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Theorem idlnegcl 34363
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 2825 . . . 4 ran 𝐺 = ran 𝐺
31, 2idlss 34357 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran 𝐺)
4 ssel2 3822 . . . . 5 ((𝐼 ⊆ ran 𝐺𝐴𝐼) → 𝐴 ∈ ran 𝐺)
5 eqid 2825 . . . . . 6 (2nd𝑅) = (2nd𝑅)
6 idlnegcl.2 . . . . . 6 𝑁 = (inv‘𝐺)
7 eqid 2825 . . . . . 6 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
81, 5, 2, 6, 7rngonegmn1l 34282 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
94, 8sylan2 588 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran 𝐺𝐴𝐼)) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
109anassrs 461 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran 𝐺) ∧ 𝐴𝐼) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
113, 10syldanl 597 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁𝐴) = ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴))
121rneqi 5584 . . . . . 6 ran 𝐺 = ran (1st𝑅)
1312, 5, 7rngo1cl 34280 . . . . 5 (𝑅 ∈ RingOps → (GId‘(2nd𝑅)) ∈ ran 𝐺)
141, 2, 6rngonegcl 34268 . . . . 5 ((𝑅 ∈ RingOps ∧ (GId‘(2nd𝑅)) ∈ ran 𝐺) → (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)
1513, 14mpdan 680 . . . 4 (𝑅 ∈ RingOps → (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)
1615ad2antrr 719 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)
171, 5, 2idllmulcl 34361 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼 ∧ (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺)) → ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴) ∈ 𝐼)
1817anassrs 461 . . 3 ((((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) ∧ (𝑁‘(GId‘(2nd𝑅))) ∈ ran 𝐺) → ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴) ∈ 𝐼)
1916, 18mpdan 680 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → ((𝑁‘(GId‘(2nd𝑅)))(2nd𝑅)𝐴) ∈ 𝐼)
2011, 19eqeltrd 2906 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁𝐴) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  wss 3798  ran crn 5343  cfv 6123  (class class class)co 6905  1st c1st 7426  2nd c2nd 7427  GIdcgi 27900  invcgn 27901  RingOpscrngo 34235  Idlcidl 34348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-1st 7428  df-2nd 7429  df-grpo 27903  df-gid 27904  df-ginv 27905  df-ablo 27955  df-ass 34184  df-exid 34186  df-mgmOLD 34190  df-sgrOLD 34202  df-mndo 34208  df-rngo 34236  df-idl 34351
This theorem is referenced by:  idlsubcl  34364
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