Theorem ovtpos 8181

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 8175	. . . . 5 ⊢ (𝑦 ∈ V → (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴⟩𝐹𝑦))
2	1	elv 3443	. . . 4 ⊢ (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴⟩𝐹𝑦)
3	2	iotabii 6471	. . 3 ⊢ (℩𝑦⟨𝐴, 𝐵⟩tpos 𝐹𝑦) = (℩𝑦⟨𝐵, 𝐴⟩𝐹𝑦)
4		df-fv 6494	. . 3 ⊢ (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑦⟨𝐴, 𝐵⟩tpos 𝐹𝑦)
5		df-fv 6494	. . 3 ⊢ (𝐹‘⟨𝐵, 𝐴⟩) = (℩𝑦⟨𝐵, 𝐴⟩𝐹𝑦)
6	3, 4, 5	3eqtr4i 2762	. 2 ⊢ (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐵, 𝐴⟩)
7		df-ov 7356	. 2 ⊢ (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘⟨𝐴, 𝐵⟩)
8		df-ov 7356	. 2 ⊢ (𝐵𝐹𝐴) = (𝐹‘⟨𝐵, 𝐴⟩)
9	6, 7, 8	3eqtr4i 2762	1 ⊢ (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴)

Syntax hints:   ↔ wb 206   = wceq 1540  Vcvv 3438  ⟨cop 4585   class class class wbr 5095  ℩cio 6440  ‘cfv 6486  (class class class)co 7353  tpos ctpos 8165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494  df-ov 7356  df-tpos 8166

This theorem is referenced by:  tpossym  8198  oppchom  17640  oppcco  17642  oppcmon  17664  funcoppc  17801  fulloppc  17850  fthoppc  17851  fthepi  17856  yonedalem22  18203  oppgplus  19247  oppglsm  19540  opprmul  20244  mamutpos  22362  mdettpos  22515  madutpos  22546  mdetpmtr2  33810  tposid  48889  tposidres  48890  tposideq  48892  oppf2  49145  cofuoppf  49155  oppcup  49212  natoppf  49234  opf12  49409