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Theorem ovtpos 7898
Description: The transposition swaps the arguments in a two-argument function. When 𝐹 is a matrix, which is to say a function from (1...𝑚) × (1...𝑛) to or some ring, tpos 𝐹 is the transposition of 𝐹, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
ovtpos (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴)

Proof of Theorem ovtpos
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 brtpos 7892 . . . . 5 (𝑦 ∈ V → (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴𝐹𝑦))
21elv 3505 . . . 4 (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴𝐹𝑦)
32iotabii 6338 . . 3 (℩𝑦𝐴, 𝐵⟩tpos 𝐹𝑦) = (℩𝑦𝐵, 𝐴𝐹𝑦)
4 df-fv 6360 . . 3 (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑦𝐴, 𝐵⟩tpos 𝐹𝑦)
5 df-fv 6360 . . 3 (𝐹‘⟨𝐵, 𝐴⟩) = (℩𝑦𝐵, 𝐴𝐹𝑦)
63, 4, 53eqtr4i 2859 . 2 (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐵, 𝐴⟩)
7 df-ov 7151 . 2 (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘⟨𝐴, 𝐵⟩)
8 df-ov 7151 . 2 (𝐵𝐹𝐴) = (𝐹‘⟨𝐵, 𝐴⟩)
96, 7, 83eqtr4i 2859 1 (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1530  Vcvv 3500  cop 4570   class class class wbr 5063  cio 6310  cfv 6352  (class class class)co 7148  tpos ctpos 7882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-fv 6360  df-ov 7151  df-tpos 7883
This theorem is referenced by:  tpossym  7915  oppchom  16975  oppcco  16977  oppcmon  16998  funcoppc  17135  fulloppc  17182  fthoppc  17183  fthepi  17188  yonedalem22  17518  oppgplus  18407  oppglsm  18687  opprmul  19296  mamutpos  20983  mdettpos  21136  madutpos  21167  mdetpmtr2  30975
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