Theorem grpodivfval 30318

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 7902	. . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
4	2, 3	eqeltrid 2832	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5		mpoexga 8074	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V)
6	4, 4, 5	syl2anc 583	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V)
7		rneq 5932	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
8	7, 2	eqtr4di 2785	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9		id 22	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑔 = 𝐺)
10		eqidd 2728	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥)
11		fveq2 6891	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = (inv‘𝐺))
12		grpdiv.2	. . . . . . . 8 ⊢ 𝑁 = (inv‘𝐺)
13	11, 12	eqtr4di 2785	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = 𝑁)
14	13	fveq1d 6893	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((inv‘𝑔)‘𝑦) = (𝑁‘𝑦))
15	9, 10, 14	oveq123d 7435	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥𝑔((inv‘𝑔)‘𝑦)) = (𝑥𝐺(𝑁‘𝑦)))
16	8, 8, 15	mpoeq123dv 7489	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))
17		df-gdiv 30280	. . . 4 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
18	16, 17	fvmptg 6997	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))
19	6, 18	mpdan 686	. 2 ⊢ (𝐺 ∈ GrpOp → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))
20	1, 19	eqtrid 2779	1 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3469  ran crn 5673  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  GrpOpcgr 30273  invcgn 30275   /_𝑔 cgs 30276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-gdiv 30280

This theorem is referenced by:  grpodivval  30319  grpodivf  30322  nvmfval  30428