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Theorem ovtpos 6967
 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 3116 . . . . 5
2 brtpos 6961 . . . . 5 tpos
31, 2ax-mp 5 . . . 4 tpos
43iotabii 5571 . . 3 tpos
5 df-fv 5594 . . 3 tpos tpos
6 df-fv 5594 . . 3
74, 5, 63eqtr4i 2506 . 2 tpos
8 df-ov 6285 . 2 tpos tpos
9 df-ov 6285 . 2
107, 8, 93eqtr4i 2506 1 tpos
 Colors of variables: wff setvar class Syntax hints:   wb 184   wceq 1379   wcel 1767  cvv 3113  cop 4033   class class class wbr 4447  cio 5547  cfv 5586  (class class class)co 6282  tpos ctpos 6951 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-ov 6285  df-tpos 6952 This theorem is referenced by:  tpossym  6984  oppchom  14967  oppcco  14969  oppcmon  14990  funcoppc  15098  fulloppc  15145  fthoppc  15146  fthepi  15151  yonedalem22  15401  oppgplus  16179  oppglsm  16458  opprmul  17059  mamutpos  18727  mdettpos  18880  madutpos  18911
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