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Metamath Proof Explorer

Definition df-plusf

Description: Define group addition function. Usually we will use +g directly instead of +f , and they have the same behavior in most cases. The main advantage of +f for any magma is that it is a guaranteed function ( mgmplusf ), while +g only has closure ( mgmcl ). (Contributed by Mario Carneiro, 14-Aug-2015)

Ref Expression
Assertion df-plusf ⊢ + 𝑓 = g ∈ V ⟼ x ∈ Base g , y ∈ Base g ⟼ x + g y

Detailed syntax breakdown

Step Hyp Ref Expression
0 cplusf class + 𝑓
1 vg setvar g
2 cvv class V
3 vx setvar x
4 cbs class Base
5 1 cv setvar g
6 5 4 cfv class Base g
7 vy setvar y
8 3 cv setvar x
9 cplusg class + 𝑔
10 5 9 cfv class + g
11 7 cv setvar y
12 8 11 10 co class x + g y
13 3 7 6 6 12 cmpo class x ∈ Base g , y ∈ Base g ⟼ x + g y
14 1 2 13 cmpt class g ∈ V ⟼ x ∈ Base g , y ∈ Base g ⟼ x + g y
15 0 14 wceq wff + 𝑓 = g ∈ V ⟼ x ∈ Base g , y ∈ Base g ⟼ x + g y