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Metamath Proof Explorer

Theorem ovtpos

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 ) X. ( 1 ... n ) to RR or some ring, tpos F is the transposition of F , which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015)

Ref Expression
Assertion ovtpos ( 𝐴 tpos 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 )

Proof

Step Hyp Ref Expression
1 brtpos ⊢ ( 𝑦 ∈ V → ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝑦 ↔ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝑦 ) )
2 1 elv ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝑦 ↔ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝑦 )
3 2 iotabii ⊢ ( ℩ 𝑦 ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝑦 ) = ( ℩ 𝑦 ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝑦 )
4 df-fv ⊢ ( tpos 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ℩ 𝑦 ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝑦 )
5 df-fv ⊢ ( 𝐹 ‘ ⟨ 𝐵 , 𝐴 ⟩ ) = ( ℩ 𝑦 ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝑦 )
6 3 4 5 3eqtr4i ⊢ ( tpos 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐹 ‘ ⟨ 𝐵 , 𝐴 ⟩ )
7 df-ov ⊢ ( 𝐴 tpos 𝐹 𝐵 ) = ( tpos 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ )
8 df-ov ⊢ ( 𝐵 𝐹 𝐴 ) = ( 𝐹 ‘ ⟨ 𝐵 , 𝐴 ⟩ )
9 6 7 8 3eqtr4i ⊢ ( 𝐴 tpos 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 )