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Theorem ovtpos 6962
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 3109 . . . . 5  |-  y  e. 
_V
2 brtpos 6956 . . . . 5  |-  ( y  e.  _V  ->  ( <. A ,  B >.tpos  F y  <->  <. B ,  A >. F y ) )
31, 2ax-mp 5 . . . 4  |-  ( <. A ,  B >.tpos  F y  <->  <. B ,  A >. F y )
43iotabii 5556 . . 3  |-  ( iota y <. A ,  B >.tpos  F y )  =  ( iota y <. B ,  A >. F y )
5 df-fv 5578 . . 3  |-  (tpos  F `  <. A ,  B >. )  =  ( iota y <. A ,  B >.tpos  F y )
6 df-fv 5578 . . 3  |-  ( F `
 <. B ,  A >. )  =  ( iota y <. B ,  A >. F y )
74, 5, 63eqtr4i 2493 . 2  |-  (tpos  F `  <. A ,  B >. )  =  ( F `
 <. B ,  A >. )
8 df-ov 6273 . 2  |-  ( Atpos 
F B )  =  (tpos  F `  <. A ,  B >. )
9 df-ov 6273 . 2  |-  ( B F A )  =  ( F `  <. B ,  A >. )
107, 8, 93eqtr4i 2493 1  |-  ( Atpos 
F B )  =  ( B F A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398    e. wcel 1823   _Vcvv 3106   <.cop 4022   class class class wbr 4439   iotacio 5532   ` cfv 5570  (class class class)co 6270  tpos ctpos 6946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578  df-ov 6273  df-tpos 6947
This theorem is referenced by:  tpossym  6979  oppchom  15206  oppcco  15208  oppcmon  15229  funcoppc  15366  fulloppc  15413  fthoppc  15414  fthepi  15419  yonedalem22  15749  oppgplus  16586  oppglsm  16864  opprmul  17473  mamutpos  19130  mdettpos  19283  madutpos  19314
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