Theorem ovtpos

Description: The transposition swaps the arguments in a two-argument function. When F is a matrix, which is to say a function from ( 1 ... m ) X. ( 1 ... n ) to RR or some ring, tpos F is the transposition of F , which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015)

	Ref	Expression
Assertion	ovtpos	|- ( A tpos F B ) = ( B F A )

Proof

Step	Hyp	Ref	Expression
1		brtpos	|- ( y e. _V -> ( <. A , B >. tpos F y <-> <. B , A >. F y ) )
2	1	elv	|- ( <. A , B >. tpos F y <-> <. B , A >. F y )
3	2	iotabii	|- ( iota y <. A , B >. tpos F y ) = ( iota y <. B , A >. F y )
4		df-fv	|- ( tpos F ` <. A , B >. ) = ( iota y <. A , B >. tpos F y )
5		df-fv	|- ( F ` <. B , A >. ) = ( iota y <. B , A >. F y )
6	3 4 5	3eqtr4i	|- ( tpos F ` <. A , B >. ) = ( F ` <. B , A >. )
7		df-ov	|- ( A tpos F B ) = ( tpos F ` <. A , B >. )
8		df-ov	|- ( B F A ) = ( F ` <. B , A >. )
9	6 7 8	3eqtr4i	|- ( A tpos F B ) = ( B F A )