Theorem grpsubfval 18902

Description: Group subtraction (division) operation. For a shorter proof using ax-rep 5275, see grpsubfvalALT 18903. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5275. (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 6881	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		grpsubval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	eqtr4di 2782	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6881	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		grpsubval.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
7	5, 6	eqtr4di 2782	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8		eqidd 2725	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥)
9		fveq2 6881	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = (invg‘𝐺))
10		grpsubval.i	. . . . . . . 8 ⊢ 𝐼 = (invg‘𝐺)
11	9, 10	eqtr4di 2782	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = 𝐼)
12	11	fveq1d 6883	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((invg‘𝑔)‘𝑦) = (𝐼‘𝑦))
13	7, 8, 12	oveq123d 7422	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)) = (𝑥 + (𝐼‘𝑦)))
14	4, 4, 13	mpoeq123dv 7476	. . . 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 18857	. . . 4 ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))))
16	3	fvexi 6895	. . . . 5 ⊢ 𝐵 ∈ V
17	6	fvexi 6895	. . . . . . 7 ⊢ + ∈ V
18	17	rnex 7896	. . . . . 6 ⊢ ran + ∈ V
19		p0ex 5372	. . . . . 6 ⊢ {∅} ∈ V
20	18, 19	unex 7726	. . . . 5 ⊢ (ran + ∪ {∅}) ∈ V
21		df-ov 7404	. . . . . . 7 ⊢ (𝑥 + (𝐼‘𝑦)) = ( + ‘⟨𝑥, (𝐼‘𝑦)⟩)
22		fvrn0 6911	. . . . . . 7 ⊢ ( + ‘⟨𝑥, (𝐼‘𝑦)⟩) ∈ (ran + ∪ {∅})
23	21, 22	eqeltri 2821	. . . . . 6 ⊢ (𝑥 + (𝐼‘𝑦)) ∈ (ran + ∪ {∅})
24	23	rgen2w 3058	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + (𝐼‘𝑦)) ∈ (ran + ∪ {∅})
25	16, 16, 20, 24	mpoexw 8058	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V
26	14, 15, 25	fvmpt 6988	. . 3 ⊢ (𝐺 ∈ V → (-g‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
27	1, 26	eqtrid 2776	. 2 ⊢ (𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
28		fvprc 6873	. . . 4 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅)
29	1, 28	eqtrid 2776	. . 3 ⊢ (¬ 𝐺 ∈ V → − = ∅)
30		fvprc 6873	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
31	3, 30	eqtrid 2776	. . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
32	31	olcd 871	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
33		0mpo0 7484	. . . 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 2767	. 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 844   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ∪ cun 3938  ∅c0 4314  {csn 4620  ⟨cop 4626  ran crn 5667  ‘cfv 6533  (class class class)co 7401   ∈ cmpo 7403  Basecbs 17142  +_gcplusg 17195  inv_gcminusg 18853  -_gcsg 18854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-sbg 18857

This theorem is referenced by:  grpsubval  18904  grpsubf  18936  grpsubpropd  18962  grpsubpropd2  18963  tgpsubcn  23915  tngtopn  24488