Theorem grpodivfval 28896

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 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
grpodivfval	⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))

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.

Step	Hyp	Ref	Expression
1		grpdiv.3	. 2 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
2		grpdiv.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
3		rnexg 7751	. . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
4	2, 3	eqeltrid 2843	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5		mpoexga 7918	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V)
6	4, 4, 5	syl2anc 584	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V)
7		rneq 5845	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
8	7, 2	eqtr4di 2796	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9		id 22	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑔 = 𝐺)
10		eqidd 2739	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥)
11		fveq2 6774	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = (inv‘𝐺))
12		grpdiv.2	. . . . . . . 8 ⊢ 𝑁 = (inv‘𝐺)
13	11, 12	eqtr4di 2796	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = 𝑁)
14	13	fveq1d 6776	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((inv‘𝑔)‘𝑦) = (𝑁‘𝑦))
15	9, 10, 14	oveq123d 7296	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥𝑔((inv‘𝑔)‘𝑦)) = (𝑥𝐺(𝑁‘𝑦)))
16	8, 8, 15	mpoeq123dv 7350	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))
17		df-gdiv 28858	. . . 4 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
18	16, 17	fvmptg 6873	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))
19	6, 18	mpdan 684	. 2 ⊢ (𝐺 ∈ GrpOp → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))
20	1, 19	eqtrid 2790	1 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ran crn 5590  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  GrpOpcgr 28851  invcgn 28853   /_𝑔 cgs 28854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-gdiv 28858

This theorem is referenced by:  grpodivval  28897  grpodivf  28900  nvmfval  29006