Theorem s3co

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

	Ref	Expression
Hypotheses	s2co.1	$$⊢\phi \to F:X⟶Y$$
	s2co.2	$$⊢\phi \to A\in X$$
	s2co.3	$$⊢\phi \to B\in X$$
	s3co.4	$$⊢\phi \to C\in X$$
Assertion	s3co	$$⊢\phi \to F\circ ⟨“ABC”⟩=⟨“F\left(A\right)F\left(B\right)F\left(C\right)”⟩$$

Proof

Step	Hyp	Ref	Expression
1		s2co.1	$$⊢\phi \to F:X⟶Y$$
2		s2co.2	$$⊢\phi \to A\in X$$
3		s2co.3	$$⊢\phi \to B\in X$$
4		s3co.4	$$⊢\phi \to C\in X$$
5		df-s3	$$⊢⟨“ABC”⟩=⟨“AB”⟩\mathrm{++}⟨“C”⟩$$
6	2 3	s2cld	$$⊢\phi \to ⟨“AB”⟩\in \mathrm{Word}X$$
7	1 2 3	s2co	$$⊢\phi \to F\circ ⟨“AB”⟩=⟨“F\left(A\right)F\left(B\right)”⟩$$
8		df-s3	$$⊢⟨“F\left(A\right)F\left(B\right)F\left(C\right)”⟩=⟨“F\left(A\right)F\left(B\right)”⟩\mathrm{++}⟨“F\left(C\right)”⟩$$
9	5 6 4 1 7 8	cats1co	$$⊢\phi \to F\circ ⟨“ABC”⟩=⟨“F\left(A\right)F\left(B\right)F\left(C\right)”⟩$$