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Theorem ovtposg 5762
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 2536 . . . . 5 y V
2 brtposg 5757 . . . . 5 ((A 𝑉 B 𝑊 y V) → (⟨A, B⟩tpos 𝐹y ↔ ⟨B, A𝐹y))
31, 2mp3an3 1208 . . . 4 ((A 𝑉 B 𝑊) → (⟨A, B⟩tpos 𝐹y ↔ ⟨B, A𝐹y))
43iotabidv 4782 . . 3 ((A 𝑉 B 𝑊) → (℩yA, B⟩tpos 𝐹y) = (℩yB, A𝐹y))
5 df-fv 4804 . . 3 (tpos 𝐹‘⟨A, B⟩) = (℩yA, B⟩tpos 𝐹y)
6 df-fv 4804 . . 3 (𝐹‘⟨B, A⟩) = (℩yB, A𝐹y)
74, 5, 63eqtr4g 2079 . 2 ((A 𝑉 B 𝑊) → (tpos 𝐹‘⟨A, B⟩) = (𝐹‘⟨B, A⟩))
8 df-ov 5408 . 2 (Atpos 𝐹B) = (tpos 𝐹‘⟨A, B⟩)
9 df-ov 5408 . 2 (B𝐹A) = (𝐹‘⟨B, A⟩)
107, 8, 93eqtr4g 2079 1 ((A 𝑉 B 𝑊) → (Atpos 𝐹B) = (B𝐹A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1373   wcel 1375  Vcvv 2533  cop 3330   class class class wbr 3716  cio 4759  cfv 4796  (class class class)co 5405  tpos ctpos 5747
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896  ax-un 4093
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-rab 2291  df-v 2535  df-sbc 2740  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-br 3717  df-opab 3771  df-mpt 3772  df-id 3983  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-res 4250  df-ima 4251  df-iota 4761  df-fun 4798  df-fn 4799  df-fv 4804  df-ov 5408  df-tpos 5748
This theorem is referenced by:  tpossym  5779
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