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Theorem ovtpos 6265
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.)
Assertion
Ref Expression
ovtpos  |-  ( Atpos 
F B )  =  ( B F A )

Proof of Theorem ovtpos
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2804 . . . . 5  |-  y  e. 
_V
2 brtpos 6259 . . . . 5  |-  ( y  e.  _V  ->  ( <. A ,  B >.tpos  F y  <->  <. B ,  A >. F y ) )
31, 2ax-mp 8 . . . 4  |-  ( <. A ,  B >.tpos  F y  <->  <. B ,  A >. F y )
43iotabii 5257 . . 3  |-  ( iota y <. A ,  B >.tpos  F y )  =  ( iota y <. B ,  A >. F y )
5 df-fv 5279 . . 3  |-  (tpos  F `  <. A ,  B >. )  =  ( iota y <. A ,  B >.tpos  F y )
6 df-fv 5279 . . 3  |-  ( F `
 <. B ,  A >. )  =  ( iota y <. B ,  A >. F y )
74, 5, 63eqtr4i 2326 . 2  |-  (tpos  F `  <. A ,  B >. )  =  ( F `
 <. B ,  A >. )
8 df-ov 5877 . 2  |-  ( Atpos 
F B )  =  (tpos  F `  <. A ,  B >. )
9 df-ov 5877 . 2  |-  ( B F A )  =  ( F `  <. B ,  A >. )
107, 8, 93eqtr4i 2326 1  |-  ( Atpos 
F B )  =  ( B F A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039   iotacio 5233   ` cfv 5271  (class class class)co 5874  tpos ctpos 6249
This theorem is referenced by:  tpossym  6282  oppchom  13634  oppcco  13636  oppcmon  13657  funcoppc  13765  fulloppc  13812  fthoppc  13813  fthepi  13818  yonedalem22  14068  oppgplus  14838  oppglsm  14969  opprmul  15424  dualded  25886  dualcat2  25887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5877  df-tpos 6250
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