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Theorem ablonncan 27539
Description: Cancellation law for group division. (nncan 10348 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
abldiv.1 𝑋 = ran 𝐺
abldiv.3 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
ablonncan ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = 𝐵)

Proof of Theorem ablonncan
StepHypRef Expression
1 id 22 . . . . 5 ((𝐴𝑋𝐴𝑋𝐵𝑋) → (𝐴𝑋𝐴𝑋𝐵𝑋))
213anidm12 1423 . . . 4 ((𝐴𝑋𝐵𝑋) → (𝐴𝑋𝐴𝑋𝐵𝑋))
3 abldiv.1 . . . . 5 𝑋 = ran 𝐺
4 abldiv.3 . . . . 5 𝐷 = ( /𝑔𝐺)
53, 4ablodivdiv 27535 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
62, 5sylan2 490 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
763impb 1279 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
8 ablogrpo 27529 . . . . 5 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
9 eqid 2651 . . . . . 6 (GId‘𝐺) = (GId‘𝐺)
103, 4, 9grpodivid 27524 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
118, 10sylan 487 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
12113adant3 1101 . . 3 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
1312oveq1d 6705 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐴)𝐺𝐵) = ((GId‘𝐺)𝐺𝐵))
143, 9grpolid 27498 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
158, 14sylan 487 . . 3 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
16153adant2 1100 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
177, 13, 163eqtrd 2689 1 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  ran crn 5144  cfv 5926  (class class class)co 6690  GrpOpcgr 27471  GIdcgi 27472   /𝑔 cgs 27474  AbelOpcablo 27526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-grpo 27475  df-gid 27476  df-ginv 27477  df-gdiv 27478  df-ablo 27527
This theorem is referenced by:  ablonnncan1  27540
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