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	$⊢ D = abs \circ -$
		cnmetcoval.f	$⊢ φ \to F : A ⟶ ℂ \times ℂ$
		cnmetcoval.b	$⊢ φ \to B \in A$
	Assertion	cnmetcoval	$⊢ φ \to (D \circ F) (B) = \|1^{st} (F (B)) - 2^{nd} (F (B))\|$

Proof

Step	Hyp	Ref	Expression
1		cnmetcoval.d	$⊢ D = abs \circ -$
2		cnmetcoval.f	$⊢ φ \to F : A ⟶ ℂ \times ℂ$
3		cnmetcoval.b	$⊢ φ \to B \in A$
4	2 3	fvovco	$⊢ φ \to (D \circ F) (B) = 1^{st} (F (B)) D 2^{nd} (F (B))$
5	2 3	ffvelcdmd	$⊢ φ \to F (B) \in (ℂ \times ℂ)$
6		xp1st	$⊢ F (B) \in (ℂ \times ℂ) \to 1^{st} (F (B)) \in ℂ$
7	5 6	syl	$⊢ φ \to 1^{st} (F (B)) \in ℂ$
8		xp2nd	$⊢ F (B) \in (ℂ \times ℂ) \to 2^{nd} (F (B)) \in ℂ$
9	5 8	syl	$⊢ φ \to 2^{nd} (F (B)) \in ℂ$
10	1	cnmetdval	$⊢ (1^{st} (F (B)) \in ℂ \land 2^{nd} (F (B)) \in ℂ) \to 1^{st} (F (B)) D 2^{nd} (F (B)) = \|1^{st} (F (B)) - 2^{nd} (F (B))\|$
11	7 9 10	syl2anc	$⊢ φ \to 1^{st} (F (B)) D 2^{nd} (F (B)) = \|1^{st} (F (B)) - 2^{nd} (F (B))\|$
12	4 11	eqtrd	$⊢ φ \to (D \circ F) (B) = \|1^{st} (F (B)) - 2^{nd} (F (B))\|$