Theorem ovtpos 8166

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 8160	. . . . 5 ⊢ (𝑦 ∈ V → (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴⟩𝐹𝑦))
2	1	elv 3441	. . . 4 ⊢ (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴⟩𝐹𝑦)
3	2	iotabii 6461	. . 3 ⊢ (℩𝑦⟨𝐴, 𝐵⟩tpos 𝐹𝑦) = (℩𝑦⟨𝐵, 𝐴⟩𝐹𝑦)
4		df-fv 6484	. . 3 ⊢ (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑦⟨𝐴, 𝐵⟩tpos 𝐹𝑦)
5		df-fv 6484	. . 3 ⊢ (𝐹‘⟨𝐵, 𝐴⟩) = (℩𝑦⟨𝐵, 𝐴⟩𝐹𝑦)
6	3, 4, 5	3eqtr4i 2764	. 2 ⊢ (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐵, 𝐴⟩)
7		df-ov 7344	. 2 ⊢ (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘⟨𝐴, 𝐵⟩)
8		df-ov 7344	. 2 ⊢ (𝐵𝐹𝐴) = (𝐹‘⟨𝐵, 𝐴⟩)
9	6, 7, 8	3eqtr4i 2764	1 ⊢ (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴)

Syntax hints:   ↔ wb 206   = wceq 1541  Vcvv 3436  ⟨cop 4577   class class class wbr 5086  ℩cio 6430  ‘cfv 6476  (class class class)co 7341  tpos ctpos 8150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-fv 6484  df-ov 7344  df-tpos 8151

This theorem is referenced by:  tpossym  8183  oppchom  17616  oppcco  17618  oppcmon  17640  funcoppc  17777  fulloppc  17826  fthoppc  17827  fthepi  17832  yonedalem22  18179  oppgplus  19256  oppglsm  19549  opprmul  20253  mamutpos  22368  mdettpos  22521  madutpos  22552  mdetpmtr2  33829  tposid  48916  tposidres  48917  tposideq  48919  oppf2  49172  cofuoppf  49182  oppcup  49239  natoppf  49261  opf12  49436