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Theorem idlnegcl 36484
Description: An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idlnegcl.1 𝐺 = (1st β€˜π‘…)
idlnegcl.2 𝑁 = (invβ€˜πΊ)
Assertion
Ref Expression
idlnegcl (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐴 ∈ 𝐼) β†’ (π‘β€˜π΄) ∈ 𝐼)

Proof of Theorem idlnegcl
StepHypRef Expression
1 idlnegcl.1 . . . 4 𝐺 = (1st β€˜π‘…)
2 eqid 2737 . . . 4 ran 𝐺 = ran 𝐺
31, 2idlss 36478 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) β†’ 𝐼 βŠ† ran 𝐺)
4 ssel2 3940 . . . . 5 ((𝐼 βŠ† ran 𝐺 ∧ 𝐴 ∈ 𝐼) β†’ 𝐴 ∈ ran 𝐺)
5 eqid 2737 . . . . . 6 (2nd β€˜π‘…) = (2nd β€˜π‘…)
6 idlnegcl.2 . . . . . 6 𝑁 = (invβ€˜πΊ)
7 eqid 2737 . . . . . 6 (GIdβ€˜(2nd β€˜π‘…)) = (GIdβ€˜(2nd β€˜π‘…))
81, 5, 2, 6, 7rngonegmn1l 36403 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺) β†’ (π‘β€˜π΄) = ((π‘β€˜(GIdβ€˜(2nd β€˜π‘…)))(2nd β€˜π‘…)𝐴))
94, 8sylan2 594 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐼 βŠ† ran 𝐺 ∧ 𝐴 ∈ 𝐼)) β†’ (π‘β€˜π΄) = ((π‘β€˜(GIdβ€˜(2nd β€˜π‘…)))(2nd β€˜π‘…)𝐴))
109anassrs 469 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 βŠ† ran 𝐺) ∧ 𝐴 ∈ 𝐼) β†’ (π‘β€˜π΄) = ((π‘β€˜(GIdβ€˜(2nd β€˜π‘…)))(2nd β€˜π‘…)𝐴))
113, 10syldanl 603 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐴 ∈ 𝐼) β†’ (π‘β€˜π΄) = ((π‘β€˜(GIdβ€˜(2nd β€˜π‘…)))(2nd β€˜π‘…)𝐴))
121rneqi 5893 . . . . . 6 ran 𝐺 = ran (1st β€˜π‘…)
1312, 5, 7rngo1cl 36401 . . . . 5 (𝑅 ∈ RingOps β†’ (GIdβ€˜(2nd β€˜π‘…)) ∈ ran 𝐺)
141, 2, 6rngonegcl 36389 . . . . 5 ((𝑅 ∈ RingOps ∧ (GIdβ€˜(2nd β€˜π‘…)) ∈ ran 𝐺) β†’ (π‘β€˜(GIdβ€˜(2nd β€˜π‘…))) ∈ ran 𝐺)
1513, 14mpdan 686 . . . 4 (𝑅 ∈ RingOps β†’ (π‘β€˜(GIdβ€˜(2nd β€˜π‘…))) ∈ ran 𝐺)
1615ad2antrr 725 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐴 ∈ 𝐼) β†’ (π‘β€˜(GIdβ€˜(2nd β€˜π‘…))) ∈ ran 𝐺)
171, 5, 2idllmulcl 36482 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ (𝐴 ∈ 𝐼 ∧ (π‘β€˜(GIdβ€˜(2nd β€˜π‘…))) ∈ ran 𝐺)) β†’ ((π‘β€˜(GIdβ€˜(2nd β€˜π‘…)))(2nd β€˜π‘…)𝐴) ∈ 𝐼)
1817anassrs 469 . . 3 ((((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐴 ∈ 𝐼) ∧ (π‘β€˜(GIdβ€˜(2nd β€˜π‘…))) ∈ ran 𝐺) β†’ ((π‘β€˜(GIdβ€˜(2nd β€˜π‘…)))(2nd β€˜π‘…)𝐴) ∈ 𝐼)
1916, 18mpdan 686 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐴 ∈ 𝐼) β†’ ((π‘β€˜(GIdβ€˜(2nd β€˜π‘…)))(2nd β€˜π‘…)𝐴) ∈ 𝐼)
2011, 19eqeltrd 2838 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) ∧ 𝐴 ∈ 𝐼) β†’ (π‘β€˜π΄) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911  ran crn 5635  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  GIdcgi 29435  invcgn 29436  RingOpscrngo 36356  Idlcidl 36469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-1st 7922  df-2nd 7923  df-grpo 29438  df-gid 29439  df-ginv 29440  df-ablo 29490  df-ass 36305  df-exid 36307  df-mgmOLD 36311  df-sgrOLD 36323  df-mndo 36329  df-rngo 36357  df-idl 36472
This theorem is referenced by:  idlsubcl  36485
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