P. 365 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bisym1 36401	A symmetry with ↔. See negsym1 36399 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓 ↔ 𝜑))
Theorem	consym1 36402	A symmetry with ∧. See negsym1 36399 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑))
Theorem	dissym1 36403	A symmetry with ∨. See negsym1 36399 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ∨ (𝜓 ∨ ⊥)) → (𝜓 ∨ 𝜑))
Theorem	nandsym1 36404	A symmetry with ⊼. See negsym1 36399 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓 ⊼ 𝜑))
Theorem	unisym1 36405	A symmetry with ∀. See negsym1 36399 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
⊢ (∀𝑥∀𝑥⊥ → ∀𝑥𝜑)
Theorem	exisym1 36406	A symmetry with ∃. See negsym1 36399 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ (∃𝑥∃𝑥⊥ → ∃𝑥𝜑)
Theorem	unqsym1 36407	A symmetry with ∃!. See negsym1 36399 for more information. (Contributed by Anthony Hart, 6-Sep-2011.)
⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)
Theorem	amosym1 36408	A symmetry with ∃*. See negsym1 36399 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
Theorem	subsym1 36409	A symmetry with [𝑥 / 𝑦]. See negsym1 36399 for more information. (Contributed by Anthony Hart, 11-Sep-2011.)
⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑)
21.15  Mathbox for Chen-Pang He
21.15.1  Ordinal topology
Theorem	ontopbas 36410	An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases)
Theorem	onsstopbas 36411	The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.)
⊢ On ⊆ TopBases
Theorem	onpsstopbas 36412	The class of ordinal numbers is a proper subclass of the class of topological bases. (Contributed by Chen-Pang He, 9-Oct-2015.)
⊢ On ⊊ TopBases
Theorem	ontgval 36413	The topology generated from an ordinal number 𝐵 is suc ∪ 𝐵. (Contributed by Chen-Pang He, 10-Oct-2015.)
⊢ (𝐵 ∈ On → (topGen‘𝐵) = suc ∪ 𝐵)
Theorem	ontgsucval 36414	The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.)
⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴)
Theorem	onsuctop 36415	A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top)
Theorem	onsuctopon 36416	One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴))
Theorem	ordtoplem 36417	Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)
⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆)    ⇒   ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆))
Theorem	ordtop 36418	An ordinal is a topology iff it is not its supremum (union), proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 1-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽))
Theorem	onsucconni 36419	A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ 𝐴 ∈ On    ⇒   ⊢ suc 𝐴 ∈ Conn
Theorem	onsucconn 36420	A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Conn)
Theorem	ordtopconn 36421	An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn))
Theorem	onintopssconn 36422	An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ (On ∩ Top) ⊆ Conn
Theorem	onsuct0 36423	A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Kol2)
Theorem	ordtopt0 36424	An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2))
Theorem	onsucsuccmpi 36425	The successor of a successor ordinal number is a compact topology, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 18-Oct-2015.)
⊢ 𝐴 ∈ On    ⇒   ⊢ suc suc 𝐴 ∈ Comp
Theorem	onsucsuccmp 36426	The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015.)
⊢ (𝐴 ∈ On → suc suc 𝐴 ∈ Comp)
Theorem	limsucncmpi 36427	The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
⊢ Lim 𝐴    ⇒   ⊢ ¬ suc 𝐴 ∈ Comp
Theorem	limsucncmp 36428	The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
Theorem	ordcmp 36429	An ordinal topology is compact iff the underlying set is its supremum (union) only when the ordinal is 1o. (Contributed by Chen-Pang He, 1-Nov-2015.)
⊢ (Ord 𝐴 → (𝐴 ∈ Comp ↔ (∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o)))
Theorem	ssoninhaus 36430	The ordinal topologies 1o and 2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
⊢ {1o, 2o} ⊆ (On ∩ Haus)
Theorem	onint1 36431	The ordinal T1 spaces are 1o and 2o, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015.)
⊢ (On ∩ Fre) = {1o, 2o}
Theorem	oninhaus 36432	The ordinal Hausdorff spaces are 1o and 2o. (Contributed by Chen-Pang He, 10-Nov-2015.)
⊢ (On ∩ Haus) = {1o, 2o}
21.16  Mathbox for Jeff Hoffman
21.16.1  Inferences for finite induction on generic function values
Theorem	fveleq 36433	Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃)))
Theorem	findfvcl 36434*	Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
⊢ (𝜑 → (𝐹‘∅) ∈ 𝑃)    &   ⊢ (𝑦 ∈ ω → (𝜑 → ((𝐹‘𝑦) ∈ 𝑃 → (𝐹‘suc 𝑦) ∈ 𝑃)))    ⇒   ⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 𝑃))
Theorem	findreccl 36435*	Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.)
⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃)    ⇒   ⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
Theorem	findabrcl 36436*	Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃)    ⇒   ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ 𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) ∈ 𝑃)
21.16.2  gdc.mm
Theorem	nnssi2 36437	Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
⊢ ℕ ⊆ 𝐷    &   ⊢ (𝐵 ∈ ℕ → 𝜑)    &   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑) → 𝜓)    ⇒   ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓)
Theorem	nnssi3 36438	Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
⊢ ℕ ⊆ 𝐷    &   ⊢ (𝐶 ∈ ℕ → 𝜑)    &   ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ∧ 𝜑) → 𝜓)    ⇒   ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓)
Theorem	nndivsub 36439	Please add description here. (Contributed by Jeff Hoffman, 17-Jun-2008.)
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴 / 𝐶) ∈ ℕ ∧ 𝐴 < 𝐵)) → ((𝐵 / 𝐶) ∈ ℕ ↔ ((𝐵 − 𝐴) / 𝐶) ∈ ℕ))
Theorem	nndivlub 36440	A factor of a positive integer cannot exceed it. (Contributed by Jeff Hoffman, 17-Jun-2008.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) ∈ ℕ → 𝐵 ≤ 𝐴))
Syntax	cgcdOLD 36441	Extend class notation to include the gdc function. (New usage is discouraged.)
class gcdOLD (𝐴, 𝐵)
Definition	df-gcdOLD 36442*	gcdOLD (𝐴, 𝐵) is the largest positive integer that evenly divides both 𝐴 and 𝐵. (Contributed by Jeff Hoffman, 17-Jun-2008.) (New usage is discouraged.)
⊢ gcdOLD (𝐴, 𝐵) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, < )
Theorem	ee7.2aOLD 36443	Lemma for Euclid's Elements, Book 7, proposition 2. The original mentions the smaller measure being 'continually subtracted' from the larger. Many authors interpret this phrase as 𝐴 mod 𝐵. Here, just one subtraction step is proved to preserve the gcdOLD. The rec function will be used in other proofs for iterated subtraction. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 → gcdOLD (𝐴, 𝐵) = gcdOLD (𝐴, (𝐵 − 𝐴))))
21.17  Mathbox for Matthew House
21.17.1  Relations on well-ordered indexed unions
Theorem	weiunlem1 36444*	Lemma for weiunpo 36447, weiunso 36448, weiunfr 36449, and weiunse 36450. (Contributed by Matthew House, 8-Sep-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ⊢ (𝐶𝑇𝐷 ↔ ((𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝐷 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝐶)𝑅(𝐹‘𝐷) ∨ ((𝐹‘𝐶) = (𝐹‘𝐷) ∧ 𝐶⦋(𝐹‘𝐶) / 𝑥⦌𝑆𝐷))))
Theorem	weiunlem2 36445*	Lemma for weiunpo 36447, weiunso 36448, weiunfr 36449, and weiunse 36450. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    ⇒   ⊢ (𝜑 → (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 ∧ ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡)))
Theorem	weiunfrlem 36446*	Lemma for weiunfr 36449. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ 𝐸 = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)    &   ⊢ (𝜑 → 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)    &   ⊢ (𝜑 → 𝑟 ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐸 ∈ (𝐹 “ 𝑟) ∧ ∀𝑡 ∈ 𝑟 ¬ (𝐹‘𝑡)𝑅𝐸 ∧ ∀𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)(𝐹‘𝑡) = 𝐸))
Theorem	weiunpo 36447*	A partial ordering on an indexed union can be constructed from a well-ordering on its index class and a collection of partial orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Po 𝐵) → 𝑇 Po ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	weiunso 36448*	A strict ordering on an indexed union can be constructed from a well-ordering on its index class and a collection of strict orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Or 𝐵) → 𝑇 Or ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	weiunfr 36449*	A well-founded relation on an indexed union can be constructed from a well-ordering on its index class and a collection of well-founded relations on its members. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) → 𝑇 Fr ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	weiunse 36450*	The relation constructed in weiunpo 36447, weiunso 36448, weiunfr 36449, and weiunwe 36451 is set-like if all members of the indexed union are sets. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → 𝑇 Se ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	weiunwe 36451*	A well-ordering on an indexed union can be constructed from a well-ordering on its index class and a collection of well-orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 We 𝐵) → 𝑇 We ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	numiunnum 36452*	An indexed union of sets is numerable if its index set is numerable and there exists a collection of well-orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
