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Definition df-sbg 17198
Description: Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)
Assertion
Ref Expression
df-sbg -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))
Distinct variable group:   𝑥,𝑔,𝑦

Detailed syntax breakdown of Definition df-sbg
StepHypRef Expression
1 csg 17195 . 2 class -g
2 vg . . 3 setvar 𝑔
3 cvv 3172 . . 3 class V
4 vx . . . 4 setvar 𝑥
5 vy . . . 4 setvar 𝑦
62cv 1473 . . . . 5 class 𝑔
7 cbs 15643 . . . . 5 class Base
86, 7cfv 5789 . . . 4 class (Base‘𝑔)
94cv 1473 . . . . 5 class 𝑥
105cv 1473 . . . . . 6 class 𝑦
11 cminusg 17194 . . . . . . 7 class invg
126, 11cfv 5789 . . . . . 6 class (invg𝑔)
1310, 12cfv 5789 . . . . 5 class ((invg𝑔)‘𝑦)
14 cplusg 15716 . . . . . 6 class +g
156, 14cfv 5789 . . . . 5 class (+g𝑔)
169, 13, 15co 6526 . . . 4 class (𝑥(+g𝑔)((invg𝑔)‘𝑦))
174, 5, 8, 8, 16cmpt2 6528 . . 3 class (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦)))
182, 3, 17cmpt 4637 . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))
191, 18wceq 1474 1 wff -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))
Colors of variables: wff setvar class
This definition is referenced by:  grpsubfval  17235
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