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Theorem grpodivfval 27695
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 7261 . . . . 5 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
42, 3syl5eqel 2841 . . . 4 (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5 mpt2exga 7412 . . . 4 ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) ∈ V)
64, 4, 5syl2anc 696 . . 3 (𝐺 ∈ GrpOp → (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) ∈ V)
7 rneq 5504 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
87, 2syl6eqr 2810 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9 id 22 . . . . . 6 (𝑔 = 𝐺𝑔 = 𝐺)
10 eqidd 2759 . . . . . 6 (𝑔 = 𝐺𝑥 = 𝑥)
11 fveq2 6350 . . . . . . . 8 (𝑔 = 𝐺 → (inv‘𝑔) = (inv‘𝐺))
12 grpdiv.2 . . . . . . . 8 𝑁 = (inv‘𝐺)
1311, 12syl6eqr 2810 . . . . . . 7 (𝑔 = 𝐺 → (inv‘𝑔) = 𝑁)
1413fveq1d 6352 . . . . . 6 (𝑔 = 𝐺 → ((inv‘𝑔)‘𝑦) = (𝑁𝑦))
159, 10, 14oveq123d 6832 . . . . 5 (𝑔 = 𝐺 → (𝑥𝑔((inv‘𝑔)‘𝑦)) = (𝑥𝐺(𝑁𝑦)))
168, 8, 15mpt2eq123dv 6880 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
17 df-gdiv 27657 . . . 4 /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
1816, 17fvmptg 6440 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) ∈ V) → ( /𝑔𝐺) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
196, 18mpdan 705 . 2 (𝐺 ∈ GrpOp → ( /𝑔𝐺) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
201, 19syl5eq 2804 1 (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2137  Vcvv 3338  ran crn 5265  cfv 6047  (class class class)co 6811  cmpt2 6813  GrpOpcgr 27650  invcgn 27652   /𝑔 cgs 27653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-1st 7331  df-2nd 7332  df-gdiv 27657
This theorem is referenced by:  grpodivval  27696  grpodivf  27699  nvmfval  27806
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