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Metamath Proof Explorer

Theorem dvhvaddcbv

Description: Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013)

		Ref	Expression
	Hypothesis	dvhvaddval.a	⊢ + = ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) ⟩ )
	Assertion	dvhvaddcbv	⊢ + = ( ℎ ∈ ( 𝑇 × 𝐸 ) , 𝑖 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ ℎ ) ∘ ( 1^st ‘ 𝑖 ) ) , ( ( 2^nd ‘ ℎ ) ⨣ ( 2^nd ‘ 𝑖 ) ) ⟩ )

Proof

Step	Hyp	Ref	Expression
1		dvhvaddval.a	⊢ + = ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) ⟩ )
2		fveq2	⊢ ( 𝑓 = ℎ → ( 1^st ‘ 𝑓 ) = ( 1^st ‘ ℎ ) )
3	2	coeq1d	⊢ ( 𝑓 = ℎ → ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) = ( ( 1^st ‘ ℎ ) ∘ ( 1^st ‘ 𝑔 ) ) )
4		fveq2	⊢ ( 𝑓 = ℎ → ( 2^nd ‘ 𝑓 ) = ( 2^nd ‘ ℎ ) )
5	4	oveq1d	⊢ ( 𝑓 = ℎ → ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) = ( ( 2^nd ‘ ℎ ) ⨣ ( 2^nd ‘ 𝑔 ) ) )
6	3 5	opeq12d	⊢ ( 𝑓 = ℎ → ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) ⟩ = ⟨ ( ( 1^st ‘ ℎ ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ( 2^nd ‘ ℎ ) ⨣ ( 2^nd ‘ 𝑔 ) ) ⟩ )
7		fveq2	⊢ ( 𝑔 = 𝑖 → ( 1^st ‘ 𝑔 ) = ( 1^st ‘ 𝑖 ) )
8	7	coeq2d	⊢ ( 𝑔 = 𝑖 → ( ( 1^st ‘ ℎ ) ∘ ( 1^st ‘ 𝑔 ) ) = ( ( 1^st ‘ ℎ ) ∘ ( 1^st ‘ 𝑖 ) ) )
9		fveq2	⊢ ( 𝑔 = 𝑖 → ( 2^nd ‘ 𝑔 ) = ( 2^nd ‘ 𝑖 ) )
10	9	oveq2d	⊢ ( 𝑔 = 𝑖 → ( ( 2^nd ‘ ℎ ) ⨣ ( 2^nd ‘ 𝑔 ) ) = ( ( 2^nd ‘ ℎ ) ⨣ ( 2^nd ‘ 𝑖 ) ) )
11	8 10	opeq12d	⊢ ( 𝑔 = 𝑖 → ⟨ ( ( 1^st ‘ ℎ ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ( 2^nd ‘ ℎ ) ⨣ ( 2^nd ‘ 𝑔 ) ) ⟩ = ⟨ ( ( 1^st ‘ ℎ ) ∘ ( 1^st ‘ 𝑖 ) ) , ( ( 2^nd ‘ ℎ ) ⨣ ( 2^nd ‘ 𝑖 ) ) ⟩ )
12	6 11	cbvmpov	⊢ ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) ⟩ ) = ( ℎ ∈ ( 𝑇 × 𝐸 ) , 𝑖 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ ℎ ) ∘ ( 1^st ‘ 𝑖 ) ) , ( ( 2^nd ‘ ℎ ) ⨣ ( 2^nd ‘ 𝑖 ) ) ⟩ )
13	1 12	eqtri	⊢ + = ( ℎ ∈ ( 𝑇 × 𝐸 ) , 𝑖 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ ℎ ) ∘ ( 1^st ‘ 𝑖 ) ) , ( ( 2^nd ‘ ℎ ) ⨣ ( 2^nd ‘ 𝑖 ) ) ⟩ )