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Theorem ovtposg 5978
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 2618 . . . . 5  |-  y  e. 
_V
2 brtposg 5973 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  y  e.  _V )  ->  ( <. A ,  B >.tpos  F y  <->  <. B ,  A >. F y ) )
31, 2mp3an3 1260 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.tpos  F y  <->  <. B ,  A >. F y ) )
43iotabidv 4967 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( iota y <. A ,  B >.tpos  F y )  =  ( iota y <. B ,  A >. F y ) )
5 df-fv 4989 . . 3  |-  (tpos  F `  <. A ,  B >. )  =  ( iota y <. A ,  B >.tpos  F y )
6 df-fv 4989 . . 3  |-  ( F `
 <. B ,  A >. )  =  ( iota y <. B ,  A >. F y )
74, 5, 63eqtr4g 2142 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  (tpos  F `  <. A ,  B >. )  =  ( F `  <. B ,  A >. ) )
8 df-ov 5616 . 2  |-  ( Atpos 
F B )  =  (tpos  F `  <. A ,  B >. )
9 df-ov 5616 . 2  |-  ( B F A )  =  ( F `  <. B ,  A >. )
107, 8, 93eqtr4g 2142 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( Atpos  F B )  =  ( B F A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436   _Vcvv 2615   <.cop 3434   class class class wbr 3820   iotacio 4944   ` cfv 4981  (class class class)co 5613  tpos ctpos 5963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-fv 4989  df-ov 5616  df-tpos 5964
This theorem is referenced by:  tpossym  5995
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