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Theorem ovtposg 5908
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 the reals 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
ovtposg ((𝐴𝑉𝐵𝑊) → (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴))

Proof of Theorem ovtposg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 2605 . . . . 5 𝑦 ∈ V
2 brtposg 5903 . . . . 5 ((𝐴𝑉𝐵𝑊𝑦 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴𝐹𝑦))
31, 2mp3an3 1258 . . . 4 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴𝐹𝑦))
43iotabidv 4918 . . 3 ((𝐴𝑉𝐵𝑊) → (℩𝑦𝐴, 𝐵⟩tpos 𝐹𝑦) = (℩𝑦𝐵, 𝐴𝐹𝑦))
5 df-fv 4940 . . 3 (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑦𝐴, 𝐵⟩tpos 𝐹𝑦)
6 df-fv 4940 . . 3 (𝐹‘⟨𝐵, 𝐴⟩) = (℩𝑦𝐵, 𝐴𝐹𝑦)
74, 5, 63eqtr4g 2139 . 2 ((𝐴𝑉𝐵𝑊) → (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐵, 𝐴⟩))
8 df-ov 5546 . 2 (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘⟨𝐴, 𝐵⟩)
9 df-ov 5546 . 2 (𝐵𝐹𝐴) = (𝐹‘⟨𝐵, 𝐴⟩)
107, 8, 93eqtr4g 2139 1 ((𝐴𝑉𝐵𝑊) → (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  Vcvv 2602  cop 3409   class class class wbr 3793  cio 4895  cfv 4932  (class class class)co 5543  tpos ctpos 5893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940  df-ov 5546  df-tpos 5894
This theorem is referenced by:  tpossym  5925
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