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Theorem idlsubcl 34147
Description: An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
idlsubcl.1 𝐺 = (1st𝑅)
idlsubcl.2 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
idlsubcl (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼)

Proof of Theorem idlsubcl
StepHypRef Expression
1 idlsubcl.1 . . . . 5 𝐺 = (1st𝑅)
2 eqid 2770 . . . . 5 ran 𝐺 = ran 𝐺
31, 2idlcl 34141 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → 𝐴 ∈ ran 𝐺)
41, 2idlcl 34141 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐵𝐼) → 𝐵 ∈ ran 𝐺)
53, 4anim12da 33831 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴 ∈ ran 𝐺𝐵 ∈ ran 𝐺))
6 eqid 2770 . . . . . 6 (inv‘𝐺) = (inv‘𝐺)
7 idlsubcl.2 . . . . . 6 𝐷 = ( /𝑔𝐺)
81, 2, 6, 7rngosub 34054 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺𝐵 ∈ ran 𝐺) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
983expb 1112 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ ran 𝐺𝐵 ∈ ran 𝐺)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
109adantlr 686 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ ran 𝐺𝐵 ∈ ran 𝐺)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
115, 10syldan 571 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
12 simprl 746 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → 𝐴𝐼)
131, 6idlnegcl 34146 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐵𝐼) → ((inv‘𝐺)‘𝐵) ∈ 𝐼)
1413adantrl 687 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → ((inv‘𝐺)‘𝐵) ∈ 𝐼)
1512, 14jca 495 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐼 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝐼))
161idladdcl 34143 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝐼)) → (𝐴𝐺((inv‘𝐺)‘𝐵)) ∈ 𝐼)
1715, 16syldan 571 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐺((inv‘𝐺)‘𝐵)) ∈ 𝐼)
1811, 17eqeltrd 2849 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  ran crn 5250  cfv 6031  (class class class)co 6792  1st c1st 7312  invcgn 27679   /𝑔 cgs 27680  RingOpscrngo 34018  Idlcidl 34131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-grpo 27681  df-gid 27682  df-ginv 27683  df-gdiv 27684  df-ablo 27733  df-ass 33967  df-exid 33969  df-mgmOLD 33973  df-sgrOLD 33985  df-mndo 33991  df-rngo 34019  df-idl 34134
This theorem is referenced by: (None)
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