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Theorem ablonnncan 27277
Description: Cancellation law for group division. (nnncan 10267 analog.) (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
abldiv.1 𝑋 = ran 𝐺
abldiv.3 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
ablonnncan ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷(𝐵𝐷𝐶))𝐷𝐶) = (𝐴𝐷𝐵))

Proof of Theorem ablonnncan
StepHypRef Expression
1 simpr1 1065 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑋)
2 ablogrpo 27268 . . . . . 6 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
3 abldiv.1 . . . . . . 7 𝑋 = ran 𝐺
4 abldiv.3 . . . . . . 7 𝐷 = ( /𝑔𝐺)
53, 4grpodivcl 27260 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐷𝐶) ∈ 𝑋)
62, 5syl3an1 1356 . . . . 5 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐷𝐶) ∈ 𝑋)
763adant3r1 1271 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐷𝐶) ∈ 𝑋)
8 simpr3 1067 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐶𝑋)
91, 7, 83jca 1240 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑋 ∧ (𝐵𝐷𝐶) ∈ 𝑋𝐶𝑋))
103, 4ablodivdiv4 27275 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋 ∧ (𝐵𝐷𝐶) ∈ 𝑋𝐶𝑋)) → ((𝐴𝐷(𝐵𝐷𝐶))𝐷𝐶) = (𝐴𝐷((𝐵𝐷𝐶)𝐺𝐶)))
119, 10syldan 487 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷(𝐵𝐷𝐶))𝐷𝐶) = (𝐴𝐷((𝐵𝐷𝐶)𝐺𝐶)))
123, 4grponpcan 27264 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → ((𝐵𝐷𝐶)𝐺𝐶) = 𝐵)
132, 12syl3an1 1356 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋𝐶𝑋) → ((𝐵𝐷𝐶)𝐺𝐶) = 𝐵)
14133adant3r1 1271 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐵𝐷𝐶)𝐺𝐶) = 𝐵)
1514oveq2d 6626 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷((𝐵𝐷𝐶)𝐺𝐶)) = (𝐴𝐷𝐵))
1611, 15eqtrd 2655 1 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷(𝐵𝐷𝐶))𝐷𝐶) = (𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  ran crn 5080  cfv 5852  (class class class)co 6610  GrpOpcgr 27210   /𝑔 cgs 27213  AbelOpcablo 27265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-grpo 27214  df-gid 27215  df-ginv 27216  df-gdiv 27217  df-ablo 27266
This theorem is referenced by: (None)
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