Theorem grpsubfval 19113

Description: Group subtraction (division) operation. For a shorter proof using ax-rep 5236, see grpsubfvalALT 19114. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5236. (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 6882	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		grpsubval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	eqtr4di 2815	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6882	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		grpsubval.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
7	5, 6	eqtr4di 2815	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8		eqidd 2763	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥)
9		fveq2 6882	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = (invg‘𝐺))
10		grpsubval.i	. . . . . . . 8 ⊢ 𝐼 = (invg‘𝐺)
11	9, 10	eqtr4di 2815	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = 𝐼)
12	11	fveq1d 6884	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((invg‘𝑔)‘𝑦) = (𝐼‘𝑦))
13	7, 8, 12	oveq123d 7438	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)) = (𝑥 + (𝐼‘𝑦)))
14	4, 4, 13	mpoeq123dv 7492	. . . 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 19068	. . . 4 ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))))
16	3	fvexi 6896	. . . . 5 ⊢ 𝐵 ∈ V
17	6	fvexi 6896	. . . . . . 7 ⊢ + ∈ V
18	17	rnex 7911	. . . . . 6 ⊢ ran + ∈ V
19		p0ex 5353	. . . . . 6 ⊢ {∅} ∈ V
20	18, 19	unex 7750	. . . . 5 ⊢ (ran + ∪ {∅}) ∈ V
21		df-ov 7420	. . . . . . 7 ⊢ (𝑥 + (𝐼‘𝑦)) = ( + ‘⟨𝑥, (𝐼‘𝑦)⟩)
22		fvrn0 6910	. . . . . . 7 ⊢ ( + ‘⟨𝑥, (𝐼‘𝑦)⟩) ∈ (ran + ∪ {∅})
23	21, 22	eqeltri 2858	. . . . . 6 ⊢ (𝑥 + (𝐼‘𝑦)) ∈ (ran + ∪ {∅})
24	23	rgen2w 3083	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + (𝐼‘𝑦)) ∈ (ran + ∪ {∅})
25	16, 16, 20, 24	mpoexw 8081	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V
26	14, 15, 25	fvmpt 6990	. . 3 ⊢ (𝐺 ∈ V → (-g‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
27	1, 26	eqtrid 2809	. 2 ⊢ (𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
28		fvprc 6874	. . . 4 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅)
29	1, 28	eqtrid 2809	. . 3 ⊢ (¬ 𝐺 ∈ V → − = ∅)
30		fvprc 6874	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
31	3, 30	eqtrid 2809	. . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
32	31	olcd 888	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
33		0mpo0 7500	. . . 4 ⊢ ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) = ∅)
34	32, 33	syl 18	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) = ∅)
35	29, 34	eqtr4d 2800	. 2 ⊢ (¬ 𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
36	27, 35	pm2.61i 184	1 ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∪ cun 3900  ∅c0 4282  {csn 4587  ⟨cop 4593  ran crn 5660  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  Basecbs 17307  +_gcplusg 17348  inv_gcminusg 19064  -_gcsg 19065

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-sbg 19068

This theorem is used by:  grpsubval  19115  grpsubf  19148  grpsubpropd  19174  grpsubpropd2  19175  tgpsubcn  24322  tngtopn  24882