P. 361 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36001-36100   *Has distinct variable group(s)

Type	Label	Description
Statement
21.6.5  Model theory
Syntax	wsmext 36001	Extend wff notation to include a predicate for standard models of the Axiom of Extensionality.
wff 𝑀 ⊨ Ax-Ext
Syntax	wsmrep 36002	Extend wff notation to include a predicate for standard models of a strong version of the Axiom Scheme of Replacement.
wff 𝑀 ⊨ Ax-Rep*
Definition	df-smext 36003*	Define the standard model of Extensionality predicate, which is true for a class that is a model of the Axiom of Extensionality under the membership predicate. Our notation is similar to that of df-prv 36198 and df-gzext 36305 (𝑀⊧AxExt), but 𝑀 ⊨ Ax-Ext uses a different double turnstile character and a different format for the axiom name. (Contributed by BTernaryTau, 2-Oct-2026.)
⊢ (𝑀 ⊨ Ax-Ext ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))
Definition	df-smrep 36004*	Define the strong Axiom Scheme of Replacement predicate, which is true for a class that is a standard model of the strong version of the Axiom Scheme of Replacement used in wfaxrep 46078 (see comments there). Our notation includes an asterisk because we cannot guarantee that a standard model of the Axiom Scheme of Replacement will also be a standard model of this strong version. However, proving that a class is a standard model of the strong Axiom Scheme of Replacement is sufficient to prove that it is also a standard model of the ZF Axiom Scheme of Replacement. The exact expression used in wfaxrep 46078 is unsuitable for use in a definition because it contains a wff metavariable that we cannot quantify over. Instead, this definition uses the specific instance where 𝜑 is replaced with (Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧). This allows the set 𝑓 to effectively take on the role of the function-like wff, and since 𝑓 is a setvar, we are free to add a universal quantifier over it. The full scheme can then be recovered from this definition using smrepsrep 36008. This approach works because we can use our ax-rep 5231 to guarantee the existence of a suitable 𝑓 in V for any 𝑥 and function-like 𝜑. Note that 𝑀 is not guaranteed to contain a suitable 𝑓 (since it might not model the necessary axioms), so we must use an unrestricted universal quantifier for 𝑓. (Contributed by BTernaryTau, 30-Sep-2026.)
⊢ (𝑀 ⊨ Ax-Rep* ↔ ∀𝑓∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧)))))
Theorem	srepdflem1 36005*	Lemma for srepdf 36007. (Contributed by BTernaryTau, 30-Sep-2026.)
⊢ Ⅎ𝑤𝜑    &   ⊢ (𝜑 → ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜓 → 𝑧 = 𝑦))    &   ⊢ 𝐹 = (𝐺 ↾ 𝑥)    &   ⊢ 𝐺 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀 ∧ ∀𝑦𝜓)}    ⇒   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑀) → (∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜓) ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝐹 ∧ 𝑤 ∈ dom 𝐹 ∧ (𝐹‘𝑤) = 𝑧))))
Theorem	srepdflem2 36006*	Lemma for srepdf 36007. (Contributed by BTernaryTau, 30-Sep-2026.)
⊢ Ⅎ𝑓𝜑    &   ⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ (𝜑 → 𝑥 ∈ 𝑀)    &   ⊢ (𝜑 → (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧)))))    &   ⊢ (𝜑 → ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜓 → 𝑧 = 𝑦))    &   ⊢ 𝐹 = (𝐺 ↾ 𝑥)    &   ⊢ 𝐺 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀 ∧ ∀𝑦𝜓)}    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜓)))
Theorem	srepdf 36007*	Derive the strong version of the Axiom Scheme of Replacement used in wfaxrep 46078 (see comments there) from a particular instance of that version of Replacement. Note that this theorem depends on ax-rep 5231, so it does not follow that we could replace ax-rep 5231 with an axiom that lacks a wff metavariable. (Contributed by BTernaryTau, 30-Sep-2026.)
⊢ Ⅎ𝑓𝜑    &   ⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧)))))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜓 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜓))))
Theorem	smrepsrep 36008*	Derive the strong version of the Axiom Scheme of Replacement used in wfaxrep 46078 (see comments there) from 𝑀 ⊨ Ax-Rep*. (Contributed by BTernaryTau, 30-Sep-2026.)
