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Theorem ovtposg 6124
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.
StepHypRef Expression
1 vex 2663 . . . . 5  |-  y  e. 
_V
2 brtposg 6119 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  y  e.  _V )  ->  ( <. A ,  B >.tpos  F y  <->  <. B ,  A >. F y ) )
31, 2mp3an3 1289 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.tpos  F y  <->  <. B ,  A >. F y ) )
43iotabidv 5079 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( iota y <. A ,  B >.tpos  F y )  =  ( iota y <. B ,  A >. F y ) )
5 df-fv 5101 . . 3  |-  (tpos  F `  <. A ,  B >. )  =  ( iota y <. A ,  B >.tpos  F y )
6 df-fv 5101 . . 3  |-  ( F `
 <. B ,  A >. )  =  ( iota y <. B ,  A >. F y )
74, 5, 63eqtr4g 2175 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  (tpos  F `  <. A ,  B >. )  =  ( F `  <. B ,  A >. ) )
8 df-ov 5745 . 2  |-  ( Atpos 
F B )  =  (tpos  F `  <. A ,  B >. )
9 df-ov 5745 . 2  |-  ( B F A )  =  ( F `  <. B ,  A >. )
107, 8, 93eqtr4g 2175 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( Atpos  F B )  =  ( B F A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   _Vcvv 2660   <.cop 3500   class class class wbr 3899   iotacio 5056   ` cfv 5093  (class class class)co 5742  tpos ctpos 6109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101  df-ov 5745  df-tpos 6110
This theorem is referenced by:  tpossym  6141
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