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Theorem grpodivval 27517
Description: Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdiv.1 𝑋 = ran 𝐺
grpdiv.2 𝑁 = (inv‘𝐺)
grpdiv.3 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
grpodivval ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))

Proof of Theorem grpodivval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpdiv.1 . . . . 5 𝑋 = ran 𝐺
2 grpdiv.2 . . . . 5 𝑁 = (inv‘𝐺)
3 grpdiv.3 . . . . 5 𝐷 = ( /𝑔𝐺)
41, 2, 3grpodivfval 27516 . . . 4 (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
54oveqd 6707 . . 3 (𝐺 ∈ GrpOp → (𝐴𝐷𝐵) = (𝐴(𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦)))𝐵))
6 oveq1 6697 . . . 4 (𝑥 = 𝐴 → (𝑥𝐺(𝑁𝑦)) = (𝐴𝐺(𝑁𝑦)))
7 fveq2 6229 . . . . 5 (𝑦 = 𝐵 → (𝑁𝑦) = (𝑁𝐵))
87oveq2d 6706 . . . 4 (𝑦 = 𝐵 → (𝐴𝐺(𝑁𝑦)) = (𝐴𝐺(𝑁𝐵)))
9 eqid 2651 . . . 4 (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦)))
10 ovex 6718 . . . 4 (𝐴𝐺(𝑁𝐵)) ∈ V
116, 8, 9, 10ovmpt2 6838 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴(𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦)))𝐵) = (𝐴𝐺(𝑁𝐵)))
125, 11sylan9eq 2705 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))
13123impb 1279 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  ran crn 5144  cfv 5926  (class class class)co 6690  cmpt2 6692  GrpOpcgr 27471  invcgn 27473   /𝑔 cgs 27474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-gdiv 27478
This theorem is referenced by:  grpodivinv  27518  grpoinvdiv  27519  grpodivdiv  27522  grpomuldivass  27523  grpodivid  27524  grponpcan  27525  ablodivdiv4  27536  nvmval  27625  rngosub  33859
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