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Theorem ablonncan 28451
 Description: Cancellation law for group division. (nncan 10966 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 1416 . . . 4 ((𝐴𝑋𝐵𝑋) → (𝐴𝑋𝐴𝑋𝐵𝑋))
3 abldiv.1 . . . . 5 𝑋 = ran 𝐺
4 abldiv.3 . . . . 5 𝐷 = ( /𝑔𝐺)
53, 4ablodivdiv 28448 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
62, 5sylan2 595 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
763impb 1112 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
8 ablogrpo 28442 . . . . 5 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
9 eqid 2758 . . . . . 6 (GId‘𝐺) = (GId‘𝐺)
103, 4, 9grpodivid 28437 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
118, 10sylan 583 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
12113adant3 1129 . . 3 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
1312oveq1d 7171 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐴)𝐺𝐵) = ((GId‘𝐺)𝐺𝐵))
143, 9grpolid 28411 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
158, 14sylan 583 . . 3 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
16153adant2 1128 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
177, 13, 163eqtrd 2797 1 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ran crn 5529  ‘cfv 6340  (class class class)co 7156  GrpOpcgr 28384  GIdcgi 28385   /𝑔 cgs 28387  AbelOpcablo 28439 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 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-grpo 28388  df-gid 28389  df-ginv 28390  df-gdiv 28391  df-ablo 28440 This theorem is referenced by:  ablonnncan1  28452
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