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Theorem ovtposg 5874
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 2560 . . . . 5  |-  y  e. 
_V
2 brtposg 5869 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  y  e.  _V )  ->  ( <. A ,  B >.tpos  F y  <->  <. B ,  A >. F y ) )
31, 2mp3an3 1221 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.tpos  F y  <->  <. B ,  A >. F y ) )
43iotabidv 4888 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( iota y <. A ,  B >.tpos  F y )  =  ( iota y <. B ,  A >. F y ) )
5 df-fv 4910 . . 3  |-  (tpos  F `  <. A ,  B >. )  =  ( iota y <. A ,  B >.tpos  F y )
6 df-fv 4910 . . 3  |-  ( F `
 <. B ,  A >. )  =  ( iota y <. B ,  A >. F y )
74, 5, 63eqtr4g 2097 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  (tpos  F `  <. A ,  B >. )  =  ( F `  <. B ,  A >. ) )
8 df-ov 5515 . 2  |-  ( Atpos 
F B )  =  (tpos  F `  <. A ,  B >. )
9 df-ov 5515 . 2  |-  ( B F A )  =  ( F `  <. B ,  A >. )
107, 8, 93eqtr4g 2097 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( Atpos  F B )  =  ( B F A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   _Vcvv 2557   <.cop 3378   class class class wbr 3764   iotacio 4865   ` cfv 4902  (class class class)co 5512  tpos ctpos 5859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-ov 5515  df-tpos 5860
This theorem is referenced by:  tpossym  5891
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