P. 367 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36601-36700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	neutru 36601	There does not exist exactly one set such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ¬ ∃!𝑥⊤
Theorem	nmotru 36602	There does not exist at most one set such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ¬ ∃*𝑥⊤
Theorem	mofal 36603	There exist at most one set such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ∃*𝑥⊥
Theorem	nrmo 36604	"At most one" restricted existential quantifier for a statement which is never true. (Contributed by Thierry Arnoux, 27-Nov-2023.)
⊢ (𝑥 ∈ 𝐴 → ¬ 𝜑)    ⇒   ⊢ ∃*𝑥 ∈ 𝐴 𝜑
21.14.3  Miscellaneous single axioms
Theorem	meran1 36605	A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜃 ∨ 𝜑) ∨ (𝜒 ∨ (𝜏 ∨ 𝜑))))
Theorem	meran2 36606	A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜏 ∨ 𝜃) ∨ (𝜒 ∨ (𝜑 ∨ 𝜃))))
Theorem	meran3 36607	A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜒 ∨ 𝜑) ∨ (𝜏 ∨ (𝜃 ∨ 𝜑))))
Theorem	waj-ax 36608	A single axiom for propositional calculus discovered by Mordchaj Wajsberg (Logical Works, Polish Academy of Sciences, 1977). See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom W on slide 8). (Contributed by Anthony Hart, 13-Aug-2011.)
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ (((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))) ⊼ (𝜑 ⊼ (𝜑 ⊼ 𝜓))))
Theorem	lukshef-ax2 36609	A single axiom for propositional calculus discovered by Jan Lukasiewicz. See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom L2 on slide 8). (Contributed by Anthony Hart, 14-Aug-2011.)
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜑)) ⊼ ((𝜃 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
Theorem	arg-ax 36610	A single axiom for propositional calculus discovered by Ken Harris and Branden Fitelson. See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom HF1 on slide 8). (Contributed by Anthony Hart, 14-Aug-2011.)
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜒 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
21.14.4  Connective Symmetry
Theorem	negsym1 36611	In the paper "On Variable Functors of Propositional Arguments", Lukasiewicz introduced a system that can handle variable connectives. This was done by introducing a variable, marked with a lowercase delta, which takes a wff as input. In the system, "delta 𝜑 " means that "something is true of 𝜑". The expression "delta 𝜑 " can be substituted with ¬ 𝜑, 𝜓 ∧ 𝜑, ∀𝑥𝜑, etc. Later on, Meredith discovered a single axiom, in the form of ( delta delta ⊥ → delta 𝜑 ). This represents the shortest theorem in the extended propositional calculus that cannot be derived as an instance of a theorem in propositional calculus. A symmetry with ¬. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ (¬ ¬ ⊥ → ¬ 𝜑)
Theorem	imsym1 36612	A symmetry with →. See negsym1 36611 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 → (𝜓 → ⊥)) → (𝜓 → 𝜑))
Theorem	bisym1 36613	A symmetry with ↔. See negsym1 36611 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓 ↔ 𝜑))
Theorem	consym1 36614	A symmetry with ∧. See negsym1 36611 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑))
Theorem	dissym1 36615	A symmetry with ∨. See negsym1 36611 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ∨ (𝜓 ∨ ⊥)) → (𝜓 ∨ 𝜑))
Theorem	nandsym1 36616	A symmetry with ⊼. See negsym1 36611 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓 ⊼ 𝜑))
Theorem	unisym1 36617	A symmetry with ∀. See negsym1 36611 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
⊢ (∀𝑥∀𝑥⊥ → ∀𝑥𝜑)
Theorem	exisym1 36618	A symmetry with ∃. See negsym1 36611 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ (∃𝑥∃𝑥⊥ → ∃𝑥𝜑)
Theorem	unqsym1 36619	A symmetry with ∃!. See negsym1 36611 for more information. (Contributed by Anthony Hart, 6-Sep-2011.)
⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)
Theorem	amosym1 36620	A symmetry with ∃*. See negsym1 36611 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
Theorem	subsym1 36621	A symmetry with [𝑥 / 𝑦]. See negsym1 36611 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 36622	An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases)
Theorem	onsstopbas 36623	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 36624	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 36625	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 36626	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 36627	A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top)
Theorem	onsuctopon 36628	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 36629	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 36630	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 36631	A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ 𝐴 ∈ On    ⇒   ⊢ suc 𝐴 ∈ Conn
Theorem	onsucconn 36632	A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Conn)
Theorem	ordtopconn 36633	An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn))
Theorem	onintopssconn 36634	An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ (On ∩ Top) ⊆ Conn
Theorem	onsuct0 36635	A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Kol2)
Theorem	ordtopt0 36636	An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2))
Theorem	onsucsuccmpi 36637	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 36638	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 36639	The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
⊢ Lim 𝐴    ⇒   ⊢ ¬ suc 𝐴 ∈ Comp
Theorem	limsucncmp 36640	The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
Theorem	ordcmp 36641	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 36642	The ordinal topologies 1o and 2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
⊢ {1o, 2o} ⊆ (On ∩ Haus)
Theorem	onint1 36643	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 36644	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 36645	Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃)))
Theorem	findfvcl 36646*	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 36647*	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 36648*	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 36649	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 36650	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 36651	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 36652	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 36653	Extend class notation to include the gdc function. (New usage is discouraged.)
