Theorem ovtposg 5815

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 2554	. . . . 5 ⊢ y ∈ V
2		brtposg 5810	. . . . 5 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ y ∈ V) → (⟨A, B⟩tpos 𝐹y ↔ ⟨B, A⟩𝐹y))
3	1, 2	mp3an3 1220	. . . 4 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (⟨A, B⟩tpos 𝐹y ↔ ⟨B, A⟩𝐹y))
4	3	iotabidv 4831	. . 3 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (℩y⟨A, B⟩tpos 𝐹y) = (℩y⟨B, A⟩𝐹y))
5		df-fv 4853	. . 3 ⊢ (tpos 𝐹‘⟨A, B⟩) = (℩y⟨A, B⟩tpos 𝐹y)
6		df-fv 4853	. . 3 ⊢ (𝐹‘⟨B, A⟩) = (℩y⟨B, A⟩𝐹y)
7	4, 5, 6	3eqtr4g 2094	. 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (tpos 𝐹‘⟨A, B⟩) = (𝐹‘⟨B, A⟩))
8		df-ov 5458	. 2 ⊢ (Atpos 𝐹B) = (tpos 𝐹‘⟨A, B⟩)
9		df-ov 5458	. 2 ⊢ (B𝐹A) = (𝐹‘⟨B, A⟩)
10	7, 8, 9	3eqtr4g 2094	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (Atpos 𝐹B) = (B𝐹A))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370   class class class wbr 3755  ℩cio 4808  ‘cfv 4845  (class class class)co 5455  tpos ctpos 5800

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-ov 5458  df-tpos 5801

This theorem is referenced by:  tpossym  5832