Theorem grpsubfvalg 13247

Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (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
grpsubfvalg	⊢ (𝐺 ∈ 𝑉 → − = (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)   𝑉(𝑥,𝑦)

Proof of Theorem grpsubfvalg

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpsubval.m	. 2 ⊢ − = (-g‘𝐺)
2		df-sbg 13207	. . 3 ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))))
3		fveq2 5561	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		grpsubval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2247	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6		fveq2 5561	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
7		grpsubval.p	. . . . . 6 ⊢ + = (+g‘𝐺)
8	6, 7	eqtr4di 2247	. . . . 5 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
9		eqidd 2197	. . . . 5 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥)
10		fveq2 5561	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = (invg‘𝐺))
11		grpsubval.i	. . . . . . 7 ⊢ 𝐼 = (invg‘𝐺)
12	10, 11	eqtr4di 2247	. . . . . 6 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = 𝐼)
13	12	fveq1d 5563	. . . . 5 ⊢ (𝑔 = 𝐺 → ((invg‘𝑔)‘𝑦) = (𝐼‘𝑦))
14	8, 9, 13	oveq123d 5946	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)) = (𝑥 + (𝐼‘𝑦)))
15	5, 5, 14	mpoeq123dv 5988	. . 3 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
16		elex 2774	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
17		basfn 12761	. . . . . 6 ⊢ Base Fn V
18		funfvex 5578	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
19	18	funfni 5361	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
20	17, 16, 19	sylancr 414	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
21	4, 20	eqeltrid 2283	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
22		mpoexga 6279	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V)
23	21, 21, 22	syl2anc 411	. . 3 ⊢ (𝐺 ∈ 𝑉 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V)
24	2, 15, 16, 23	fvmptd3 5658	. 2 ⊢ (𝐺 ∈ 𝑉 → (-g‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
25	1, 24	eqtrid 2241	1 ⊢ (𝐺 ∈ 𝑉 → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  Vcvv 2763   Fn wfn 5254  ‘cfv 5259  (class class class)co 5925   ∈ cmpo 5927  Basecbs 12703  +_gcplusg 12780  inv_gcminusg 13203  -_gcsg 13204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-sbg 13207

This theorem is referenced by:  grpsubval  13248  grpsubf  13281  grpsubpropdg  13306  grpsubpropd2  13307