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Metamath Proof Explorer

Theorem comfval2

Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017)

Ref Expression
Hypotheses comfffval2.o ⊢ 𝑂 = ( compf ‘ 𝐶 )
comfffval2.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
comfffval2.h ⊢ 𝐻 = ( Homf ‘ 𝐶 )
comfffval2.x ⊢ · = ( comp ‘ 𝐶 )
comffval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
comffval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
comffval2.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
comfval2.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
comfval2.g ⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 𝐻 𝑍 ) )
Assertion comfval2 ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) 𝐹 ) = ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) )

Proof

Step Hyp Ref Expression
1 comfffval2.o ⊢ 𝑂 = ( compf ‘ 𝐶 )
2 comfffval2.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 comfffval2.h ⊢ 𝐻 = ( Homf ‘ 𝐶 )
4 comfffval2.x ⊢ · = ( comp ‘ 𝐶 )
5 comffval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 comffval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 comffval2.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
8 comfval2.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
9 comfval2.g ⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 𝐻 𝑍 ) )
10 eqid ⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
11 3 2 10 5 6 homfval ⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
12 8 11 eleqtrd ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
13 3 2 10 6 7 homfval ⊢ ( 𝜑 → ( 𝑌 𝐻 𝑍 ) = ( 𝑌 ( Hom ‘ 𝐶 ) 𝑍 ) )
14 9 13 eleqtrd ⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑍 ) )
15 1 2 10 4 5 6 7 12 14 comfval ⊢ ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) 𝐹 ) = ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) )