Metamath Proof Explorer

Theorem vcrel

Description: The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007) (New usage is discouraged.)

		Ref	Expression
	Assertion	vcrel	\|- Rel CVecOLD

Proof

Step	Hyp	Ref	Expression
1		df-vc	\|- CVecOLD = { <. g , s >. \| ( g e. AbelOp /\ s : ( CC X. ran g ) --> ran g /\ A. x e. ran g ( ( 1 s x ) = x /\ A. y e. CC ( A. z e. ran g ( y s ( x g z ) ) = ( ( y s x ) g ( y s z ) ) /\ A. z e. CC ( ( ( y + z ) s x ) = ( ( y s x ) g ( z s x ) ) /\ ( ( y x. z ) s x ) = ( y s ( z s x ) ) ) ) ) ) }
2	1	relopabiv	\|- Rel CVecOLD

Step

Hyp

Ref

Expression

df-vc

 |-  CVecOLD = { <. g , s >. | ( g e. AbelOp /\ s : ( CC X. ran g ) --> ran g /\ A. x e. ran g ( ( 1 s x ) = x /\ A. y e. CC ( A. z e. ran g ( y s ( x g z ) ) = ( ( y s x ) g ( y s z ) ) /\ A. z e. CC ( ( ( y + z ) s x ) = ( ( y s x ) g ( z s x ) ) /\ ( ( y x. z ) s x ) = ( y s ( z s x ) ) ) ) ) ) }

relopabiv

 |-  Rel CVecOLD