Theorem s3co

Description: Mapping a length 3 string by a function. (Contributed by Mario Carneiro, 27-Feb-2016)

	Ref	Expression
Hypotheses	s2co.1	⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ 𝑌 )
	s2co.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
	s2co.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
	s3co.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑋 )
Assertion	s3co	⊢ ( 𝜑 → ( 𝐹 ∘ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ) = ⟨“ ( 𝐹 ‘ 𝐴 ) ( 𝐹 ‘ 𝐵 ) ( 𝐹 ‘ 𝐶 ) ”⟩ )

Proof

Step	Hyp	Ref	Expression
1		s2co.1	⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ 𝑌 )
2		s2co.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
3		s2co.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
4		s3co.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑋 )
5		df-s3	⊢ ⟨“ 𝐴 𝐵 𝐶 ”⟩ = ( ⟨“ 𝐴 𝐵 ”⟩ ++ ⟨“ 𝐶 ”⟩ )
6	2 3	s2cld	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 ”⟩ ∈ Word 𝑋 )
7	1 2 3	s2co	⊢ ( 𝜑 → ( 𝐹 ∘ ⟨“ 𝐴 𝐵 ”⟩ ) = ⟨“ ( 𝐹 ‘ 𝐴 ) ( 𝐹 ‘ 𝐵 ) ”⟩ )
8		df-s3	⊢ ⟨“ ( 𝐹 ‘ 𝐴 ) ( 𝐹 ‘ 𝐵 ) ( 𝐹 ‘ 𝐶 ) ”⟩ = ( ⟨“ ( 𝐹 ‘ 𝐴 ) ( 𝐹 ‘ 𝐵 ) ”⟩ ++ ⟨“ ( 𝐹 ‘ 𝐶 ) ”⟩ )
9	5 6 4 1 7 8	cats1co	⊢ ( 𝜑 → ( 𝐹 ∘ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ) = ⟨“ ( 𝐹 ‘ 𝐴 ) ( 𝐹 ‘ 𝐵 ) ( 𝐹 ‘ 𝐶 ) ”⟩ )