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Theorem grpodivval 28328
 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 28327 . . . 4 (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
54oveqd 7153 . . 3 (𝐺 ∈ GrpOp → (𝐴𝐷𝐵) = (𝐴(𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦)))𝐵))
6 oveq1 7143 . . . 4 (𝑥 = 𝐴 → (𝑥𝐺(𝑁𝑦)) = (𝐴𝐺(𝑁𝑦)))
7 fveq2 6646 . . . . 5 (𝑦 = 𝐵 → (𝑁𝑦) = (𝑁𝐵))
87oveq2d 7152 . . . 4 (𝑦 = 𝐵 → (𝐴𝐺(𝑁𝑦)) = (𝐴𝐺(𝑁𝐵)))
9 eqid 2798 . . . 4 (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦)))
10 ovex 7169 . . . 4 (𝐴𝐺(𝑁𝐵)) ∈ V
116, 8, 9, 10ovmpo 7291 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴(𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦)))𝐵) = (𝐴𝐺(𝑁𝐵)))
125, 11sylan9eq 2853 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))
13123impb 1112 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ran crn 5521  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  GrpOpcgr 28282  invcgn 28284   /𝑔 cgs 28285 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-gdiv 28289 This theorem is referenced by:  grpodivinv  28329  grpoinvdiv  28330  grpodivdiv  28333  grpomuldivass  28334  grpodivid  28335  grponpcan  28336  ablodivdiv4  28347  nvmval  28435  rngosub  35387
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