⊢ (𝑀 ⊨ Ax-Rep* → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
Theorem	vsmext 36009	The universe is a standard model of the Axiom of Extensionality. (Contributed by BTernaryTau, 2-Oct-2026.)
⊢ V ⊨ Ax-Ext
Theorem	vsmrep 36010	The universe is a standard model of the strong Axiom Scheme of Replacement. (Contributed by BTernaryTau, 2-Oct-2026.)
⊢ V ⊨ Ax-Rep*
21.7  Mathbox for Mario Carneiro
21.7.1  Predicate calculus with all distinct variables
Axiom	ax-7d 36011*	Distinct variable version of ax-11 2193. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Axiom	ax-8d 36012*	Distinct variable version of ax-7 2040. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
Axiom	ax-9d1 36013	Distinct variable version of ax-6 1999, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥
Axiom	ax-9d2 36014*	Distinct variable version of ax-6 1999, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Axiom	ax-10d 36015*	Distinct variable version of axc11n 2455. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Axiom	ax-11d 36016*	Distinct variable version of ax-12 2212. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
21.7.2  Miscellaneous stuff
Theorem	quartfull 36017	The quartic equation, written out in full. This actually makes a fairly good Metamath stress test. Note that the length of this formula could be shortened significantly if the intermediate expressions were expanded and simplified, but it's not like this theorem will be used anyway. (Contributed by Mario Carneiro, 6-May-2015.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)) ≠ 0)    &   ⊢ (𝜑 → -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3) ≠ 0)    ⇒   ⊢ (𝜑 → ((((𝑋↑4) + (𝐴 · (𝑋↑3))) + ((𝐵 · (𝑋↑2)) + ((𝐶 · 𝑋) + 𝐷))) = 0 ↔ ((𝑋 = (( -(𝐴 / 4) − ((√‘ -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) + (√‘(( -(((√‘ -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) + ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘ -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2))))) ∨ 𝑋 = (( -(𝐴 / 4) − ((√‘ -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) − (√‘(( -(((√‘ -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) + ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘ -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)))))) ∨ (𝑋 = (( -(𝐴 / 4) + ((√‘ -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) + (√‘(( -(((√‘ -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) − ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘ -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2))))) ∨ 𝑋 = (( -(𝐴 / 4) + ((√‘ -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) − (√‘(( -(((√‘ -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) − ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘ -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))) / ((((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))) + (√‘(((( -(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (;27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (;72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (;12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)))))))))
21.7.3  Derangements and the Subfactorial
Theorem	deranglem 36018*	Lemma for derangements. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ (𝐴 ∈ Fin → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ 𝜑)} ∈ Fin)
Theorem	derangval 36019*	Define the derangement function, which counts the number of bijections from a set to itself such that no element is mapped to itself. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    ⇒   ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}))
Theorem	derangf 36020*	The derangement number is a function from finite sets to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    ⇒   ⊢ 𝐷:Fin⟶ℕ0
Theorem	derang0 36021*	The derangement number of the empty set. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    ⇒   ⊢ (𝐷‘∅) = 1
Theorem	derangsn 36022*	The derangement number of a singleton. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐷‘{𝐴}) = 0)
Theorem	derangenlem 36023*	One half of derangen 36024. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    ⇒   ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐴) ≤ (𝐷‘𝐵))
