P. 495 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49401-49500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opnneieqv 49401*	The equivalence between neighborhood and open neighborhood. See opnneieqvv 49402 for different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓)))
Theorem	opnneieqvv 49402*	The equivalence between neighborhood and open neighborhood. A variant of opnneieqv 49401 with two dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
21.49.10.3  Subspace topologies
Theorem	restcls2lem 49403	A closed set in a subspace topology is a subset of the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    ⇒   ⊢ (𝜑 → 𝑆 ⊆ 𝑌)
Theorem	restcls2 49404	A closed set in a subspace topology is the closure in the original topology intersecting with the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    ⇒   ⊢ (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
Theorem	restclsseplem 49405	Lemma for restclssep 49406. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑌)    ⇒   ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
Theorem	restclssep 49406	Two disjoint closed sets in a subspace topology are separated in the original topology. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐾))    ⇒   ⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))
21.49.10.4  Limits and continuity in topological spaces
Theorem	cnneiima 49407	Given a continuous function, the preimage of a neighborhood is a neighborhood. To be precise, the preimage of a neighborhood of a subset 𝑇 of the codomain of a continuous function is a neighborhood of any subset of the preimage of 𝑇. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑁 ∈ ((nei‘𝐾)‘𝑇))    &   ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝑇))    ⇒   ⊢ (𝜑 → (◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆))
21.49.10.5  Topological definitions using the reals
Theorem	iooii 49408	Open intervals are open sets of II. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ ((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴(,)𝐵) ∈ II)
Theorem	icccldii 49409	Closed intervals are closed sets of II. Note that iccss 13361, iccordt 23192, and ordtresticc 23201 are proved from ixxss12 13312, ordtcld3 23177, and ordtrest2 23182, respectively. An alternate proof uses restcldi 23151, dfii2 24862, and icccld 24744. (Contributed by Zhi Wang, 8-Sep-2024.)
⊢ ((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴[,]𝐵) ∈ (Clsd‘II))
Theorem	i0oii 49410	(0[,)𝐴) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝐴 ≤ 1 → (0[,)𝐴) ∈ II)
Theorem	io1ii 49411	(𝐴(,]1) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (0 ≤ 𝐴 → (𝐴(,]1) ∈ II)
21.49.10.6  Separated sets
Theorem	sepnsepolem1 49412*	Lemma for sepnsepo 49414. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥 ∈ 𝐽 (𝜑 ∧ ∃𝑦 ∈ 𝐽 (𝜓 ∧ 𝜒)))
Theorem	sepnsepolem2 49413*	Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 49414. Proof could be shortened by 1 step using ssdisjdr 49299. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    ⇒   ⊢ (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
Theorem	sepnsepo 49414*	Open neighborhood and neighborhood is equivalent regarding disjointness for both sides. Namely, separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
Theorem	sepdisj 49415	Separated sets are disjoint. Note that in general separatedness also requires 𝑇 ⊆ ∪ 𝐽 and (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ as well but they are unnecessary here. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)    ⇒   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
Theorem	seposep 49416*	If two sets are separated by (open) neighborhoods, then they are separated subsets of the underlying set. Note that separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. See sepnsepo 49414. The relationship between separatedness and closure is also seen in isnrm 23313, isnrm2 23336, isnrm3 23337. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))    ⇒   ⊢ (𝜑 → ((𝑆 ⊆ ∪ 𝐽 ∧ 𝑇 ⊆ ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))
Theorem	sepcsepo 49417*	If two sets are separated by closed neighborhoods, then they are separated by (open) neighborhoods. See sepnsepo 49414 for the equivalence between separatedness by open neighborhoods and separatedness by neighborhoods. Although 𝐽 ∈ Top might be redundant here, it is listed for explicitness. 𝐽 ∈ Top can be obtained from neircl 49395, adantr 480, and rexlimiva 3131. (Contributed by Zhi Wang, 8-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
Theorem	sepfsepc 49418*	If two sets are separated by a continuous function, then they are separated by closed neighborhoods. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))
Theorem	seppsepf 49419	If two sets are precisely separated by a continuous function, then they are separated by the continuous function. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})))    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))
Theorem	seppcld 49420*	If two sets are precisely separated by a continuous function, then they are closed. An alternate proof involves II ∈ Fre. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})))    ⇒   ⊢ (𝜑 → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽)))
21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...
