Theorem addrfv

Description: Vector addition at a value. The operation takes each vector A and B and forms a new vector whose values are the sum of each of the values of A and B . (Contributed by Andrew Salmon, 27-Jan-2012)

	Ref	Expression
Assertion	addrfv	⊢ ( ( 𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 +𝑟 𝐵 ) ‘ 𝐶 ) = ( ( 𝐴 ‘ 𝐶 ) + ( 𝐵 ‘ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		addrval	⊢ ( ( 𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ) → ( 𝐴 +𝑟 𝐵 ) = ( 𝑥 ∈ ℝ ↦ ( ( 𝐴 ‘ 𝑥 ) + ( 𝐵 ‘ 𝑥 ) ) ) )
2	1	fveq1d	⊢ ( ( 𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ) → ( ( 𝐴 +𝑟 𝐵 ) ‘ 𝐶 ) = ( ( 𝑥 ∈ ℝ ↦ ( ( 𝐴 ‘ 𝑥 ) + ( 𝐵 ‘ 𝑥 ) ) ) ‘ 𝐶 ) )
3		fveq2	⊢ ( 𝑥 = 𝐶 → ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ 𝐶 ) )
4		fveq2	⊢ ( 𝑥 = 𝐶 → ( 𝐵 ‘ 𝑥 ) = ( 𝐵 ‘ 𝐶 ) )
5	3 4	oveq12d	⊢ ( 𝑥 = 𝐶 → ( ( 𝐴 ‘ 𝑥 ) + ( 𝐵 ‘ 𝑥 ) ) = ( ( 𝐴 ‘ 𝐶 ) + ( 𝐵 ‘ 𝐶 ) ) )
6		eqid	⊢ ( 𝑥 ∈ ℝ ↦ ( ( 𝐴 ‘ 𝑥 ) + ( 𝐵 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( ( 𝐴 ‘ 𝑥 ) + ( 𝐵 ‘ 𝑥 ) ) )
7		ovex	⊢ ( ( 𝐴 ‘ 𝐶 ) + ( 𝐵 ‘ 𝐶 ) ) ∈ V
8	5 6 7	fvmpt	⊢ ( 𝐶 ∈ ℝ → ( ( 𝑥 ∈ ℝ ↦ ( ( 𝐴 ‘ 𝑥 ) + ( 𝐵 ‘ 𝑥 ) ) ) ‘ 𝐶 ) = ( ( 𝐴 ‘ 𝐶 ) + ( 𝐵 ‘ 𝐶 ) ) )
9	2 8	sylan9eq	⊢ ( ( ( 𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 +𝑟 𝐵 ) ‘ 𝐶 ) = ( ( 𝐴 ‘ 𝐶 ) + ( 𝐵 ‘ 𝐶 ) ) )
10	9	3impa	⊢ ( ( 𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 +𝑟 𝐵 ) ‘ 𝐶 ) = ( ( 𝐴 ‘ 𝐶 ) + ( 𝐵 ‘ 𝐶 ) ) )