Theorem subf

Description: Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007) (Revised by Mario Carneiro, 16-Nov-2013)

	Ref	Expression
Assertion	subf	|- - : ( CC X. CC ) --> CC

Proof

Step	Hyp	Ref	Expression
1		subval	|- ( ( x e. CC /\ y e. CC ) -> ( x - y ) = ( iota_ z e. CC ( y + z ) = x ) )
2		subcl	|- ( ( x e. CC /\ y e. CC ) -> ( x - y ) e. CC )
3	1 2	eqeltrrd	|- ( ( x e. CC /\ y e. CC ) -> ( iota_ z e. CC ( y + z ) = x ) e. CC )
4	3	rgen2	|- A. x e. CC A. y e. CC ( iota_ z e. CC ( y + z ) = x ) e. CC
5		df-sub	|- - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )
6	5	fmpo	|- ( A. x e. CC A. y e. CC ( iota_ z e. CC ( y + z ) = x ) e. CC <-> - : ( CC X. CC ) --> CC )
7	4 6	mpbi	|- - : ( CC X. CC ) --> CC