MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpodivval Structured version   Visualization version   GIF version

Theorem grpodivval 30564
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 30563 . . . 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 6907 . . . . 5 (𝑦 = 𝐵 → (𝑁𝑦) = (𝑁𝐵))
87oveq2d 7447 . . . 4 (𝑦 = 𝐵 → (𝐴𝐺(𝑁𝑦)) = (𝐴𝐺(𝑁𝐵)))
9 eqid 2735 . . . 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 2795 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))
13123impb 1114 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  ran crn 5690  cfv 6563  (class class class)co 7431  cmpo 7433  GrpOpcgr 30518  invcgn 30520   /𝑔 cgs 30521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-gdiv 30525
This theorem is referenced by:  grpodivinv  30565  grpoinvdiv  30566  grpodivdiv  30569  grpomuldivass  30570  grpodivid  30571  grponpcan  30572  ablodivdiv4  30583  nvmval  30671  rngosub  37917
  Copyright terms: Public domain W3C validator