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Theorem ablonnncan 27750
Description: Cancellation law for group division. (nnncan 10518 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 1233 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑋)
2 ablogrpo 27741 . . . . . 6 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
3 abldiv.1 . . . . . . 7 𝑋 = ran 𝐺
4 abldiv.3 . . . . . . 7 𝐷 = ( /𝑔𝐺)
53, 4grpodivcl 27733 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐷𝐶) ∈ 𝑋)
62, 5syl3an1 1166 . . . . 5 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐷𝐶) ∈ 𝑋)
763adant3r1 1197 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐷𝐶) ∈ 𝑋)
8 simpr3 1237 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐶𝑋)
91, 7, 83jca 1122 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑋 ∧ (𝐵𝐷𝐶) ∈ 𝑋𝐶𝑋))
103, 4ablodivdiv4 27748 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋 ∧ (𝐵𝐷𝐶) ∈ 𝑋𝐶𝑋)) → ((𝐴𝐷(𝐵𝐷𝐶))𝐷𝐶) = (𝐴𝐷((𝐵𝐷𝐶)𝐺𝐶)))
119, 10syldan 579 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷(𝐵𝐷𝐶))𝐷𝐶) = (𝐴𝐷((𝐵𝐷𝐶)𝐺𝐶)))
123, 4grponpcan 27737 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → ((𝐵𝐷𝐶)𝐺𝐶) = 𝐵)
132, 12syl3an1 1166 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋𝐶𝑋) → ((𝐵𝐷𝐶)𝐺𝐶) = 𝐵)
14133adant3r1 1197 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐵𝐷𝐶)𝐺𝐶) = 𝐵)
1514oveq2d 6809 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷((𝐵𝐷𝐶)𝐺𝐶)) = (𝐴𝐷𝐵))
1611, 15eqtrd 2805 1 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷(𝐵𝐷𝐶))𝐷𝐶) = (𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  ran crn 5250  cfv 6031  (class class class)co 6793  GrpOpcgr 27683   /𝑔 cgs 27686  AbelOpcablo 27738
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 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-grpo 27687  df-gid 27688  df-ginv 27689  df-gdiv 27690  df-ablo 27739
This theorem is referenced by: (None)
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