Theorem	derangen 36024*	The derangement number is a cardinal invariant, i.e. it only depends on the size of a set and not on its contents. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    ⇒   ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐴) = (𝐷‘𝐵))
Theorem	subfacval 36025*	The subfactorial is defined as the number of derangements (see derangval 36019) of the set (1...𝑁). (Contributed by Mario Carneiro, 21-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (𝐷‘(1...𝑁)))
Theorem	derangen2 36026*	Write the derangement number in terms of the subfactorial. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    ⇒   ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (𝑆‘(♯‘𝐴)))
Theorem	subfacf 36027*	The subfactorial is a function from nonnegative integers to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    ⇒   ⊢ 𝑆:ℕ0⟶ℕ0
Theorem	subfaclefac 36028*	The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) ≤ (!‘𝑁))
Theorem	subfac0 36029*	The subfactorial at zero. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    ⇒   ⊢ (𝑆‘0) = 1
Theorem	subfac1 36030*	The subfactorial at one. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    ⇒   ⊢ (𝑆‘1) = 0
Theorem	subfacp1lem1 36031*	Lemma for subfacp1 36038. The set 𝐾 together with {1, 𝑀} partitions the set 1...(𝑁 + 1). (Contributed by Mario Carneiro, 23-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))    &   ⊢ 𝑀 ∈ V    &   ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    ⇒   ⊢ (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1)))
Theorem	subfacp1lem2a 36032*	Lemma for subfacp1 36038. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 ∪ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))    &   ⊢ 𝑀 ∈ V    &   ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   ⊢ 𝐹 = (𝐺 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})    &   ⊢ (𝜑 → 𝐺:𝐾–1-1-onto→𝐾)    ⇒   ⊢ (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹‘𝑀) = 1))
Theorem	subfacp1lem2b 36033*	Lemma for subfacp1 36038. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 ∪ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))    &   ⊢ 𝑀 ∈ V    &   ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   ⊢ 𝐹 = (𝐺 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})    &   ⊢ (𝜑 → 𝐺:𝐾–1-1-onto→𝐾)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝐺‘𝑋))
Theorem	subfacp1lem3 36034*	Lemma for subfacp1 36038. In subfacp1lem6 36037 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (𝑓‘(𝑓‘1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements that satisfy this for fixed 𝑀 = (𝑓‘1) is in bijection with 𝑁 − 1 derangements, by simply dropping the 𝑥 = 1 and 𝑥 = 𝑀 points from the function to get a derangement on 𝐾 = (1...(𝑁 − 1)) ∖ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))    &   ⊢ 𝑀 ∈ V    &   ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   ⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1)}    &   ⊢ 𝐶 = {𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)}    ⇒   ⊢ (𝜑 → (♯‘𝐵) = (𝑆‘(𝑁 − 1)))
Theorem	subfacp1lem4 36035*	Lemma for subfacp1 36038. The function 𝐹, which swaps 1 with 𝑀 and leaves all other elements alone, is a bijection of order 2, i.e. it is its own inverse. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))    &   ⊢ 𝑀 ∈ V    &   ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   ⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) ≠ 1)}    &   ⊢ 𝐹 = (( I ↾ 𝐾) ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})    ⇒   ⊢ (𝜑 → ◡𝐹 = 𝐹)
Theorem	subfacp1lem5 36036*	Lemma for subfacp1 36038. In subfacp1lem6 36037 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (𝑓‘(𝑓‘1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements with (𝑓‘(𝑓‘1)) ≠ 1 for fixed 𝑀 = (𝑓‘1) is in bijection with derangements of 2...(𝑁 + 1), because pre-composing with the function 𝐹 swaps 1 and 𝑀 and turns the function into a bijection with (𝑓‘1) = 1 and (𝑓‘𝑥) ≠ 𝑥 for all other 𝑥, so dropping the point at 1 yields a derangement on the 𝑁 remaining points. (Contributed by Mario Carneiro, 23-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))    &   ⊢ 𝑀 ∈ V    &   ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   ⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) ≠ 1)}    &   ⊢ 𝐹 = (( I ↾ 𝐾) ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})    &   ⊢ 𝐶 = {𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}    ⇒   ⊢ (𝜑 → (♯‘𝐵) = (𝑆‘𝑁))
