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Theorem idlnegcl 35418
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 2822 . . . 4 ran 𝐺 = ran 𝐺
31, 2idlss 35412 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran 𝐺)
4 ssel2 3937 . . . . 5 ((𝐼 ⊆ ran 𝐺𝐴𝐼) → 𝐴 ∈ ran 𝐺)
5 eqid 2822 . . . . . 6 (2nd𝑅) = (2nd𝑅)
6 idlnegcl.2 . . . . . 6 𝑁 = (inv‘𝐺)
7 eqid 2822 . . . . . 6 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
81, 5, 2, 6, 7rngonegmn1l 35337 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
94, 8sylan2 595 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran 𝐺𝐴𝐼)) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
109anassrs 471 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran 𝐺) ∧ 𝐴𝐼) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
113, 10syldanl 604 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
121rneqi 5784 . . . . . 6 ran 𝐺 = ran (1st𝑅)
1312, 5, 7rngo1cl 35335 . . . . 5 (𝑅 ∈ RingOps → (GId‘(2nd𝑅)) ∈ ran 𝐺)
141, 2, 6rngonegcl 35323 . . . . 5 ((𝑅 ∈ RingOps ∧ (GId‘(2nd𝑅)) ∈ ran 𝐺) → (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)
1513, 14mpdan 686 . . . 4 (𝑅 ∈ RingOps → (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)
1615ad2antrr 725 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)
171, 5, 2idllmulcl 35416 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼 ∧ (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)) → ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴) ∈ 𝐼)
1817anassrs 471 . . 3 ((((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) ∧ (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺) → ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴) ∈ 𝐼)
1916, 18mpdan 686 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴) ∈ 𝐼)
2011, 19eqeltrd 2914 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁𝐴) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wss 3908  ran crn 5533  cfv 6334  (class class class)co 7140  1st c1st 7673  2nd c2nd 7674  GIdcgi 28271  invcgn 28272  RingOpscrngo 35290  Idlcidl 35403
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-1st 7675  df-2nd 7676  df-grpo 28274  df-gid 28275  df-ginv 28276  df-ablo 28326  df-ass 35239  df-exid 35241  df-mgmOLD 35245  df-sgrOLD 35257  df-mndo 35263  df-rngo 35291  df-idl 35406
This theorem is referenced by:  idlsubcl  35419
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