Theorem grpsubfval 18898

Description: Group subtraction (division) operation. For a shorter proof using ax-rep 5219, see grpsubfvalALT 18899. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5219. (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 6828	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		grpsubval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	eqtr4di 2786	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6828	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		grpsubval.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
7	5, 6	eqtr4di 2786	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8		eqidd 2734	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥)
9		fveq2 6828	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = (invg‘𝐺))
10		grpsubval.i	. . . . . . . 8 ⊢ 𝐼 = (invg‘𝐺)
11	9, 10	eqtr4di 2786	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = 𝐼)
12	11	fveq1d 6830	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((invg‘𝑔)‘𝑦) = (𝐼‘𝑦))
13	7, 8, 12	oveq123d 7373	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)) = (𝑥 + (𝐼‘𝑦)))
14	4, 4, 13	mpoeq123dv 7427	. . . 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 18853	. . . 4 ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))))
16	3	fvexi 6842	. . . . 5 ⊢ 𝐵 ∈ V
17	6	fvexi 6842	. . . . . . 7 ⊢ + ∈ V
18	17	rnex 7846	. . . . . 6 ⊢ ran + ∈ V
19		p0ex 5324	. . . . . 6 ⊢ {∅} ∈ V
20	18, 19	unex 7683	. . . . 5 ⊢ (ran + ∪ {∅}) ∈ V
21		df-ov 7355	. . . . . . 7 ⊢ (𝑥 + (𝐼‘𝑦)) = ( + ‘⟨𝑥, (𝐼‘𝑦)⟩)
22		fvrn0 6856	. . . . . . 7 ⊢ ( + ‘⟨𝑥, (𝐼‘𝑦)⟩) ∈ (ran + ∪ {∅})
23	21, 22	eqeltri 2829	. . . . . 6 ⊢ (𝑥 + (𝐼‘𝑦)) ∈ (ran + ∪ {∅})
24	23	rgen2w 3053	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + (𝐼‘𝑦)) ∈ (ran + ∪ {∅})
25	16, 16, 20, 24	mpoexw 8016	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V
26	14, 15, 25	fvmpt 6935	. . 3 ⊢ (𝐺 ∈ V → (-g‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
27	1, 26	eqtrid 2780	. 2 ⊢ (𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
28		fvprc 6820	. . . 4 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅)
29	1, 28	eqtrid 2780	. . 3 ⊢ (¬ 𝐺 ∈ V → − = ∅)
30		fvprc 6820	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
31	3, 30	eqtrid 2780	. . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
32	31	olcd 874	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
33		0mpo0 7435	. . . 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 2771	. 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 1541   ∈ wcel 2113  Vcvv 3437   ∪ cun 3896  ∅c0 4282  {csn 4575  ⟨cop 4581  ran crn 5620  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354  Basecbs 17122  +_gcplusg 17163  inv_gcminusg 18849  -_gcsg 18850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-sbg 18853

This theorem is referenced by:  grpsubval  18900  grpsubf  18934  grpsubpropd  18960  grpsubpropd2  18961  tgpsubcn  24006  tngtopn  24566