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Theorem ovtposg 5796
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  V  W tpos  F  F

Proof of Theorem ovtposg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2538 . . . . 5 
_V
2 brtposg 5791 . . . . 5  V  W  _V  <. ,  >.tpos  F  <. ,  >. F
31, 2mp3an3 1206 . . . 4  V  W  <. ,  >.tpos  F  <. ,  >. F
43iotabidv 4815 . . 3  V  W  iota <. ,  >.tpos  F  iota <. ,  >. F
5 df-fv 4837 . . 3 tpos  F `  <. ,  >.  iota <. ,  >.tpos  F
6 df-fv 4837 . . 3  F `
 <. ,  >.  iota <. ,  >. F
74, 5, 63eqtr4g 2079 . 2  V  W tpos  F `  <. ,  >.  F `  <. ,  >.
8 df-ov 5439 . 2 tpos 
F tpos  F `  <. ,  >.
9 df-ov 5439 . 2  F  F `  <. ,  >.
107, 8, 93eqtr4g 2079 1  V  W tpos  F  F
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1228   wcel 1374   _Vcvv 2535   <.cop 3353   class class class wbr 3738   iotacio 4792   ` cfv 4829  (class class class)co 5436  tpos ctpos 5781
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837  df-ov 5439  df-tpos 5782
This theorem is referenced by:  tpossym  5813
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