MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovtpos Structured version   Visualization version   GIF version

Theorem ovtpos 7412
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 vex 3234 . . . . 5 𝑦 ∈ V
2 brtpos 7406 . . . . 5 (𝑦 ∈ V → (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴𝐹𝑦))
31, 2ax-mp 5 . . . 4 (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴𝐹𝑦)
43iotabii 5911 . . 3 (℩𝑦𝐴, 𝐵⟩tpos 𝐹𝑦) = (℩𝑦𝐵, 𝐴𝐹𝑦)
5 df-fv 5934 . . 3 (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑦𝐴, 𝐵⟩tpos 𝐹𝑦)
6 df-fv 5934 . . 3 (𝐹‘⟨𝐵, 𝐴⟩) = (℩𝑦𝐵, 𝐴𝐹𝑦)
74, 5, 63eqtr4i 2683 . 2 (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐵, 𝐴⟩)
8 df-ov 6693 . 2 (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘⟨𝐴, 𝐵⟩)
9 df-ov 6693 . 2 (𝐵𝐹𝐴) = (𝐹‘⟨𝐵, 𝐴⟩)
107, 8, 93eqtr4i 2683 1 (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wcel 2030  Vcvv 3231  cop 4216   class class class wbr 4685  cio 5887  cfv 5926  (class class class)co 6690  tpos ctpos 7396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-ov 6693  df-tpos 7397
This theorem is referenced by:  tpossym  7429  oppchom  16422  oppcco  16424  oppcmon  16445  funcoppc  16582  fulloppc  16629  fthoppc  16630  fthepi  16635  yonedalem22  16965  oppgplus  17825  oppglsm  18103  opprmul  18672  mamutpos  20312  mdettpos  20465  madutpos  20496  mdetpmtr2  30018
  Copyright terms: Public domain W3C validator