Theorem	subfacp1lem6 36037*	Lemma for subfacp1 36038. By induction, we cut up the set of all derangements on 𝑁 + 1 according to the 𝑁 possible values of (𝑓‘1) (since (𝑓‘1) ≠ 1), and for each set for fixed 𝑀 = (𝑓‘1), the subset of derangements with (𝑓‘𝑀) = 1 has size 𝑆(𝑁 − 1) (by subfacp1lem3 36034), while the subset with (𝑓‘𝑀) ≠ 1 has size 𝑆(𝑁) (by subfacp1lem5 36036). Adding it all up yields the desired equation 𝑁(𝑆(𝑁) + 𝑆(𝑁 − 1)) for the number of derangements on 𝑁 + 1. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}    ⇒   ⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))
Theorem	subfacp1 36038*	A two-term recurrence for the subfactorial. This theorem allows to forget the combinatorial definition of the derangement number in favor of the recursive definition provided by this theorem and subfac0 36029, subfac1 36030. (Contributed by Mario Carneiro, 23-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    ⇒   ⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1)))))
Theorem	subfacval2 36039*	A closed-form expression for the subfactorial. (Contributed by Mario Carneiro, 23-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(( -1↑𝑘) / (!‘𝑘))))
Theorem	subfaclim 36040*	The subfactorial converges rapidly to 𝑁! / e. This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    ⇒   ⊢ (𝑁 ∈ ℕ → (abs‘(((!‘𝑁) / e) − (𝑆‘𝑁))) < (1 / 𝑁))
Theorem	subfacval3 36041*	Another closed form expression for the subfactorial. The expression ⌊‘(𝑥 + 1 / 2) is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    ⇒   ⊢ (𝑁 ∈ ℕ → (𝑆‘𝑁) = (⌊‘(((!‘𝑁) / e) + (1 / 2))))
Theorem	derangfmla 36042*	The derangements formula, which expresses the number of derangements of a finite nonempty set in terms of the factorial. The expression ⌊‘(𝑥 + 1 / 2) is a way of saying "rounded to the nearest integer". This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.)
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))    ⇒   ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (𝐷‘𝐴) = (⌊‘(((!‘(♯‘𝐴)) / e) + (1 / 2))))
21.7.4  The Erdős-Szekeres theorem
Theorem	erdszelem1 36043*	Lemma for erdsze 36054. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ 𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}    ⇒   ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋))
Theorem	erdszelem2 36044*	Lemma for erdsze 36054. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ 𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}    ⇒   ⊢ ((♯ “ 𝑆) ∈ Fin ∧ (♯ “ 𝑆) ⊆ ℕ)
Theorem	erdszelem3 36045*	Lemma for erdsze 36054. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)    &   ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))    ⇒   ⊢ (𝐴 ∈ (1...𝑁) → (𝐾‘𝐴) = sup((♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}), ℝ, < ))
Theorem	erdszelem4 36046*	Lemma for erdsze 36054. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)    &   ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))    &   ⊢ 𝑂 Or ℝ    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ (1...𝑁)) → {𝐴} ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)})
Theorem	erdszelem5 36047*	Lemma for erdsze 36054. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)    &   ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))    &   ⊢ 𝑂 Or ℝ    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ (1...𝑁)) → (𝐾‘𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
Theorem	erdszelem6 36048*	Lemma for erdsze 36054. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)    &   ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))    &   ⊢ 𝑂 Or ℝ    ⇒   ⊢ (𝜑 → 𝐾:(1...𝑁)⟶ℕ)
Theorem	erdszelem7 36049*	Lemma for erdsze 36054. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)    &   ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))    &   ⊢ 𝑂 Or ℝ    &   ⊢ (𝜑 → 𝐴 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → ¬ (𝐾‘𝐴) ∈ (1...(𝑅 − 1)))    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠))))
Theorem	erdszelem8 36050*	Lemma for erdsze 36054. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)    &   ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))    &   ⊢ 𝑂 Or ℝ    &   ⊢ (𝜑 → 𝐴 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐵 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → ((𝐾‘𝐴) = (𝐾‘𝐵) → ¬ (𝐹‘𝐴)𝑂(𝐹‘𝐵)))
