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Theorem grpodivfval 30553
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 7924 . . . . 5 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
42, 3eqeltrid 2845 . . . 4 (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5 mpoexga 8102 . . . 4 ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) ∈ V)
64, 4, 5syl2anc 584 . . 3 (𝐺 ∈ GrpOp → (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) ∈ V)
7 rneq 5947 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
87, 2eqtr4di 2795 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9 id 22 . . . . . 6 (𝑔 = 𝐺𝑔 = 𝐺)
10 eqidd 2738 . . . . . 6 (𝑔 = 𝐺𝑥 = 𝑥)
11 fveq2 6906 . . . . . . . 8 (𝑔 = 𝐺 → (inv‘𝑔) = (inv‘𝐺))
12 grpdiv.2 . . . . . . . 8 𝑁 = (inv‘𝐺)
1311, 12eqtr4di 2795 . . . . . . 7 (𝑔 = 𝐺 → (inv‘𝑔) = 𝑁)
1413fveq1d 6908 . . . . . 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 30515 . . . 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 2789 1 (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  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:  grpodivval  30554  grpodivf  30557  nvmfval  30663
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