Theorem grpsubfval 18966

Description: Group subtraction (division) operation. For a shorter proof using ax-rep 5249, see grpsubfvalALT 18967. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5249. (Revised by Rohan Ridenour, 17-Aug-2023.) (Proof shortened by AV, 19-Feb-2024.)

Hypotheses

Ref	Expression
grpsubval.b	⊢ 𝐵 = (Base‘𝐺)
grpsubval.p	⊢ + = (+g‘𝐺)
grpsubval.i	⊢ 𝐼 = (invg‘𝐺)
grpsubval.m	⊢ − = (-g‘𝐺)

Assertion

Ref	Expression
grpsubfval	⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥,𝐼,𝑦   𝑥, + ,𝑦

Allowed substitution hints:   − (𝑥,𝑦)

Proof of Theorem grpsubfval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpsubval.m	. . 3 ⊢ − = (-g‘𝐺)
2		fveq2 6876	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		grpsubval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	eqtr4di 2788	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6876	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		grpsubval.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
7	5, 6	eqtr4di 2788	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8		eqidd 2736	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥)
9		fveq2 6876	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = (invg‘𝐺))
10		grpsubval.i	. . . . . . . 8 ⊢ 𝐼 = (invg‘𝐺)
11	9, 10	eqtr4di 2788	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = 𝐼)
12	11	fveq1d 6878	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((invg‘𝑔)‘𝑦) = (𝐼‘𝑦))
13	7, 8, 12	oveq123d 7426	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)) = (𝑥 + (𝐼‘𝑦)))
14	4, 4, 13	mpoeq123dv 7482	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
15		df-sbg 18921	. . . 4 ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))))
16	3	fvexi 6890	. . . . 5 ⊢ 𝐵 ∈ V
17	6	fvexi 6890	. . . . . . 7 ⊢ + ∈ V
18	17	rnex 7906	. . . . . 6 ⊢ ran + ∈ V
19		p0ex 5354	. . . . . 6 ⊢ {∅} ∈ V
20	18, 19	unex 7738	. . . . 5 ⊢ (ran + ∪ {∅}) ∈ V
21		df-ov 7408	. . . . . . 7 ⊢ (𝑥 + (𝐼‘𝑦)) = ( + ‘⟨𝑥, (𝐼‘𝑦)⟩)
22		fvrn0 6906	. . . . . . 7 ⊢ ( + ‘⟨𝑥, (𝐼‘𝑦)⟩) ∈ (ran + ∪ {∅})
23	21, 22	eqeltri 2830	. . . . . 6 ⊢ (𝑥 + (𝐼‘𝑦)) ∈ (ran + ∪ {∅})
24	23	rgen2w 3056	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + (𝐼‘𝑦)) ∈ (ran + ∪ {∅})
25	16, 16, 20, 24	mpoexw 8077	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V
26	14, 15, 25	fvmpt 6986	. . 3 ⊢ (𝐺 ∈ V → (-g‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
27	1, 26	eqtrid 2782	. 2 ⊢ (𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
28		fvprc 6868	. . . 4 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅)
29	1, 28	eqtrid 2782	. . 3 ⊢ (¬ 𝐺 ∈ V → − = ∅)
30		fvprc 6868	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
31	3, 30	eqtrid 2782	. . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
32	31	olcd 874	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
33		0mpo0 7490	. . . 4 ⊢ ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) = ∅)
34	32, 33	syl 17	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) = ∅)
35	29, 34	eqtr4d 2773	. 2 ⊢ (¬ 𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
36	27, 35	pm2.61i 182	1 ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))

Syntax hints:  ¬ wn 3   ∨ wo 847   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ∪ cun 3924  ∅c0 4308  {csn 4601  ⟨cop 4607  ran crn 5655  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  Basecbs 17228  +_gcplusg 17271  inv_gcminusg 18917  -_gcsg 18918

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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-sbg 18921

This theorem is referenced by:  grpsubval  18968  grpsubf  19002  grpsubpropd  19028  grpsubpropd2  19029  tgpsubcn  24028  tngtopn  24589