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Theorem ovtpos 6249
 Description: The transposition swaps the arguments in a two-argument function. When is a matrix, which is to say a function from to or some ring, tpos is the transposition of , which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
ovtpos tpos

Proof of Theorem ovtpos
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . 5
2 brtpos 6243 . . . . 5 tpos
31, 2ax-mp 8 . . . 4 tpos
43iotabii 5241 . . 3 tpos
5 df-fv 5263 . . 3 tpos tpos
6 df-fv 5263 . . 3
74, 5, 63eqtr4i 2313 . 2 tpos
8 df-ov 5861 . 2 tpos tpos
9 df-ov 5861 . 2
107, 8, 93eqtr4i 2313 1 tpos
 Colors of variables: wff set class Syntax hints:   wb 176   wceq 1623   wcel 1684  cvv 2788  cop 3643   class class class wbr 4023  cio 5217  cfv 5255  (class class class)co 5858  tpos ctpos 6233 This theorem is referenced by:  tpossym  6266  oppchom  13618  oppcco  13620  oppcmon  13641  funcoppc  13749  fulloppc  13796  fthoppc  13797  fthepi  13802  yonedalem22  14052  oppgplus  14822  oppglsm  14953  opprmul  15408  dualded  25783  dualcat2  25784 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861  df-tpos 6234
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