Theorem icoreclin 37612

Description: The set of closed-below, open-above intervals of reals is closed under finite intersection. (Contributed by ML, 27-Jul-2020.)

Hypothesis

Ref	Expression
isbasisrelowl.1	⊢ 𝐼 = ([,) “ (ℝ × ℝ))

Assertion

Ref	Expression
icoreclin	⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼)

Distinct variable group:   𝑥,𝐼,𝑦

Proof of Theorem icoreclin

Dummy variables 𝑧 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isbasisrelowl.1	. . . 4 ⊢ 𝐼 = ([,) “ (ℝ × ℝ))
2	1	icoreelrnab 37609	. . 3 ⊢ (𝑦 ∈ 𝐼 ↔ ∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})
3	1	icoreelrnab 37609	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})
4	1	isbasisrelowllem1 37610	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
5	4	ex 412	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → ((𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑) → (𝑥 ∩ 𝑦) ∈ 𝐼))
6	1	isbasisrelowllem2 37611	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
7	6	ex 412	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → ((𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏) → (𝑥 ∩ 𝑦) ∈ 𝐼))
8	5, 7	jaod 860	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → (((𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑) ∨ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼))
9		incom 4163	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
10	1	isbasisrelowllem2 37611	. . . . . . . . . . . . . . 15 ⊢ ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})) ∧ (𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑)) → (𝑦 ∩ 𝑥) ∈ 𝐼)
11	9, 10	eqeltrrid 2842	. . . . . . . . . . . . . 14 ⊢ ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})) ∧ (𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
12	11	ancom1s 654	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
13	12	ex 412	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → ((𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑) → (𝑥 ∩ 𝑦) ∈ 𝐼))
14	1	isbasisrelowllem1 37610	. . . . . . . . . . . . . . 15 ⊢ ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})) ∧ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏)) → (𝑦 ∩ 𝑥) ∈ 𝐼)
15	9, 14	eqeltrrid 2842	. . . . . . . . . . . . . 14 ⊢ ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})) ∧ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
16	15	ancom1s 654	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
17	16	ex 412	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → ((𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏) → (𝑥 ∩ 𝑦) ∈ 𝐼))
18	13, 17	jaod 860	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → (((𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑) ∨ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼))
19		3simpa 1149	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) → (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ))
20		3simpa 1149	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ))
21		letric 11245	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑐 ∨ 𝑐 ≤ 𝑎))
22		letric 11245	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑏 ≤ 𝑑 ∨ 𝑑 ≤ 𝑏))
23	21, 22	anim12i 614	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑎 ≤ 𝑐 ∨ 𝑐 ≤ 𝑎) ∧ (𝑏 ≤ 𝑑 ∨ 𝑑 ≤ 𝑏)))
24		anddi 1013	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ≤ 𝑐 ∨ 𝑐 ≤ 𝑎) ∧ (𝑏 ≤ 𝑑 ∨ 𝑑 ≤ 𝑏)) ↔ (((𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑) ∨ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) ∨ ((𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑) ∨ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏))))
25	23, 24	sylib 218	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑) ∨ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) ∨ ((𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑) ∨ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏))))
26	25	an4s 661	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑) ∨ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) ∨ ((𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑) ∨ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏))))
27	19, 20, 26	syl2an 597	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → (((𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑) ∨ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) ∨ ((𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑) ∨ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏))))
28	8, 18, 27	mpjaod 861	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → (𝑥 ∩ 𝑦) ∈ 𝐼)
29	28	ex 412	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑥 ∩ 𝑦) ∈ 𝐼))
30	29	3expia 1122	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑥 ∩ 𝑦) ∈ 𝐼)))
31	30	rexlimivv 3180	. . . . . . 7 ⊢ (∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑥 ∩ 𝑦) ∈ 𝐼))
32	3, 31	sylbi 217	. . . . . 6 ⊢ (𝑥 ∈ 𝐼 → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑥 ∩ 𝑦) ∈ 𝐼))
33	32	com12 32	. . . . 5 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼))
34	33	3expia 1122	. . . 4 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)} → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼)))
35	34	rexlimivv 3180	. . 3 ⊢ (∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)} → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼))
36	2, 35	sylbi 217	. 2 ⊢ (𝑦 ∈ 𝐼 → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼))
37	36	impcom 407	1 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3401   ∩ cin 3902   class class class wbr 5100   × cxp 5630   “ cima 5635  ℝcr 11037   < clt 11178   ≤ cle 11179  [,)cico 13275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-pre-lttri 11112  ax-pre-lttrn 11113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-ico 13279

This theorem is referenced by:  isbasisrelowl  37613