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Theorem ablonncan 30394
Description: Cancellation law for group division. (nncan 11529 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 30391 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
62, 5sylan2 591 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
763impb 1112 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
8 ablogrpo 30385 . . . . 5 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
9 eqid 2728 . . . . . 6 (GId‘𝐺) = (GId‘𝐺)
103, 4, 9grpodivid 30380 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
118, 10sylan 578 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
12113adant3 1129 . . 3 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
1312oveq1d 7441 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐴)𝐺𝐵) = ((GId‘𝐺)𝐺𝐵))
143, 9grpolid 30354 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
158, 14sylan 578 . . 3 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
16153adant2 1128 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
177, 13, 163eqtrd 2772 1 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  ran crn 5683  cfv 6553  (class class class)co 7426  GrpOpcgr 30327  GIdcgi 30328   /𝑔 cgs 30330  AbelOpcablo 30382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002  df-grpo 30331  df-gid 30332  df-ginv 30333  df-gdiv 30334  df-ablo 30383
This theorem is referenced by:  ablonnncan1  30395
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