⊢ ((𝐴 ∈ dom card ∧ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ dom card)
21.18  Mathbox for Asger C. Ipsen
21.18.1  Continuous nowhere differentiable functions
Theorem	dnival 36453*	Value of the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    ⇒   ⊢ (𝐴 ∈ ℝ → (𝑇‘𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))
Theorem	dnicld1 36454	Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ ℝ)
Theorem	dnicld2 36455*	Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑇‘𝐴) ∈ ℝ)
Theorem	dnif 36456	The "distance to nearest integer" function is a function. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    ⇒   ⊢ 𝑇:ℝ⟶ℝ
Theorem	dnizeq0 36457*	The distance to nearest integer is zero for integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝑇‘𝐴) = 0)
Theorem	dnizphlfeqhlf 36458*	The distance to nearest integer is a half for half-integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝑇‘(𝐴 + (1 / 2))) = (1 / 2))
Theorem	rddif2 36459	Variant of rddif 15375. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝐴 ∈ ℝ → 0 ≤ ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))
Theorem	dnibndlem1 36460*	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ 𝑆 ↔ (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ 𝑆))
Theorem	dnibndlem2 36461*	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) = (⌊‘(𝐴 + (1 / 2))))    ⇒   ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴)))
Theorem	dnibndlem3 36462	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) = ((⌊‘(𝐴 + (1 / 2))) + 1))    ⇒   ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) = (abs‘((𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))) + (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴))))
Theorem	dnibndlem4 36463	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝐵 ∈ ℝ → 0 ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))))
Theorem	dnibndlem5 36464	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝐴 ∈ ℝ → 0 < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴))
Theorem	dnibndlem6 36465	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) + ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))))
Theorem	dnibndlem7 36466	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))))
Theorem	dnibndlem8 36467	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ≤ (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴))
Theorem	dnibndlem9 36468*	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) = ((⌊‘(𝐴 + (1 / 2))) + 1))    ⇒   ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴)))
Theorem	dnibndlem10 36469	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2))))    ⇒   ⊢ (𝜑 → 1 ≤ (𝐵 − 𝐴))
Theorem	dnibndlem11 36470	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (1 / 2))
Theorem	dnibndlem12 36471*	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2))))    ⇒   ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴)))
Theorem	dnibndlem13 36472*	Lemma for dnibnd 36473. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2))))    ⇒   ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴)))
Theorem	dnibnd 36473*	The "distance to nearest integer" function is 1-Lipschitz continuous, i.e., is a short map. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴)))
Theorem	dnicn 36474	The "distance to nearest integer" function is continuous. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    ⇒   ⊢ 𝑇 ∈ (ℝ–cn→ℝ)
Theorem	knoppcnlem1 36475*	Lemma for knoppcn 36486. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴)‘𝑀) = ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))))
Theorem	knoppcnlem2 36476*	Lemma for knoppcn 36486. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))) ∈ ℝ)
Theorem	knoppcnlem3 36477*	Lemma for knoppcn 36486. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴)‘𝑀) ∈ ℝ)
Theorem	knoppcnlem4 36478*	Lemma for knoppcn 36486. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (abs‘((𝐹‘𝐴)‘𝑀)) ≤ ((𝑚 ∈ ℕ0 ↦ ((abs‘𝐶)↑𝑚))‘𝑀))
Theorem	knoppcnlem5 36479*	Lemma for knoppcn 36486. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ0⟶(ℂ ↑m ℝ))
Theorem	knoppcnlem6 36480*	Lemma for knoppcn 36486. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐶) < 1)    ⇒   ⊢ (𝜑 → seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))) ∈ dom (⇝𝑢‘ℝ))
Theorem	knoppcnlem7 36481*	Lemma for knoppcn 36486. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))‘𝑀) = (𝑤 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑤))‘𝑀)))
