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Metamath Proof Explorer

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 )