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Theorem ovtposg 5793
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 ((A 𝑉 B 𝑊) → (Atpos 𝐹B) = (B𝐹A))

Proof of Theorem ovtposg
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 vex 2537 . . . . 5 y V
2 brtposg 5788 . . . . 5 ((A 𝑉 B 𝑊 y V) → (⟨A, B⟩tpos 𝐹y ↔ ⟨B, A𝐹y))
31, 2mp3an3 1206 . . . 4 ((A 𝑉 B 𝑊) → (⟨A, B⟩tpos 𝐹y ↔ ⟨B, A𝐹y))
43iotabidv 4813 . . 3 ((A 𝑉 B 𝑊) → (℩yA, B⟩tpos 𝐹y) = (℩yB, A𝐹y))
5 df-fv 4835 . . 3 (tpos 𝐹‘⟨A, B⟩) = (℩yA, B⟩tpos 𝐹y)
6 df-fv 4835 . . 3 (𝐹‘⟨B, A⟩) = (℩yB, A𝐹y)
74, 5, 63eqtr4g 2080 . 2 ((A 𝑉 B 𝑊) → (tpos 𝐹‘⟨A, B⟩) = (𝐹‘⟨B, A⟩))
8 df-ov 5437 . 2 (Atpos 𝐹B) = (tpos 𝐹‘⟨A, B⟩)
9 df-ov 5437 . 2 (B𝐹A) = (𝐹‘⟨B, A⟩)
107, 8, 93eqtr4g 2080 1 ((A 𝑉 B 𝑊) → (Atpos 𝐹B) = (B𝐹A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1375  Vcvv 2534  cop 3352   class class class wbr 3737  cio 4790  cfv 4827  (class class class)co 5434  tpos ctpos 5778
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917  ax-un 4118
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-rab 2292  df-v 2536  df-sbc 2741  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-uni 3554  df-br 3738  df-opab 3792  df-mpt 3793  df-id 4003  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-res 4282  df-ima 4283  df-iota 4792  df-fun 4829  df-fn 4830  df-fv 4835  df-ov 5437  df-tpos 5779
This theorem is referenced by:  tpossym  5810
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