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Theorem ovtposg 5812
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 2554 . . . . 5 y V
2 brtposg 5807 . . . . 5 ((A 𝑉 B 𝑊 y V) → (⟨A, B⟩tpos 𝐹y ↔ ⟨B, A𝐹y))
31, 2mp3an3 1220 . . . 4 ((A 𝑉 B 𝑊) → (⟨A, B⟩tpos 𝐹y ↔ ⟨B, A𝐹y))
43iotabidv 4830 . . 3 ((A 𝑉 B 𝑊) → (℩yA, B⟩tpos 𝐹y) = (℩yB, A𝐹y))
5 df-fv 4852 . . 3 (tpos 𝐹‘⟨A, B⟩) = (℩yA, B⟩tpos 𝐹y)
6 df-fv 4852 . . 3 (𝐹‘⟨B, A⟩) = (℩yB, A𝐹y)
74, 5, 63eqtr4g 2094 . 2 ((A 𝑉 B 𝑊) → (tpos 𝐹‘⟨A, B⟩) = (𝐹‘⟨B, A⟩))
8 df-ov 5455 . 2 (Atpos 𝐹B) = (tpos 𝐹‘⟨A, B⟩)
9 df-ov 5455 . 2 (B𝐹A) = (𝐹‘⟨B, A⟩)
107, 8, 93eqtr4g 2094 1 ((A 𝑉 B 𝑊) → (Atpos 𝐹B) = (B𝐹A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551  cop 3369   class class class wbr 3754  cio 4807  cfv 4844  (class class class)co 5452  tpos ctpos 5797
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934  ax-un 4135
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-opab 3809  df-mpt 3810  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-fv 4852  df-ov 5455  df-tpos 5798
This theorem is referenced by:  tpossym  5829
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