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Theorem ovtpos 6989
Description: The transposition swaps the arguments in a two-argument function. When is a matrix, which is to say a function from (1 )X.(1 ) to or some ring, is the transposition of , which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
ovtpos

Proof of Theorem ovtpos
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 3112 . . . . 5
2 brtpos 6983 . . . . 5
31, 2ax-mp 5 . . . 4
43iotabii 5578 . . 3
5 df-fv 5601 . . 3
6 df-fv 5601 . . 3
74, 5, 63eqtr4i 2496 . 2
8 df-ov 6299 . 2
9 df-ov 6299 . 2
107, 8, 93eqtr4i 2496 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  =wceq 1395  e.wcel 1818   cvv 3109  <.cop 4035   class class class wbr 4452  iotacio 5554  `cfv 5593  (class class class)co 6296  tposctpos 6973
This theorem is referenced by:  tpossym  7006  oppchom  15110  oppcco  15112  oppcmon  15133  funcoppc  15244  fulloppc  15291  fthoppc  15292  fthepi  15297  yonedalem22  15547  oppgplus  16384  oppglsm  16662  opprmul  17275  mamutpos  18960  mdettpos  19113  madutpos  19144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601  df-ov 6299  df-tpos 6974
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