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Theorem idlsubcl 35458
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 2798 . . . . 5 ran 𝐺 = ran 𝐺
31, 2idlcl 35452 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → 𝐴 ∈ ran 𝐺)
41, 2idlcl 35452 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐵𝐼) → 𝐵 ∈ ran 𝐺)
53, 4anim12dan 621 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴 ∈ ran 𝐺𝐵 ∈ ran 𝐺))
6 eqid 2798 . . . . . 6 (inv‘𝐺) = (inv‘𝐺)
7 idlsubcl.2 . . . . . 6 𝐷 = ( /𝑔𝐺)
81, 2, 6, 7rngosub 35365 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺𝐵 ∈ ran 𝐺) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
983expb 1117 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ ran 𝐺𝐵 ∈ ran 𝐺)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
109adantlr 714 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ ran 𝐺𝐵 ∈ ran 𝐺)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
115, 10syldan 594 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
12 simprl 770 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → 𝐴𝐼)
131, 6idlnegcl 35457 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐵𝐼) → ((inv‘𝐺)‘𝐵) ∈ 𝐼)
1413adantrl 715 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → ((inv‘𝐺)‘𝐵) ∈ 𝐼)
1512, 14jca 515 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐼 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝐼))
161idladdcl 35454 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝐼)) → (𝐴𝐺((inv‘𝐺)‘𝐵)) ∈ 𝐼)
1715, 16syldan 594 . 2 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐺((inv‘𝐺)‘𝐵)) ∈ 𝐼)
1811, 17eqeltrd 2890 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  ran crn 5520  cfv 6324  (class class class)co 7135  1st c1st 7669  invcgn 28274   /𝑔 cgs 28275  RingOpscrngo 35329  Idlcidl 35442
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-grpo 28276  df-gid 28277  df-ginv 28278  df-gdiv 28279  df-ablo 28328  df-ass 35278  df-exid 35280  df-mgmOLD 35284  df-sgrOLD 35296  df-mndo 35302  df-rngo 35330  df-idl 35445
This theorem is referenced by: (None)
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