MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-sub Unicode version

Definition df-sub 9055
Description: Define subtraction. Theorem subval 9059 shows its value (and describes how this definition works), theorem subaddi 9149 relates it to addition, and theorems subcli 9138 and resubcli 9125 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
Assertion
Ref Expression
df-sub  |-  -  =  ( x  e.  CC ,  y  e.  CC  |->  ( iota_ z  e.  CC ( y  +  z )  =  x ) )
Distinct variable group:    x, y, z

Detailed syntax breakdown of Definition df-sub
StepHypRef Expression
1 cmin 9053 . 2  class  -
2 vx . . 3  set  x
3 vy . . 3  set  y
4 cc 8751 . . 3  class  CC
53cv 1631 . . . . . 6  class  y
6 vz . . . . . . 7  set  z
76cv 1631 . . . . . 6  class  z
8 caddc 8756 . . . . . 6  class  +
95, 7, 8co 5874 . . . . 5  class  ( y  +  z )
102cv 1631 . . . . 5  class  x
119, 10wceq 1632 . . . 4  wff  ( y  +  z )  =  x
1211, 6, 4crio 6313 . . 3  class  ( iota_ z  e.  CC ( y  +  z )  =  x )
132, 3, 4, 4, 12cmpt2 5876 . 2  class  ( x  e.  CC ,  y  e.  CC  |->  ( iota_ z  e.  CC ( y  +  z )  =  x ) )
141, 13wceq 1632 1  wff  -  =  ( x  e.  CC ,  y  e.  CC  |->  ( iota_ z  e.  CC ( y  +  z )  =  x ) )
Colors of variables: wff set class
This definition is referenced by:  subval  9059  subf  9069
  Copyright terms: Public domain W3C validator