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 AtposFB=BFA

Proof

Step Hyp Ref Expression
1 brtpos yVABtposFyBAFy
2 1 elv ABtposFyBAFy
3 2 iotabii ιy|ABtposFy=ιy|BAFy
4 df-fv tposFAB=ιy|ABtposFy
5 df-fv FBA=ιy|BAFy
6 3 4 5 3eqtr4i tposFAB=FBA
7 df-ov AtposFB=tposFAB
8 df-ov BFA=FBA
9 6 7 8 3eqtr4i AtposFB=BFA