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Theorem ablonncan 27752
Description: Cancellation law for group division. (nncan 10513 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 1529 . . . 4 ((𝐴𝑋𝐵𝑋) → (𝐴𝑋𝐴𝑋𝐵𝑋))
3 abldiv.1 . . . . 5 𝑋 = ran 𝐺
4 abldiv.3 . . . . 5 𝐷 = ( /𝑔𝐺)
53, 4ablodivdiv 27748 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
62, 5sylan2 574 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
763impb 1107 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
8 ablogrpo 27742 . . . . 5 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
9 eqid 2771 . . . . . 6 (GId‘𝐺) = (GId‘𝐺)
103, 4, 9grpodivid 27737 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
118, 10sylan 563 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
12113adant3 1126 . . 3 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
1312oveq1d 6809 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐴)𝐺𝐵) = ((GId‘𝐺)𝐺𝐵))
143, 9grpolid 27711 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
158, 14sylan 563 . . 3 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
16153adant2 1125 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
177, 13, 163eqtrd 2809 1 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  ran crn 5251  cfv 6032  (class class class)co 6794  GrpOpcgr 27684  GIdcgi 27685   /𝑔 cgs 27687  AbelOpcablo 27739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-1st 7316  df-2nd 7317  df-grpo 27688  df-gid 27689  df-ginv 27690  df-gdiv 27691  df-ablo 27740
This theorem is referenced by:  ablonnncan1  27753
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