Theorem	erdszelem9 36051*	Lemma for erdsze 36054. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)    &   ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))    &   ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))    &   ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩)    ⇒   ⊢ (𝜑 → 𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
Theorem	erdszelem10 36052*	Lemma for erdsze 36054. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)    &   ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))    &   ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))    &   ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)    ⇒   ⊢ (𝜑 → ∃𝑚 ∈ (1...𝑁)(¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))))
Theorem	erdszelem11 36053*	Lemma for erdsze 36054. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)    &   ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))    &   ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))    &   ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))
Theorem	erdsze 36054*	The Erdős-Szekeres theorem. For any injective sequence 𝐹 on the reals of length at least (𝑅 − 1) · (𝑆 − 1) + 1, there is either a subsequence of length at least 𝑅 on which 𝐹 is increasing (i.e. a < , < order isomorphism) or a subsequence of length at least 𝑆 on which 𝐹 is decreasing (i.e. a < , ◡ < order isomorphism, recalling that ◡ < is the "greater than" relation). This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))
Theorem	erdsze2lem1 36055*	Lemma for erdsze2 36057. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1))    &   ⊢ (𝜑 → 𝑁 < (♯‘𝐴))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
Theorem	erdsze2lem2 36056*	Lemma for erdsze2 36057. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1))    &   ⊢ (𝜑 → 𝑁 < (♯‘𝐴))    &   ⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))–1-1→𝐴)    &   ⊢ (𝜑 → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺))    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))
Theorem	erdsze2 36057*	Generalize the statement of the Erdős-Szekeres theorem erdsze 36054 to "sequences" indexed by an arbitrary subset of ℝ, which can be infinite. This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < (♯‘𝐴))    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))
21.7.5  The Kuratowski closure-complement theorem
Theorem	kur14lem1 36058	Lemma for kur14 36068. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ 𝐴 ⊆ 𝑋    &   ⊢ (𝑋 ∖ 𝐴) ∈ 𝑇    &   ⊢ (𝐾‘𝐴) ∈ 𝑇    ⇒   ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
Theorem	kur14lem2 36059	Lemma for kur14 36068. Write interior in terms of closure and complement: 𝑖𝐴 = 𝑐𝑘𝑐𝐴 where 𝑐 is complement and 𝑘 is closure. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐽 ∈ Top    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝐼 = (int‘𝐽)    &   ⊢ 𝐴 ⊆ 𝑋    ⇒   ⊢ (𝐼‘𝐴) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴)))
Theorem	kur14lem3 36060	Lemma for kur14 36068. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐽 ∈ Top    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝐼 = (int‘𝐽)    &   ⊢ 𝐴 ⊆ 𝑋    ⇒   ⊢ (𝐾‘𝐴) ⊆ 𝑋
Theorem	kur14lem4 36061	Lemma for kur14 36068. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐽 ∈ Top    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝐼 = (int‘𝐽)    &   ⊢ 𝐴 ⊆ 𝑋    ⇒   ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴
Theorem	kur14lem5 36062	Lemma for kur14 36068. Closure is an idempotent operation in the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐽 ∈ Top    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝐼 = (int‘𝐽)    &   ⊢ 𝐴 ⊆ 𝑋    ⇒   ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴)
Theorem	kur14lem6 36063	Lemma for kur14 36068. If 𝑘 is the complementation operator and 𝑘 is the closure operator, this expresses the identity 𝑘𝑐𝑘𝐴 = 𝑘𝑐𝑘𝑐𝑘𝑐𝑘𝐴 for any subset 𝐴 of the topological space. This is the key result that lets us cut down long enough sequences of 𝑐𝑘𝑐𝑘... that arise when applying closure and complement repeatedly to 𝐴, and explains why we end up with a number as large as 14, yet no larger. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐽 ∈ Top    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝐼 = (int‘𝐽)    &   ⊢ 𝐴 ⊆ 𝑋    &   ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴))    ⇒   ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = (𝐾‘𝐵)