Theorem	isnrm4 49421*	A topological space is normal iff any two disjoint closed sets are separated by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥 ∩ 𝑦) = ∅)))
Theorem	dfnrm2 49422*	A topological space is normal if any disjoint closed sets can be separated by open neighborhoods. An alternate definition of df-nrm 23295. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝑗 ∃𝑦 ∈ 𝑗 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))}
Theorem	dfnrm3 49423*	A topological space is normal if any disjoint closed sets can be separated by neighborhoods. An alternate definition of df-nrm 23295. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝑗)‘𝑐)∃𝑦 ∈ ((nei‘𝑗)‘𝑑)(𝑥 ∩ 𝑦) = ∅)}
Theorem	iscnrm3lem1 49424*	Lemma for iscnrm3 49442. Subspace topology is a topology. (Contributed by Zhi Wang, 3-Sep-2024.)
⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ((𝐽 ↾t 𝑥) ∈ Top ∧ 𝜑)))
Theorem	iscnrm3lem2 49425*	Lemma for iscnrm3 49442 proving a biconditional on restricted universal quantifications. (Contributed by Zhi Wang, 3-Sep-2024.)
⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 → ((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒)))    &   ⊢ (𝜑 → (∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 ↔ ∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒))
Theorem	iscnrm3lem4 49426	Lemma for iscnrm3lem5 49427 and iscnrm3r 49438. (Contributed by Zhi Wang, 4-Sep-2024.)
⊢ (𝜂 → (𝜓 → 𝜁))    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜂)    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → (𝜁 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
Theorem	iscnrm3lem5 49427*	Lemma for iscnrm3l 49441. (Contributed by Zhi Wang, 3-Sep-2024.)
⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → (𝜑 ↔ 𝜓))    &   ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → (𝜒 ↔ 𝜃))    &   ⊢ ((𝜏 ∧ 𝜂 ∧ 𝜁) → (𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊))    &   ⊢ ((𝜏 ∧ 𝜂 ∧ 𝜁) → ((𝜓 → 𝜃) → 𝜎))    ⇒   ⊢ (𝜏 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 (𝜑 → 𝜒) → (𝜂 → (𝜁 → 𝜎))))
Theorem	iscnrm3lem6 49428*	Lemma for iscnrm3lem7 49429. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑊 𝜓 → 𝜒))
Theorem	iscnrm3lem7 49429*	Lemma for iscnrm3rlem8 49437 and iscnrm3llem2 49440 involving restricted existential quantifications. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝑧 = 𝑍 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑤 = 𝑊 → (𝜃 ↔ 𝜏))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → (𝑍 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝜏))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒))
Theorem	iscnrm3rlem1 49430	Lemma for iscnrm3rlem2 49431. The hypothesis could be generalized to (𝜑 → (𝑆 ∖ 𝑇) ⊆ 𝑋). (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 ∖ 𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))))
Theorem	iscnrm3rlem2 49431	Lemma for iscnrm3rlem3 49432. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    ⇒   ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ 𝑇)))))
Theorem	iscnrm3rlem3 49432	Lemma for iscnrm3r 49438. The designed subspace is a subset of the original set; the two sets are closed sets in the subspace. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → ((∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 ∪ 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))))
Theorem	iscnrm3rlem4 49433	Lemma for iscnrm3rlem8 49437. Given two disjoint subsets 𝑆 and 𝑇 of the underlying set of a topology 𝐽, if 𝑁 is a superset of (((cls‘𝐽)‘𝑆) ∖ 𝑇), then it is a superset of 𝑆. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ⊆ 𝑁)    ⇒   ⊢ (𝜑 → 𝑆 ⊆ 𝑁)
Theorem	iscnrm3rlem5 49434	Lemma for iscnrm3rlem6 49435. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)    ⇒   ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽)
Theorem	iscnrm3rlem6 49435	Lemma for iscnrm3rlem7 49436. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑂 ⊆ (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))    ⇒   ⊢ (𝜑 → (𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ↔ 𝑂 ∈ 𝐽))
Theorem	iscnrm3rlem7 49436	Lemma for iscnrm3rlem8 49437. Open neighborhoods in the subspace topology are open neighborhoods in the original topology given that the subspace is an open set in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))    ⇒   ⊢ (𝜑 → 𝑂 ∈ 𝐽)
Theorem	iscnrm3rlem8 49437*	Lemma for iscnrm3r 49438. Disjoint open neighborhoods in the subspace topology are disjoint open neighborhoods in the original topology given that the subspace is an open set in the original topology. Therefore, given any two sets separated in the original topology and separated by open neighborhoods in the subspace topology, they must be separated by open neighborhoods in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) → (∃𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
Theorem	iscnrm3r 49438*	Lemma for iscnrm3 49442. If all subspaces of a topology are normal, i.e., two disjoint closed sets can be separated by open neighborhoods, then in the original topology two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 ∪ 𝐽∀𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → ((𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) → (((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))))
Theorem	iscnrm3llem1 49439	Lemma for iscnrm3l 49441. Closed sets in the subspace are subsets of the underlying set of the original topology. (Contributed by Zhi Wang, 4-Sep-2024.)
⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (𝐶 ∈ 𝒫 ∪ 𝐽 ∧ 𝐷 ∈ 𝒫 ∪ 𝐽))
Theorem	iscnrm3llem2 49440*	Lemma for iscnrm3l 49441. If there exist disjoint open neighborhoods in the original topology for two disjoint closed sets in a subspace, then they can be separated by open neighborhoods in the subspace topology. (Could shorten proof with ssin0 45507.) (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))
Theorem	iscnrm3l 49441*	Lemma for iscnrm3 49442. Given a topology 𝐽, if two separated sets can be separated by open neighborhoods, then all subspaces of the topology 𝐽 are normal, i.e., two disjoint closed sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝐽 ∈ Top → (∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) → ((𝐶 ∩ 𝐷) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))))
Theorem	iscnrm3 49442*	A completely normal topology is a topology in which two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
Theorem	iscnrm3v 49443*	A topology is completely normal iff two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 10-Sep-2024.)
⊢ (𝐽 ∈ Top → (𝐽 ∈ CNrm ↔ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
Theorem	iscnrm4 49444*	A completely normal topology is a topology in which two separated sets can be separated by neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛 ∩ 𝑚) = ∅)))
21.49.11  Preordered sets and directed sets using extensible structures
Theorem	isprsd 49445*	Property of being a preordered set (deduction form). (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
21.49.12  Posets and lattices using extensible structures
21.49.12.1  Posets
Theorem	lubeldm2 49446*	Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
Theorem	glbeldm2 49447*	Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
Theorem	lubeldm2d 49448*	Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 28-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
Theorem	glbeldm2d 49449*	Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
Theorem	lubsscl 49450	If a subset of 𝑆 contains the LUB of 𝑆, then the two sets have the same LUB. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)    &   ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝑇)    ⇒   ⊢ (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈‘𝑇) = (𝑈‘𝑆)))
Theorem	glbsscl 49451	If a subset of 𝑆 contains the GLB of 𝑆, then the two sets have the same GLB. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ dom 𝐺)    &   ⊢ (𝜑 → (𝐺‘𝑆) ∈ 𝑇)    ⇒   ⊢ (𝜑 → (𝑇 ∈ dom 𝐺 ∧ (𝐺‘𝑇) = (𝐺‘𝑆)))
Theorem	lubprlem 49452	Lemma for lubprdm 49453 and lubpr 49454. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ∧ (𝑈‘𝑆) = 𝑌))
Theorem	lubprdm 49453	The set of two comparable elements in a poset has LUB. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
Theorem	lubpr 49454	The LUB of the set of two comparable elements in a poset is the greater one of the two. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ (𝜑 → (𝑈‘𝑆) = 𝑌)
Theorem	glbprlem 49455	Lemma for glbprdm 49456 and glbpr 49457. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ∧ (𝐺‘𝑆) = 𝑋))
Theorem	glbprdm 49456	The set of two comparable elements in a poset has GLB. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ (𝜑 → 𝑆 ∈ dom 𝐺)
Theorem	glbpr 49457	The GLB of the set of two comparable elements in a poset is the less one of the two. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ (𝜑 → (𝐺‘𝑆) = 𝑋)
Theorem	joindm2 49458*	The join of any two elements always exists iff all unordered pairs have LUB. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈))
Theorem	joindm3 49459*	The join of any two elements always exists iff all unordered pairs have LUB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))
Theorem	meetdm2 49460*	The meet of any two elements always exists iff all unordered pairs have GLB. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
Theorem	meetdm3 49461*	The meet of any two elements always exists iff all unordered pairs have GLB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))
Theorem	posjidm 49462	Poset join is idempotent. latjidm 18422 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = 𝑋)
Theorem	posmidm 49463	Poset meet is idempotent. latmidm 18434 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) = 𝑋)
Theorem	resiposbas 49464	Construct a poset (resipos 49465) for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐾))
