Theorem ovtposg 6523

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 the reals or some ring, tpos F is the transposition of F, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
ovtposg	|- ( ( A e. V /\ B e. W ) -> ( Atpos F B ) = ( B F A ) )

Proof of Theorem ovtposg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2824	. . . . 5 |- y e. _V
2		brtposg 6518	. . . . 5 |- ( ( A e. V /\ B e. W /\ y e. _V ) -> ( <. A , B >.tpos F y <-> <. B , A >. F y ) )
3	1, 2	mp3an3 1367	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >.tpos F y <-> <. B , A >. F y ) )
4	3	iotabidv 5358	. . 3 |- ( ( A e. V /\ B e. W ) -> ( iota y <. A , B >.tpos F y ) = ( iota y <. B , A >. F y ) )
5		df-fv 5383	. . 3 |- (tpos F ` <. A , B >. ) = ( iota y <. A , B >.tpos F y )
6		df-fv 5383	. . 3 |- ( F ` <. B , A >. ) = ( iota y <. B , A >. F y )
7	4, 5, 6	3eqtr4g 2296	. 2 |- ( ( A e. V /\ B e. W ) -> (tpos F ` <. A , B >. ) = ( F ` <. B , A >. ) )
8		df-ov 6081	. 2 |- ( Atpos F B ) = (tpos F ` <. A , B >. )
9		df-ov 6081	. 2 |- ( B F A ) = ( F ` <. B , A >. )
10	7, 8, 9	3eqtr4g 2296	1 |- ( ( A e. V /\ B e. W ) -> ( Atpos F B ) = ( B F A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   <.cop 3711   class class class wbr 4128   iotacio 5333   ` cfv 5375  (class class class)co 6078  tpos ctpos 6508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ov 6081  df-tpos 6509

This theorem is referenced by:  tpossym  6540  opprmulg  14352