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Theorem grpodivval 30554
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 30553 . . . 4 (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
54oveqd 7448 . . 3 (𝐺 ∈ GrpOp → (𝐴𝐷𝐵) = (𝐴(𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦)))𝐵))
6 oveq1 7438 . . . 4 (𝑥 = 𝐴 → (𝑥𝐺(𝑁𝑦)) = (𝐴𝐺(𝑁𝑦)))
7 fveq2 6906 . . . . 5 (𝑦 = 𝐵 → (𝑁𝑦) = (𝑁𝐵))
87oveq2d 7447 . . . 4 (𝑦 = 𝐵 → (𝐴𝐺(𝑁𝑦)) = (𝐴𝐺(𝑁𝐵)))
9 eqid 2737 . . . 4 (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦)))
10 ovex 7464 . . . 4 (𝐴𝐺(𝑁𝐵)) ∈ V
116, 8, 9, 10ovmpo 7593 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴(𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦)))𝐵) = (𝐴𝐺(𝑁𝐵)))
125, 11sylan9eq 2797 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))
13123impb 1115 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  ran crn 5686  cfv 6561  (class class class)co 7431  cmpo 7433  GrpOpcgr 30508  invcgn 30510   /𝑔 cgs 30511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-gdiv 30515
This theorem is referenced by:  grpodivinv  30555  grpoinvdiv  30556  grpodivdiv  30559  grpomuldivass  30560  grpodivid  30561  grponpcan  30562  ablodivdiv4  30573  nvmval  30661  rngosub  37937
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