Metamath Proof Explorer

Theorem cnmetcoval

Description: Value of the distance function of the metric space of complex numbers, composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021)

		Ref	Expression
	Hypotheses	cnmetcoval.d	⊢ 𝐷 = ( abs ∘ − )
		cnmetcoval.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ( ℂ × ℂ ) )
		cnmetcoval.b	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
	Assertion	cnmetcoval	⊢ ( 𝜑 → ( ( 𝐷 ∘ 𝐹 ) ‘ 𝐵 ) = ( abs ‘ ( ( 1^st ‘ ( 𝐹 ‘ 𝐵 ) ) − ( 2^nd ‘ ( 𝐹 ‘ 𝐵 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cnmetcoval.d	⊢ 𝐷 = ( abs ∘ − )
2		cnmetcoval.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ( ℂ × ℂ ) )
3		cnmetcoval.b	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
4	2 3	fvovco	⊢ ( 𝜑 → ( ( 𝐷 ∘ 𝐹 ) ‘ 𝐵 ) = ( ( 1^st ‘ ( 𝐹 ‘ 𝐵 ) ) 𝐷 ( 2^nd ‘ ( 𝐹 ‘ 𝐵 ) ) ) )
5	2 3	ffvelrnd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ( ℂ × ℂ ) )
6		xp1st	⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ ( ℂ × ℂ ) → ( 1^st ‘ ( 𝐹 ‘ 𝐵 ) ) ∈ ℂ )
7	5 6	syl	⊢ ( 𝜑 → ( 1^st ‘ ( 𝐹 ‘ 𝐵 ) ) ∈ ℂ )
8		xp2nd	⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ ( ℂ × ℂ ) → ( 2^nd ‘ ( 𝐹 ‘ 𝐵 ) ) ∈ ℂ )
9	5 8	syl	⊢ ( 𝜑 → ( 2^nd ‘ ( 𝐹 ‘ 𝐵 ) ) ∈ ℂ )
10	1	cnmetdval	⊢ ( ( ( 1^st ‘ ( 𝐹 ‘ 𝐵 ) ) ∈ ℂ ∧ ( 2^nd ‘ ( 𝐹 ‘ 𝐵 ) ) ∈ ℂ ) → ( ( 1^st ‘ ( 𝐹 ‘ 𝐵 ) ) 𝐷 ( 2^nd ‘ ( 𝐹 ‘ 𝐵 ) ) ) = ( abs ‘ ( ( 1^st ‘ ( 𝐹 ‘ 𝐵 ) ) − ( 2^nd ‘ ( 𝐹 ‘ 𝐵 ) ) ) ) )
11	7 9 10	syl2anc	⊢ ( 𝜑 → ( ( 1^st ‘ ( 𝐹 ‘ 𝐵 ) ) 𝐷 ( 2^nd ‘ ( 𝐹 ‘ 𝐵 ) ) ) = ( abs ‘ ( ( 1^st ‘ ( 𝐹 ‘ 𝐵 ) ) − ( 2^nd ‘ ( 𝐹 ‘ 𝐵 ) ) ) ) )
12	4 11	eqtrd	⊢ ( 𝜑 → ( ( 𝐷 ∘ 𝐹 ) ‘ 𝐵 ) = ( abs ‘ ( ( 1^st ‘ ( 𝐹 ‘ 𝐵 ) ) − ( 2^nd ‘ ( 𝐹 ‘ 𝐵 ) ) ) ) )