Theorem	kur14lem7 36064	Lemma for kur14 36068: main proof. The set 𝑇 here contains all the distinct combinations of 𝑘 and 𝑐 that can arise, and we prove here that applying 𝑘 or 𝑐 to any element of 𝑇 yields another element of 𝑇. In operator shorthand, we have 𝑇 = {𝐴, 𝑐𝐴, 𝑘𝐴 , 𝑐𝑘𝐴, 𝑘𝑐𝐴, 𝑐𝑘𝑐𝐴, 𝑘𝑐𝑘𝐴, 𝑐𝑘𝑐𝑘𝐴, 𝑘𝑐𝑘𝑐𝐴, 𝑐𝑘𝑐𝑘𝑐𝐴, 𝑘𝑐𝑘𝑐𝑘𝐴, 𝑐𝑘𝑐𝑘𝑐𝑘𝐴, 𝑘𝑐𝑘𝑐𝑘𝑐𝐴, 𝑐𝑘𝑐𝑘𝑐𝑘𝑐𝐴}. From the identities 𝑐𝑐𝐴 = 𝐴 and 𝑘𝑘𝐴 = 𝑘𝐴, we can reduce any operator combination containing two adjacent identical operators, which is why the list only contains alternating sequences. The reason the sequences don't keep going after a certain point is due to the identity 𝑘𝑐𝑘𝐴 = 𝑘𝑐𝑘𝑐𝑘𝑐𝑘𝐴, proved in kur14lem6 36063. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐽 ∈ Top    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝐼 = (int‘𝐽)    &   ⊢ 𝐴 ⊆ 𝑋    &   ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴))    &   ⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴))    &   ⊢ 𝐷 = (𝐼‘(𝐾‘𝐴))    &   ⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))    ⇒   ⊢ (𝑁 ∈ 𝑇 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
Theorem	kur14lem8 36065	Lemma for kur14 36068. Show that the set 𝑇 contains at most 14 elements. (It could be less if some of the operators take the same value for a given set, but Kuratowski showed that this upper bound of 14 is tight in the sense that there exist topological spaces and subsets of these spaces for which all 14 generated sets are distinct, and indeed the real numbers form such a topological space.) (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐽 ∈ Top    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝐼 = (int‘𝐽)    &   ⊢ 𝐴 ⊆ 𝑋    &   ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴))    &   ⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴))    &   ⊢ 𝐷 = (𝐼‘(𝐾‘𝐴))    &   ⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))    ⇒   ⊢ (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ ;14)
Theorem	kur14lem9 36066*	Lemma for kur14 36068. Since the set 𝑇 is closed under closure and complement, it contains the minimal set 𝑆 as a subset, so 𝑆 also has at most 14 elements. (Indeed 𝑆 = 𝑇, and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐽 ∈ Top    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝐼 = (int‘𝐽)    &   ⊢ 𝐴 ⊆ 𝑋    &   ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴))    &   ⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴))    &   ⊢ 𝐷 = (𝐼‘(𝐾‘𝐴))    &   ⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))    &   ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}    ⇒   ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14)
Theorem	kur14lem10 36067*	Lemma for kur14 36068. Discharge the set 𝑇. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐽 ∈ Top    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}    &   ⊢ 𝐴 ⊆ 𝑋    ⇒   ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14)
Theorem	kur14 36068*	Kuratowski's closure-complement theorem. There are at most 14 sets which can be obtained by the application of the closure and complement operations to a set in a topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14))
21.7.6  Retracts and sections
Syntax	cretr 36069	Extend class notation with the retract relation.
class Retr
Definition	df-retr 36070*	Define the set of retractions on two topological spaces. We say that 𝑅 is a retraction from 𝐽 to 𝐾. or 𝑅 ∈ (𝐽 Retr 𝐾) iff there is an 𝑆 such that 𝑅:𝐽⟶𝐾, 𝑆:𝐾⟶𝐽 are continuous functions called the retraction and section respectively, and their composite 𝑅 ∘ 𝑆 is homotopic to the identity map. If a retraction exists, we say 𝐽 is a retract of 𝐾. (This terminology is borrowed from HoTT and appears to be nonstandard, although it has similaries to the concept of retract in the category of topological spaces and to a deformation retract in general topology.) Two topological spaces that are retracts of each other are called homotopy equivalent. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅})
21.7.7  Path-connected and simply connected spaces
Syntax	cpconn 36071	Extend class notation with the class of path-connected topologies.
class PConn
Syntax	csconn 36072	Extend class notation with the class of simply connected topologies.
