Theorem ovtposg 6314

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 2763	. . . . 5 |- y e. _V
2		brtposg 6309	. . . . 5 |- ( ( A e. V /\ B e. W /\ y e. _V ) -> ( <. A , B >.tpos F y <-> <. B , A >. F y ) )
3	1, 2	mp3an3 1337	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >.tpos F y <-> <. B , A >. F y ) )
4	3	iotabidv 5238	. . 3 |- ( ( A e. V /\ B e. W ) -> ( iota y <. A , B >.tpos F y ) = ( iota y <. B , A >. F y ) )
5		df-fv 5263	. . 3 |- (tpos F ` <. A , B >. ) = ( iota y <. A , B >.tpos F y )
6		df-fv 5263	. . 3 |- ( F ` <. B , A >. ) = ( iota y <. B , A >. F y )
7	4, 5, 6	3eqtr4g 2251	. 2 |- ( ( A e. V /\ B e. W ) -> (tpos F ` <. A , B >. ) = ( F ` <. B , A >. ) )
8		df-ov 5922	. 2 |- ( Atpos F B ) = (tpos F ` <. A , B >. )
9		df-ov 5922	. 2 |- ( B F A ) = ( F ` <. B , A >. )
10	7, 8, 9	3eqtr4g 2251	1 |- ( ( A e. V /\ B e. W ) -> ( Atpos F B ) = ( B F A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   <.cop 3622   class class class wbr 4030   iotacio 5214   ` cfv 5255  (class class class)co 5919  tpos ctpos 6299

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-tpos 6300

This theorem is referenced by:  tpossym  6331  opprmulg  13570