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Theorem ovtpos 6494
 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 2959 . . . . 5
2 brtpos 6488 . . . . 5 tpos
31, 2ax-mp 8 . . . 4 tpos
43iotabii 5440 . . 3 tpos
5 df-fv 5462 . . 3 tpos tpos
6 df-fv 5462 . . 3
74, 5, 63eqtr4i 2466 . 2 tpos
8 df-ov 6084 . 2 tpos tpos
9 df-ov 6084 . 2
107, 8, 93eqtr4i 2466 1 tpos
 Colors of variables: wff set class Syntax hints:   wb 177   wceq 1652   wcel 1725  cvv 2956  cop 3817   class class class wbr 4212  cio 5416  cfv 5454  (class class class)co 6081  tpos ctpos 6478 This theorem is referenced by:  tpossym  6511  oppchom  13941  oppcco  13943  oppcmon  13964  funcoppc  14072  fulloppc  14119  fthoppc  14120  fthepi  14125  yonedalem22  14375  oppgplus  15145  oppglsm  15276  opprmul  15731 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-ov 6084  df-tpos 6479
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