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Theorem grponpcan 26781
 Description: Cancellation law for group division. (npcan 10169 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1 𝑋 = ran 𝐺
grpdivf.3 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
grponpcan ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)

Proof of Theorem grponpcan
StepHypRef Expression
1 grpdivf.1 . . . 4 𝑋 = ran 𝐺
2 eqid 2610 . . . 4 (inv‘𝐺) = (inv‘𝐺)
3 grpdivf.3 . . . 4 𝐷 = ( /𝑔𝐺)
41, 2, 3grpodivval 26773 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
54oveq1d 6564 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵))
6 simp1 1054 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐺 ∈ GrpOp)
7 simp2 1055 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐴𝑋)
81, 2grpoinvcl 26762 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → ((inv‘𝐺)‘𝐵) ∈ 𝑋)
983adant2 1073 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((inv‘𝐺)‘𝐵) ∈ 𝑋)
10 simp3 1056 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐵𝑋)
111grpoass 26741 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋𝐵𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵) = (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)))
126, 7, 9, 10, 11syl13anc 1320 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵) = (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)))
13 eqid 2610 . . . . . . 7 (GId‘𝐺) = (GId‘𝐺)
141, 13, 2grpolinv 26764 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (((inv‘𝐺)‘𝐵)𝐺𝐵) = (GId‘𝐺))
1514oveq2d 6565 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)) = (𝐴𝐺(GId‘𝐺)))
16153adant2 1073 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)) = (𝐴𝐺(GId‘𝐺)))
171, 13grporid 26755 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
18173adant3 1074 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
1916, 18eqtrd 2644 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)) = 𝐴)
2012, 19eqtrd 2644 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵) = 𝐴)
215, 20eqtrd 2644 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ran crn 5039  ‘cfv 5804  (class class class)co 6549  GrpOpcgr 26727  GIdcgi 26728  invcgn 26729   /𝑔 cgs 26730 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-grpo 26731  df-gid 26732  df-ginv 26733  df-gdiv 26734 This theorem is referenced by:  ablonnncan  26794  grpoeqdivid  32850  ghomdiv  32861
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