Metamath Proof Explorer


Definition df-addr

Description: Define the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012)

Ref Expression
Assertion df-addr
|- +r = ( x e. _V , y e. _V |-> ( v e. RR |-> ( ( x ` v ) + ( y ` v ) ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cplusr
 |-  +r
1 vx
 |-  x
2 cvv
 |-  _V
3 vy
 |-  y
4 vv
 |-  v
5 cr
 |-  RR
6 1 cv
 |-  x
7 4 cv
 |-  v
8 7 6 cfv
 |-  ( x ` v )
9 caddc
 |-  +
10 3 cv
 |-  y
11 7 10 cfv
 |-  ( y ` v )
12 8 11 9 co
 |-  ( ( x ` v ) + ( y ` v ) )
13 4 5 12 cmpt
 |-  ( v e. RR |-> ( ( x ` v ) + ( y ` v ) ) )
14 1 3 2 2 13 cmpo
 |-  ( x e. _V , y e. _V |-> ( v e. RR |-> ( ( x ` v ) + ( y ` v ) ) ) )
15 0 14 wceq
 |-  +r = ( x e. _V , y e. _V |-> ( v e. RR |-> ( ( x ` v ) + ( y ` v ) ) ) )