hoicoto2

Description: The half-open interval expressed using a composition of a function into ( RR X. RR ) and using two distinct real-valued functions for the borders. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Hypotheses	hoicoto2.i	$⊢ φ \to I : X ⟶ ℝ^{2}$
	hoicoto2.a	$⊢ A = (k \in X ⟼ 1^{st} (I (k)))$
	hoicoto2.b	$⊢ B = (k \in X ⟼ 2^{nd} (I (k)))$
Assertion	hoicoto2	$⊢ φ \to \underset{k \in X}{⨉} ([.) \circ I) (k) = \underset{k \in X}{⨉} [A (k), B (k))$

Proof

Step	Hyp	Ref	Expression
1		hoicoto2.i	$⊢ φ \to I : X ⟶ ℝ^{2}$
2		hoicoto2.a	$⊢ A = (k \in X ⟼ 1^{st} (I (k)))$
3		hoicoto2.b	$⊢ B = (k \in X ⟼ 2^{nd} (I (k)))$
4	1	adantr	$⊢ (φ \land k \in X) \to I : X ⟶ ℝ^{2}$
5		simpr	$⊢ (φ \land k \in X) \to k \in X$
6	4 5	fvovco	$⊢ (φ \land k \in X) \to ([.) \circ I) (k) = [1^{st} (I (k)), 2^{nd} (I (k)))$
7	1	ffvelcdmda	$⊢ (φ \land k \in X) \to I (k) \in ℝ^{2}$
8		xp1st	$⊢ I (k) \in ℝ^{2} \to 1^{st} (I (k)) \in ℝ$
9	7 8	syl	$⊢ (φ \land k \in X) \to 1^{st} (I (k)) \in ℝ$
10	9	elexd	$⊢ (φ \land k \in X) \to 1^{st} (I (k)) \in V$
11	2	fvmpt2	$⊢ (k \in X \land 1^{st} (I (k)) \in V) \to A (k) = 1^{st} (I (k))$
12	5 10 11	syl2anc	$⊢ (φ \land k \in X) \to A (k) = 1^{st} (I (k))$
13	12	eqcomd	$⊢ (φ \land k \in X) \to 1^{st} (I (k)) = A (k)$
14		xp2nd	$⊢ I (k) \in ℝ^{2} \to 2^{nd} (I (k)) \in ℝ$
15	7 14	syl	$⊢ (φ \land k \in X) \to 2^{nd} (I (k)) \in ℝ$
16	15	elexd	$⊢ (φ \land k \in X) \to 2^{nd} (I (k)) \in V$
17	3	fvmpt2	$⊢ (k \in X \land 2^{nd} (I (k)) \in V) \to B (k) = 2^{nd} (I (k))$
18	5 16 17	syl2anc	$⊢ (φ \land k \in X) \to B (k) = 2^{nd} (I (k))$
19	18	eqcomd	$⊢ (φ \land k \in X) \to 2^{nd} (I (k)) = B (k)$
20	13 19	oveq12d	$⊢ (φ \land k \in X) \to [1^{st} (I (k)), 2^{nd} (I (k))) = [A (k), B (k))$
21	6 20	eqtrd	$⊢ (φ \land k \in X) \to ([.) \circ I) (k) = [A (k), B (k))$
22	21	ixpeq2dva	$⊢ φ \to \underset{k \in X}{⨉} ([.) \circ I) (k) = \underset{k \in X}{⨉} [A (k), B (k))$

Metamath Proof Explorer

Theorem hoicoto2

Proof