Theorem grpodivfval 28471

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 7637	. . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
4	2, 3	eqeltrid 2837	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5		mpoexga 7803	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V)
6	4, 4, 5	syl2anc 587	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V)
7		rneq 5779	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
8	7, 2	eqtr4di 2791	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9		id 22	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑔 = 𝐺)
10		eqidd 2739	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥)
11		fveq2 6676	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = (inv‘𝐺))
12		grpdiv.2	. . . . . . . 8 ⊢ 𝑁 = (inv‘𝐺)
13	11, 12	eqtr4di 2791	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = 𝑁)
14	13	fveq1d 6678	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((inv‘𝑔)‘𝑦) = (𝑁‘𝑦))
15	9, 10, 14	oveq123d 7193	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥𝑔((inv‘𝑔)‘𝑦)) = (𝑥𝐺(𝑁‘𝑦)))
16	8, 8, 15	mpoeq123dv 7245	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))
17		df-gdiv 28433	. . . 4 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
18	16, 17	fvmptg 6775	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))
19	6, 18	mpdan 687	. 2 ⊢ (𝐺 ∈ GrpOp → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))
20	1, 19	syl5eq 2785	1 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3398  ran crn 5526  ‘cfv 6339  (class class class)co 7172   ∈ cmpo 7174  GrpOpcgr 28426  invcgn 28428   /_𝑔 cgs 28429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7175  df-oprab 7176  df-mpo 7177  df-1st 7716  df-2nd 7717  df-gdiv 28433

This theorem is referenced by:  grpodivval  28472  grpodivf  28475  nvmfval  28581