81, 5, 2, 6, 7rngonegmn1l 33369 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
94, 8sylan2 491 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran 𝐺𝐴𝐼)) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
109anassrs 679 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran 𝐺) ∧ 𝐴𝐼) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
113, 10syldanl 734 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
121rneqi 5312 . . . . . 6 ran 𝐺 = ran (1st𝑅)
1312, 5, 7rngo1cl 33367 . . . . 5 (𝑅 ∈ RingOps → (GId‘(2nd𝑅)) ∈ ran 𝐺)
141, 2, 6rngonegcl 33355 . . . . 5 ((𝑅 ∈ RingOps ∧ (GId‘(2nd𝑅)) ∈ ran 𝐺) → (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)
1513, 14mpdan 701 . . . 4 (𝑅 ∈ RingOps → (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)
1615ad2antrr 761 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)
171, 5, 2idllmulcl 33448 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼 ∧ (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)) → ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴) ∈ 𝐼)
1817anassrs 679 . . 3 ((((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) ∧ (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺) → ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴) ∈ 𝐼)
1916, 18mpdan 701 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴) ∈ 𝐼)
2011, 19eqeltrd 2698 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁𝐴) ∈ 𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ⊆ wss 3555  ran crn 5075  ‘cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  GIdcgi 27190  invcgn 27191  RingOpscrngo 33322  Idlcidl 33435 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-1st 7113  df-2nd 7114  df-grpo 27193  df-gid 27194  df-ginv 27195  df-ablo 27245  df-ass 33271  df-exid 33273  df-mgmOLD 33277  df-sgrOLD 33289  df-mndo 33295  df-rngo 33323  df-idl 33438 This theorem is referenced by:  idlsubcl  33451
 Copyright terms: Public domain W3C validator