Theorem ovtpos 8251

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 ℝ 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.

Step	Hyp	Ref	Expression
1		brtpos 8245	. . . . 5 ⊢ (𝑦 ∈ V → (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴⟩𝐹𝑦))
2	1	elv 3456	. . . 4 ⊢ (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴⟩𝐹𝑦)
3	2	iotabii 6522	. . 3 ⊢ (℩𝑦⟨𝐴, 𝐵⟩tpos 𝐹𝑦) = (℩𝑦⟨𝐵, 𝐴⟩𝐹𝑦)
4		df-fv 6545	. . 3 ⊢ (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑦⟨𝐴, 𝐵⟩tpos 𝐹𝑦)
5		df-fv 6545	. . 3 ⊢ (𝐹‘⟨𝐵, 𝐴⟩) = (℩𝑦⟨𝐵, 𝐴⟩𝐹𝑦)
6	3, 4, 5	3eqtr4i 2794	. 2 ⊢ (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐵, 𝐴⟩)
7		df-ov 7421	. 2 ⊢ (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘⟨𝐴, 𝐵⟩)
8		df-ov 7421	. 2 ⊢ (𝐵𝐹𝐴) = (𝐹‘⟨𝐵, 𝐴⟩)
9	6, 7, 8	3eqtr4i 2794	1 ⊢ (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  Vcvv 3451  ⟨cop 4590   class class class wbr 5103  ℩cio 6491  ‘cfv 6537  (class class class)co 7418  tpos ctpos 8235

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-ov 7421  df-tpos 8236

This theorem is used by:  tpossym  8268  oppchom  17882  oppcco  17884  oppcmon  17906  funcoppc  18043  fulloppc  18092  fthoppc  18093  fthepi  18098  yonedalem22  18445  oppgplus  19556  oppglsm  19849  opprmul  20563  mamutpos  22766  mdettpos  22919  madutpos  22950  mdetpmtr2  34449  tposid  49962  tposidres  49963  tposideq  49965  oppf2  50217  cofuoppf  50227  oppcup  50284  natoppf  50306  opf12  50481