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Theorem ovtpos 6967
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 3116 . . . . 5  |-  y  e. 
_V
2 brtpos 6961 . . . . 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 5571 . . 3  |-  ( iota y <. A ,  B >.tpos  F y )  =  ( iota y <. B ,  A >. F y )
5 df-fv 5594 . . 3  |-  (tpos  F `  <. A ,  B >. )  =  ( iota y <. A ,  B >.tpos  F y )
6 df-fv 5594 . . 3  |-  ( F `
 <. B ,  A >. )  =  ( iota y <. B ,  A >. F y )
74, 5, 63eqtr4i 2506 . 2  |-  (tpos  F `  <. A ,  B >. )  =  ( F `
 <. B ,  A >. )
8 df-ov 6285 . 2  |-  ( Atpos 
F B )  =  (tpos  F `  <. A ,  B >. )
9 df-ov 6285 . 2  |-  ( B F A )  =  ( F `  <. B ,  A >. )
107, 8, 93eqtr4i 2506 1  |-  ( Atpos 
F B )  =  ( B F A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033   class class class wbr 4447   iotacio 5547   ` cfv 5586  (class class class)co 6282  tpos ctpos 6951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-ov 6285  df-tpos 6952
This theorem is referenced by:  tpossym  6984  oppchom  14967  oppcco  14969  oppcmon  14990  funcoppc  15098  fulloppc  15145  fthoppc  15146  fthepi  15151  yonedalem22  15401  oppgplus  16179  oppglsm  16458  opprmul  17059  mamutpos  18727  mdettpos  18880  madutpos  18911
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