Theorem	resipos 49465	A set equipped with an order where no distinct elements are comparable is a poset. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐾 ∈ Poset)
Theorem	exbaspos 49466*	There exists a poset for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝐵 ∈ 𝑉 → ∃𝑘 ∈ Poset 𝐵 = (Base‘𝑘))
Theorem	exbasprs 49467*	There exists a preordered set for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝐵 ∈ 𝑉 → ∃𝑘 ∈ Proset 𝐵 = (Base‘𝑘))
Theorem	basresposfo 49468	The base function restricted to the class of posets maps the class of posets onto the universal class. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (Base ↾ Poset):Poset–onto→V
Theorem	basresprsfo 49469	The base function restricted to the class of preordered sets maps the class of preordered sets onto the universal class. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (Base ↾ Proset ): Proset –onto→V
Theorem	posnex 49470	The class of posets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ Poset ∉ V
Theorem	prsnex 49471	The class of preordered sets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ Proset ∉ V
21.49.12.2  Lattices
Theorem	toslat 49472	A toset is a lattice. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝐾 ∈ Toset → 𝐾 ∈ Lat)
Theorem	isclatd 49473*	The predicate "is a complete lattice" (deduction form). (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐾))    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝑈)    &   ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝐺)    ⇒   ⊢ (𝜑 → 𝐾 ∈ CLat)
21.49.12.3  Subset order structures
Theorem	intubeu 49474*	Existential uniqueness of the least upper bound. (Contributed by Zhi Wang, 28-Sep-2024.)
⊢ (𝐶 ∈ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) ↔ 𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}))
Theorem	unilbeu 49475*	Existential uniqueness of the greatest lower bound. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ (𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) ↔ 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}))
Theorem	ipolublem 49476*	Lemma for ipolubdm 49477 and ipolub 49478. (Contributed by Zhi Wang, 28-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ ≤ = (le‘𝐼)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((∪ 𝑆 ⊆ 𝑋 ∧ ∀𝑧 ∈ 𝐹 (∪ 𝑆 ⊆ 𝑧 → 𝑋 ⊆ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑋 ≤ 𝑧))))
Theorem	ipolubdm 49477*	The domain of the LUB of the inclusion poset. (Contributed by Zhi Wang, 28-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐼))    &   ⊢ (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥})    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ 𝑇 ∈ 𝐹))
Theorem	ipolub 49478*	The LUB of the inclusion poset. (hypotheses "ipolub.s" and "ipolub.t" could be eliminated with 𝑆 ∈ dom 𝑈.) Could be significantly shortened if poslubdg 18372 is in quantified form. mrelatlub 18522 could potentially be shortened using this. See mrelatlubALT 49485. (Contributed by Zhi Wang, 28-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐼))    &   ⊢ (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥})    &   ⊢ (𝜑 → 𝑇 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇)
Theorem	ipoglblem 49479*	Lemma for ipoglbdm 49480 and ipoglb 49481. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ ≤ = (le‘𝐼)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((𝑋 ⊆ ∩ 𝑆 ∧ ∀𝑧 ∈ 𝐹 (𝑧 ⊆ ∩ 𝑆 → 𝑧 ⊆ 𝑋)) ↔ (∀𝑦 ∈ 𝑆 𝑋 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑋))))
Theorem	ipoglbdm 49480*	The domain of the GLB of the inclusion poset. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐼))    &   ⊢ (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆})    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ 𝑇 ∈ 𝐹))
Theorem	ipoglb 49481*	The GLB of the inclusion poset. (hypotheses "ipolub.s" and "ipoglb.t" could be eliminated with 𝑆 ∈ dom 𝐺.) Could be significantly shortened if posglbdg 18373 is in quantified form. mrelatglb 18520 could potentially be shortened using this. See mrelatglbALT 49486. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐼))    &   ⊢ (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆})    &   ⊢ (𝜑 → 𝑇 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇)
Theorem	ipolub0 49482	The LUB of the empty set is the intersection of the base. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐼))    &   ⊢ (𝜑 → ∩ 𝐹 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑈‘∅) = ∩ 𝐹)
