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Theorem grpodivfval 30563
Description: Group division (or subtraction) operation. (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
grpodivfval (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐷(𝑥,𝑦)

Proof of Theorem grpodivfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpdiv.3 . 2 𝐷 = ( /𝑔𝐺)
2 grpdiv.1 . . . . 5 𝑋 = ran 𝐺
3 rnexg 7925 . . . . 5 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
42, 3eqeltrid 2843 . . . 4 (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5 mpoexga 8101 . . . 4 ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) ∈ V)
64, 4, 5syl2anc 584 . . 3 (𝐺 ∈ GrpOp → (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) ∈ V)
7 rneq 5950 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
87, 2eqtr4di 2793 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9 id 22 . . . . . 6 (𝑔 = 𝐺𝑔 = 𝐺)
10 eqidd 2736 . . . . . 6 (𝑔 = 𝐺𝑥 = 𝑥)
11 fveq2 6907 . . . . . . . 8 (𝑔 = 𝐺 → (inv‘𝑔) = (inv‘𝐺))
12 grpdiv.2 . . . . . . . 8 𝑁 = (inv‘𝐺)
1311, 12eqtr4di 2793 . . . . . . 7 (𝑔 = 𝐺 → (inv‘𝑔) = 𝑁)
1413fveq1d 6909 . . . . . 6 (𝑔 = 𝐺 → ((inv‘𝑔)‘𝑦) = (𝑁𝑦))
159, 10, 14oveq123d 7452 . . . . 5 (𝑔 = 𝐺 → (𝑥𝑔((inv‘𝑔)‘𝑦)) = (𝑥𝐺(𝑁𝑦)))
168, 8, 15mpoeq123dv 7508 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
17 df-gdiv 30525 . . . 4 /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
1816, 17fvmptg 7014 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) ∈ V) → ( /𝑔𝐺) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
196, 18mpdan 687 . 2 (𝐺 ∈ GrpOp → ( /𝑔𝐺) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
201, 19eqtrid 2787 1 (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  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:  grpodivval  30564  grpodivf  30567  nvmfval  30673
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