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Theorem ovtposg 5874
 Description: The transposition swaps the arguments in a two-argument function. When is a matrix, which is to say a function from ( 1 ... m ) ( 1 ... n ) to the reals 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
ovtposg tpos

Proof of Theorem ovtposg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2560 . . . . 5
2 brtposg 5869 . . . . 5 tpos
31, 2mp3an3 1221 . . . 4 tpos
43iotabidv 4888 . . 3 tpos
5 df-fv 4910 . . 3 tpos tpos
6 df-fv 4910 . . 3
74, 5, 63eqtr4g 2097 . 2 tpos
8 df-ov 5515 . 2 tpos tpos
9 df-ov 5515 . 2
107, 8, 93eqtr4g 2097 1 tpos
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243   wcel 1393  cvv 2557  cop 3378   class class class wbr 3764  cio 4865  cfv 4902  (class class class)co 5512  tpos ctpos 5859 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-ov 5515  df-tpos 5860 This theorem is referenced by:  tpossym  5891
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