Theorem grpsubfvalg 12748

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 12713	. . 3 ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))))
3		fveq2 5496	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		grpsubval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2221	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6		fveq2 5496	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
7		grpsubval.p	. . . . . 6 ⊢ + = (+g‘𝐺)
8	6, 7	eqtr4di 2221	. . . . 5 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
9		eqidd 2171	. . . . 5 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥)
10		fveq2 5496	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = (invg‘𝐺))
11		grpsubval.i	. . . . . . 7 ⊢ 𝐼 = (invg‘𝐺)
12	10, 11	eqtr4di 2221	. . . . . 6 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = 𝐼)
13	12	fveq1d 5498	. . . . 5 ⊢ (𝑔 = 𝐺 → ((invg‘𝑔)‘𝑦) = (𝐼‘𝑦))
14	8, 9, 13	oveq123d 5874	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)) = (𝑥 + (𝐼‘𝑦)))
15	5, 5, 14	mpoeq123dv 5915	. . 3 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
16		elex 2741	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
17		basfn 12473	. . . . . 6 ⊢ Base Fn V
18		funfvex 5513	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
19	18	funfni 5298	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
20	17, 16, 19	sylancr 412	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
21	4, 20	eqeltrid 2257	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
22		mpoexga 6191	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V)
23	21, 21, 22	syl2anc 409	. . 3 ⊢ (𝐺 ∈ 𝑉 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V)
24	2, 15, 16, 23	fvmptd3 5589	. 2 ⊢ (𝐺 ∈ 𝑉 → (-g‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
25	1, 24	eqtrid 2215	1 ⊢ (𝐺 ∈ 𝑉 → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  Vcvv 2730   Fn wfn 5193  ‘cfv 5198  (class class class)co 5853   ∈ cmpo 5855  Basecbs 12416  +_gcplusg 12480  inv_gcminusg 12709  -_gcsg 12710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-inn 8879  df-ndx 12419  df-slot 12420  df-base 12422  df-sbg 12713

This theorem is referenced by:  grpsubval  12749