class gcdOLD (𝐴, 𝐵)
Definition	df-gcdOLD 36654*	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 36655	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	weiunval 36656*	Value of the relation constructed in weiunpo 36659, weiunso 36660, weiunfr 36661, and weiunse 36662. (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	weiunlem 36657*	Lemma for weiunpo 36659, weiunso 36660, weiunfr 36661, and weiunse 36662. (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 36658*	Lemma for weiunfr 36661. (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 36659*	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 36660*	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 36661*	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 36662*	The relation constructed in weiunpo 36659, weiunso 36660, weiunfr 36661, and weiunwe 36663 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 36663*	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 36664*	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)
21.17.2  Stronger axioms of regularity This section contains some experiments related to the Axiom of Regularity ax-reg 9497. As written, ax-reg 9497 cannot guarantee that all sets are well-founded unless we further assume ax-inf 9547 / ax-inf2 9550; in particular, ax-reg 9497 alone is insufficient to assert that every set has a transitive closure (tz9.1 9638), even though this is true among the hereditarily finite sets. The underlying cause of this issue is that ax-reg 9497 requires a witness set to detect non-well-foundedness, but if all sets are hereditarily finite, then there may be no such witness set for an infinite descending ∈-chain. The question is, how can we strengthen ax-reg 9497 so that we get a true "Axiom of Foundation" even in the absence of ax-inf 9547 / ax-inf2 9550 (e.g., so that we can prove unir1 9725 ∪ (𝑅1 “ On) = V)? There are a few possible solutions. First, we can directly strengthen ax-reg 9497 into ax-regs 35282, which asserts that every class {𝑥 ∣ 𝜑} has an ∈-minimal element. Second, we can keep ax-reg 9497 and add exeltr 36665, which asserts that every set is a member of a transitive set. Third, we can replace ax-reg 9497 with a set-induction axiom mh-setind 36666. Fourth, we can take unir1 9725 as an axiom and derive everything from that. This list is far from exhaustive. In this section, we prove that these four listed principles are equivalent. We see that ax-regs 35282 implies the other three principles: ax-reg 9497 + exeltr 36665 via axreg 35283 + tz9.1regs 35290, mh-setind 36666 via setindregs 35286, and unir1 9725 via unir1regs 35291. So we just have to show that ax-regs 35282 is implied by each of the other three. Some questions: When expanded to primitives, what is the shortest single axiom equivalent to these, over ZF minus ax-reg 9497 and ax-inf 9547 / ax-inf2 9550? One candidate is mh-setind 36666, with 19 primitives. What is the shortest single axiom not using any wff variables? The conjunction of ax-reg 9497 + exeltr 36665, expanded and slightly simplified, comes out to 46 primitives. Can we do better?
Theorem	exeltr 36665*	Every set is a member of a transitive set. This requires ax-inf2 9550 to prove, see tz9.1 9638. (Contributed by Matthew House, 4-Mar-2026.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
Theorem	mh-setind 36666*	Principle of set induction setind 9656, written with primitive symbols. (Contributed by Matthew House, 4-Mar-2026.)
⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → 𝜑)
Theorem	mh-setindnd 36667	A version of mh-setind 36666 with no distinct variable conditions. (Contributed by Matthew House, 5-Mar-2026.) (New usage is discouraged.)
⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑)
Theorem	regsfromregtr 36668*	Derivation of ax-regs 35282 from ax-reg 9497 + exeltr 36665. (Contributed by Matthew House, 4-Mar-2026.)
⊢ (∃𝑦 𝑦 ∈ 𝑤 → ∃𝑦(𝑦 ∈ 𝑤 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤)))    &   ⊢ ∃𝑢(𝑣 ∈ 𝑢 ∧ ∀𝑡(𝑡 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢)))    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
Theorem	regsfromsetind 36669*	Derivation of ax-regs 35282 from mh-setind 36666. (Contributed by Matthew House, 4-Mar-2026.)
⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) → ¬ 𝜑)    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
Theorem	regsfromunir1 36670*	Derivation of ax-regs 35282 from unir1 9725. (Contributed by Matthew House, 4-Mar-2026.)
⊢ ∪ (𝑅1 “ On) = V    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
21.18  Mathbox for Asger C. Ipsen
21.18.1  Continuous nowhere differentiable functions
Theorem	dnival 36671*	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 36672	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 36673*	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 36674	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 36675*	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 36676*	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 36677	Variant of rddif 15264. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝐴 ∈ ℝ → 0 ≤ ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))
Theorem	dnibndlem1 36678*	Lemma for dnibnd 36691. (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 36679*	Lemma for dnibnd 36691. (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 36680	Lemma for dnibnd 36691. (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 36681	Lemma for dnibnd 36691. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝐵 ∈ ℝ → 0 ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))))
Theorem	dnibndlem5 36682	Lemma for dnibnd 36691. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝐴 ∈ ℝ → 0 < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴))
Theorem	dnibndlem6 36683	Lemma for dnibnd 36691. (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 36684	Lemma for dnibnd 36691. (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 36685	Lemma for dnibnd 36691. (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 36686*	Lemma for dnibnd 36691. (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 36687	Lemma for dnibnd 36691. (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 36688	Lemma for dnibnd 36691. (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 36689*	Lemma for dnibnd 36691. (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 36690*	Lemma for dnibnd 36691. (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 36691*	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 36692	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 36693*	Lemma for knoppcn 36704. (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 36694*	Lemma for knoppcn 36704. (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 36695*	Lemma for knoppcn 36704. (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 36696*	Lemma for knoppcn 36704. (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 36697*	Lemma for knoppcn 36704. (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 36698*	Lemma for knoppcn 36704. (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 36699*	Lemma for knoppcn 36704. (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 36700*	Lemma for knoppcn 36704. (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))