class SConn
Definition	df-pconn 36073*	Define the class of path-connected topologies. A topology is path-connected if there is a path (a continuous function from the closed unit interval) that goes from 𝑥 to 𝑦 for any points 𝑥, 𝑦 in the space. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ PConn = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}
Definition	df-sconn 36074*	Define the class of simply connected topologies. A topology is simply connected if it is path-connected and every loop (continuous path with identical start and endpoint) is contractible to a point (path-homotopic to a constant function). (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}))}
Theorem	ispconn 36075*	The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
Theorem	pconncn 36076*	The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ PConn ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵))
Theorem	pconntop 36077	A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top)
Theorem	issconn 36078*	The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
Theorem	sconnpconn 36079	A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)
Theorem	sconntop 36080	A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top)
Theorem	sconnpht 36081	A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ ((𝐽 ∈ SConn ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
Theorem	cnpconn 36082	An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn)
Theorem	pconnconn 36083	A path-connected space is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐽 ∈ PConn → 𝐽 ∈ Conn)
Theorem	txpconn 36084	The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)
Theorem	ptpconn 36085	The topological product of a collection of path-connected spaces is path-connected. The proof uses the axiom of choice. (Contributed by Mario Carneiro, 17-Feb-2015.) Remove dependency on ax-reg 9590 and ax-inf2 9646. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → (∏t‘𝐹) ∈ PConn)
Theorem	indispconn 36086	The indiscrete topology (or trivial topology) on any set is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ {∅, 𝐴} ∈ PConn
Theorem	connpconn 36087	A connected and locally path-connected space is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.)
⊢ ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) → 𝐽 ∈ PConn)
Theorem	qtoppconn 36088	A quotient of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ PConn ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ PConn)
Theorem	pconnpi1 36089	All fundamental groups in a path-connected space are isomorphic. (Contributed by Mario Carneiro, 12-Feb-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑃 = (𝐽 π1 𝐴)    &   ⊢ 𝑄 = (𝐽 π1 𝐵)    &   ⊢ 𝑆 = (Base‘𝑃)    &   ⊢ 𝑇 = (Base‘𝑄)    ⇒   ⊢ ((𝐽 ∈ PConn ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑃 ≃𝑔 𝑄)
Theorem	sconnpht2 36090	Any two paths in a simply connected space with the same start and end point are path-homotopic. (Contributed by Mario Carneiro, 12-Feb-2015.)
⊢ (𝜑 → 𝐽 ∈ SConn)    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0))    &   ⊢ (𝜑 → (𝐹‘1) = (𝐺‘1))    ⇒   ⊢ (𝜑 → 𝐹( ≃ph‘𝐽)𝐺)
Theorem	sconnpi1 36091	A path-connected topological space is simply connected iff its fundamental group is trivial. (Contributed by Mario Carneiro, 12-Feb-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) → (𝐽 ∈ SConn ↔ (Base‘(𝐽 π1 𝑌)) ≈ 1o))
Theorem	txsconnlem 36092	Lemma for txsconn 36093. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ (𝜑 → 𝑅 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ∈ Top)    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn (𝑅 ×t 𝑆)))    &   ⊢ 𝐴 = ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)    &   ⊢ 𝐵 = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})))    ⇒   ⊢ (𝜑 → 𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}))
Theorem	txsconn 36093	The topological product of two simply connected spaces is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn)
Theorem	cvxpconn 36094*	A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11284. (Revised by GG, 19-Apr-2025.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝑆)    ⇒   ⊢ (𝜑 → 𝐾 ∈ PConn)
Theorem	cvxsconn 36095*	A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11284. (Revised by GG, 19-Apr-2025.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝑆)    ⇒   ⊢ (𝜑 → 𝐾 ∈ SConn)
Theorem	blsconn 36096	An open ball in the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝑆 = (𝑃(ball‘(abs ∘ − ))𝑅)    &   ⊢ 𝐾 = (𝐽 ↾t 𝑆)    ⇒   ⊢ ((𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) → 𝐾 ∈ SConn)
Theorem	cnllysconn 36097	The topology of the complex numbers is locally simply connected. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Locally SConn
Theorem	resconn 36098	A subset of ℝ is simply connected iff it is connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐽 = ((topGen‘ran (,)) ↾t 𝐴)    ⇒   ⊢ (𝐴 ⊆ ℝ → (𝐽 ∈ SConn ↔ 𝐽 ∈ Conn))
Theorem	ioosconn 36099	An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ ((topGen‘ran (,)) ↾t (𝐴(,)𝐵)) ∈ SConn
Theorem	iccsconn 36100	A closed interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ SConn)