Theorem	knoppcnlem8 36482*	Lemma for knoppcn 36486. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))):ℕ0⟶(ℂ ↑m ℝ))
Theorem	knoppcnlem9 36483*	Lemma for knoppcn 36486. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐶) < 1)    ⇒   ⊢ (𝜑 → seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))(⇝𝑢‘ℝ)𝑊)
Theorem	knoppcnlem10 36484*	Lemma for knoppcn 36486. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) Avoid ax-mulf 11232. (Revised by GG, 19-Apr-2025.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑀)) ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂfld)))
Theorem	knoppcnlem11 36485*	Lemma for knoppcn 36486. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))):ℕ0⟶(ℝ–cn→ℂ))
Theorem	knoppcn 36486*	The continuous nowhere differentiable function 𝑊 ( Knopp, K. (1918). Math. Z. 2, 1-26 ) is, in fact, continuous. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐶) < 1)    ⇒   ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℂ))
Theorem	knoppcld 36487*	Closure theorem for Knopp's function. (Contributed by Asger C. Ipsen, 26-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐶) < 1)    ⇒   ⊢ (𝜑 → (𝑊‘𝐴) ∈ ℂ)
Theorem	unblimceq0lem 36488*	Lemma for unblimceq0 36489. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → ∀𝑏 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥))))    ⇒   ⊢ (𝜑 → ∀𝑐 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑦 ∈ 𝑆 (𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑦))))
Theorem	unblimceq0 36489*	If 𝐹 is unbounded near 𝐴 it has no limit at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → ∀𝑏 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥))))    ⇒   ⊢ (𝜑 → (𝐹 limℂ 𝐴) = ∅)
Theorem	unbdqndv1 36490*	If the difference quotient (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴)) is unbounded near 𝐴 then 𝐹 is not differentiable at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴)))    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → ∀𝑏 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑥 ∈ (𝑋 ∖ {𝐴})((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐺‘𝑥))))    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ dom (𝑆 D 𝐹))
Theorem	unbdqndv2lem1 36491	Lemma for unbdqndv2 36493. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐷 ≠ 0)    &   ⊢ (𝜑 → (2 · 𝐸) ≤ (abs‘((𝐴 − 𝐵) / 𝐷)))    ⇒   ⊢ (𝜑 → ((𝐸 · (abs‘𝐷)) ≤ (abs‘(𝐴 − 𝐶)) ∨ (𝐸 · (abs‘𝐷)) ≤ (abs‘(𝐵 − 𝐶))))
Theorem	unbdqndv2lem2 36492*	Lemma for unbdqndv2 36493. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴)))    &   ⊢ 𝑊 = if((𝐵 · (𝑉 − 𝑈)) ≤ (abs‘((𝐹‘𝑈) − (𝐹‘𝐴))), 𝑈, 𝑉)    &   ⊢ (𝜑 → 𝑋 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑈 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≤ 𝑉)    &   ⊢ (𝜑 → (𝑉 − 𝑈) < 𝐷)    &   ⊢ (𝜑 → (2 · 𝐵) ≤ ((abs‘((𝐹‘𝑉) − (𝐹‘𝑈))) / (𝑉 − 𝑈)))    ⇒   ⊢ (𝜑 → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊 − 𝐴)) < 𝐷 ∧ 𝐵 ≤ (abs‘(𝐺‘𝑊)))))
Theorem	unbdqndv2 36493*	Variant of unbdqndv1 36490 with the hypothesis that (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) is unbounded where 𝑥 ≤ 𝐴 and 𝐴 ≤ 𝑦. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ (𝜑 → 𝑋 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → ∀𝑏 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑥 ≤ 𝐴 ∧ 𝐴 ≤ 𝑦) ∧ ((𝑦 − 𝑥) < 𝑑 ∧ 𝑥 ≠ 𝑦) ∧ 𝑏 ≤ ((abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) / (𝑦 − 𝑥))))    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ dom (ℝ D 𝐹))
Theorem	knoppndvlem1 36494	Lemma for knoppndv 36516. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    ⇒   ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ)
Theorem	knoppndvlem2 36495	Lemma for knoppndv 36516. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ ℤ)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐽 < 𝐼)    ⇒   ⊢ (𝜑 → (((2 · 𝑁)↑𝐼) · ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)) ∈ ℤ)
Theorem	knoppndvlem3 36496	Lemma for knoppndv 36516. (Contributed by Asger C. Ipsen, 15-Jun-2021.)
⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    ⇒   ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
Theorem	knoppndvlem4 36497*	Lemma for knoppndv 36516. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → seq0( + , (𝐹‘𝐴)) ⇝ (𝑊‘𝐴))
Theorem	knoppndvlem5 36498*	Lemma for knoppndv 36516. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) ∈ ℝ)
Theorem	knoppndvlem6 36499*	Lemma for knoppndv 36516. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝑊‘𝐴) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖))
Theorem	knoppndvlem7 36500*	Lemma for knoppndv 36516. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = ((𝐶↑𝐽) · (𝑇‘(𝑀 / 2))))