Theorem ovtposg 5837

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.

Step	Hyp	Ref	Expression
1		vex 2557	. . . . 5 ⊢ 𝑦 ∈ V
2		brtposg 5832	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴⟩𝐹𝑦))
3	1, 2	mp3an3 1221	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴⟩𝐹𝑦))
4	3	iotabidv 4851	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (℩𝑦⟨𝐴, 𝐵⟩tpos 𝐹𝑦) = (℩𝑦⟨𝐵, 𝐴⟩𝐹𝑦))
5		df-fv 4873	. . 3 ⊢ (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑦⟨𝐴, 𝐵⟩tpos 𝐹𝑦)
6		df-fv 4873	. . 3 ⊢ (𝐹‘⟨𝐵, 𝐴⟩) = (℩𝑦⟨𝐵, 𝐴⟩𝐹𝑦)
7	4, 5, 6	3eqtr4g 2097	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐵, 𝐴⟩))
8		df-ov 5478	. 2 ⊢ (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘⟨𝐴, 𝐵⟩)
9		df-ov 5478	. 2 ⊢ (𝐵𝐹𝐴) = (𝐹‘⟨𝐵, 𝐴⟩)
10	7, 8, 9	3eqtr4g 2097	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  Vcvv 2554  ⟨cop 3375   class class class wbr 3761  ℩cio 4828  ‘cfv 4865  (class class class)co 5475  tpos ctpos 5822

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941  ax-un 4142

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-br 3762  df-opab 3816  df-mpt 3817  df-id 4027  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-fv 4873  df-ov 5478  df-tpos 5823

This theorem is referenced by:  tpossym  5854