Theorem	ipolub00 49483	The LUB of the empty set is the empty set if it is contained. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐼))    &   ⊢ (𝜑 → ∅ ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝑈‘∅) = ∅)
Theorem	ipoglb0 49484	The GLB of the empty set is the union of the base. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐼))    &   ⊢ (𝜑 → ∪ 𝐹 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺‘∅) = ∪ 𝐹)
Theorem	mrelatlubALT 49485	Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐹 = (mrCls‘𝐶)    &   ⊢ 𝐿 = (lub‘𝐼)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))
Theorem	mrelatglbALT 49486	Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐺 = (glb‘𝐼)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈)
Theorem	mreclat 49487	A Moore space is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐶)    ⇒   ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat)
Theorem	topclat 49488	A topology is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐼 ∈ CLat)
Theorem	toplatglb0 49489	The empty intersection in a topology is realized by the base set. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ 𝐺 = (glb‘𝐼)    ⇒   ⊢ (𝜑 → (𝐺‘∅) = ∪ 𝐽)
Theorem	toplatlub 49490	Least upper bounds in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐽)    &   ⊢ 𝑈 = (lub‘𝐼)    ⇒   ⊢ (𝜑 → (𝑈‘𝑆) = ∪ 𝑆)
Theorem	toplatglb 49491	Greatest lower bounds in a topology are realized by the interior of the intersection. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐽)    &   ⊢ 𝐺 = (glb‘𝐼)    &   ⊢ (𝜑 → 𝑆 ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐺‘𝑆) = ((int‘𝐽)‘∩ 𝑆))
Theorem	toplatjoin 49492	Joins in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐽)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐽)    &   ⊢ ∨ = (join‘𝐼)    ⇒   ⊢ (𝜑 → (𝐴 ∨ 𝐵) = (𝐴 ∪ 𝐵))
Theorem	toplatmeet 49493	Meets in a topology are realized by intersections. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐽)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐽)    &   ⊢ ∧ = (meet‘𝐼)    ⇒   ⊢ (𝜑 → (𝐴 ∧ 𝐵) = (𝐴 ∩ 𝐵))
Theorem	topdlat 49494	A topology is a distributive lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐼 ∈ DLat)
21.49.13  Rings
21.49.13.1  Multiplicative Group
Theorem	elmgpcntrd 49495*	The center of a ring. (Contributed by Zhi Wang, 11-Sep-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑍 = (Cntr‘𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(.r‘𝑅)𝑦) = (𝑦(.r‘𝑅)𝑋))    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝑍)
21.49.14  Associative algebras
21.49.14.1  Definition and basic properties
Theorem	asclelbasALT 49496	Alternate proof of asclelbas 21876. (Contributed by Zhi Wang, 11-Sep-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Base‘𝑊))
Theorem	asclcntr 49497	The algebra scalar lifting function maps into the center of the algebra. Equivalently, a lifted scalar is a center of the algebra. (Contributed by Zhi Wang, 11-Sep-2025.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ 𝑀 = (mulGrp‘𝑊)    ⇒   ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Cntr‘𝑀))
Theorem	asclcom 49498	Scalars are commutative after being lifted. However, the scalars themselves are not necessarily commutative if the algebra is not a faithful module. For example, Let 𝐹 be the 2 by 2 upper triangular matrix algebra over a commutative ring 𝑊. It is provable that 𝐹 is in general non-commutative. Define scalar multiplication 𝐶 · 𝑋 as multipying the top-left entry, which is a "vector" element of 𝑊, of the "scalar" 𝐶, which is now an upper triangular matrix, with the "vector" 𝑋 ∈ (Base‘𝑊). Equivalently, the algebra scalar lifting function is not necessarily injective unless the algebra is faithful. Therefore, all "scalar injection" was renamed. Alternate proof involves assa2ass 21856, assa2ass2 21857, and asclval 21872, by setting 𝑋 and 𝑌 the multiplicative identity of the algebra. (Contributed by Zhi Wang, 11-Sep-2025.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ ∗ = (.r‘𝐹)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = (𝐴‘(𝐷 ∗ 𝐶)))
21.49.15  Categories
21.49.15.1  Categories
Theorem	homf0 49499	The base is empty iff the functionalized Hom-set operation is empty. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ ((Base‘𝐶) = ∅ ↔ (Homf ‘𝐶) = ∅)
Theorem	catprslem 49500*	Lemma for catprs 49501. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))