P. 352 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35101-35200   *Has distinct variable group(s)

Type	Label	Description
Statement
21.5.4  Graph theory
Theorem	lfuhgr 35101*	A hypergraph is loop-free if and only if every edge connects at least two vertices. (Contributed by BTernaryTau, 15-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
Theorem	lfuhgr2 35102*	A hypergraph is loop-free if and only if every edge is not a loop. (Contributed by BTernaryTau, 15-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1))
Theorem	lfuhgr3 35103*	A hypergraph is loop-free if and only if none of its edges connect to only one vertex. (Contributed by BTernaryTau, 15-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ¬ ∃𝑎{𝑎} ∈ (Edg‘𝐺)))
Theorem	cplgredgex 35104*	Any two (distinct) vertices in a complete graph are connected to each other by at least one edge. (Contributed by BTernaryTau, 2-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝐺 ∈ ComplGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))
Theorem	cusgredgex 35105	Any two (distinct) vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 3-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝐺 ∈ ComplUSGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → {𝐴, 𝐵} ∈ 𝐸))
Theorem	cusgredgex2 35106	Any two distinct vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 4-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝐺 ∈ ComplUSGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ 𝐸))
Theorem	pfxwlk 35107	A prefix of a walk is a walk. (Contributed by BTernaryTau, 2-Dec-2023.)
⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿)(Walks‘𝐺)(𝑃 prefix (𝐿 + 1)))
Theorem	revwlk 35108	The reverse of a walk is a walk. (Contributed by BTernaryTau, 30-Nov-2023.)
⊢ (𝐹(Walks‘𝐺)𝑃 → (reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃))
Theorem	revwlkb 35109	Two words represent a walk if and only if their reverses also represent a walk. (Contributed by BTernaryTau, 4-Dec-2023.)
⊢ ((𝐹 ∈ Word 𝑊 ∧ 𝑃 ∈ Word 𝑈) → (𝐹(Walks‘𝐺)𝑃 ↔ (reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃)))
Theorem	swrdwlk 35110	Two matching subwords of a walk also represent a walk. (Contributed by BTernaryTau, 7-Dec-2023.)
⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐵 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 substr ⟨𝐵, 𝐿⟩)(Walks‘𝐺)(𝑃 substr ⟨𝐵, (𝐿 + 1)⟩))
Theorem	pthhashvtx 35111	A graph containing a path has at least as many vertices as there are edges in the path. (Contributed by BTernaryTau, 5-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐹(Paths‘𝐺)𝑃 → (♯‘𝐹) ≤ (♯‘𝑉))
Theorem	pthisspthorcycl 35112	A path is either a simple path or a cycle (or both). (Contributed by BTernaryTau, 20-Oct-2023.)
⊢ (𝐹(Paths‘𝐺)𝑃 → (𝐹(SPaths‘𝐺)𝑃 ∨ 𝐹(Cycles‘𝐺)𝑃))
Theorem	spthcycl 35113	A walk is a trivial path if and only if it is both a simple path and a cycle. (Contributed by BTernaryTau, 8-Oct-2023.)
⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 𝐹 = ∅) ↔ (𝐹(SPaths‘𝐺)𝑃 ∧ 𝐹(Cycles‘𝐺)𝑃))
Theorem	usgrgt2cycl 35114	A non-trivial cycle in a simple graph has a length greater than 2. (Contributed by BTernaryTau, 24-Sep-2023.)
⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 < (♯‘𝐹))
Theorem	usgrcyclgt2v 35115	A simple graph with a non-trivial cycle must have at least 3 vertices. (Contributed by BTernaryTau, 5-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 < (♯‘𝑉))
Theorem	subgrwlk 35116	If a walk exists in a subgraph of a graph 𝐺, then that walk also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
⊢ (𝑆 SubGraph 𝐺 → (𝐹(Walks‘𝑆)𝑃 → 𝐹(Walks‘𝐺)𝑃))
Theorem	subgrtrl 35117	If a trail exists in a subgraph of a graph 𝐺, then that trail also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
⊢ (𝑆 SubGraph 𝐺 → (𝐹(Trails‘𝑆)𝑃 → 𝐹(Trails‘𝐺)𝑃))
Theorem	subgrpth 35118	If a path exists in a subgraph of a graph 𝐺, then that path also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
⊢ (𝑆 SubGraph 𝐺 → (𝐹(Paths‘𝑆)𝑃 → 𝐹(Paths‘𝐺)𝑃))
Theorem	subgrcycl 35119	If a cycle exists in a subgraph of a graph 𝐺, then that cycle also exists in 𝐺. (Contributed by BTernaryTau, 23-Oct-2023.)
⊢ (𝑆 SubGraph 𝐺 → (𝐹(Cycles‘𝑆)𝑃 → 𝐹(Cycles‘𝐺)𝑃))
Theorem	cusgr3cyclex 35120*	Every complete simple graph with more than two vertices has a 3-cycle. (Contributed by BTernaryTau, 4-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 2 < (♯‘𝑉)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
Theorem	loop1cycl 35121*	A hypergraph has a cycle of length one if and only if it has a loop. (Contributed by BTernaryTau, 13-Oct-2023.)
⊢ (𝐺 ∈ UHGraph → (∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) ↔ {𝐴} ∈ (Edg‘𝐺)))
Theorem	2cycld 35122	Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    &   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))    &   ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃)
Theorem	2cycl2d 35123	Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐴”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐾)))    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    ⇒   ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃)
Theorem	umgr2cycllem 35124*	Lemma for umgr2cycl 35125. (Contributed by BTernaryTau, 17-Oct-2023.)
⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ UMGraph)    &   ⊢ (𝜑 → 𝐽 ∈ dom 𝐼)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    &   ⊢ (𝜑 → (𝐼‘𝐽) = (𝐼‘𝐾))    ⇒   ⊢ (𝜑 → ∃𝑝 𝐹(Cycles‘𝐺)𝑝)
Theorem	umgr2cycl 35125*	A multigraph with two distinct edges that connect the same vertices has a 2-cycle. (Contributed by BTernaryTau, 17-Oct-2023.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ ∃𝑗 ∈ dom 𝐼∃𝑘 ∈ dom 𝐼((𝐼‘𝑗) = (𝐼‘𝑘) ∧ 𝑗 ≠ 𝑘)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 2))
21.5.4.1  Acyclic graphs
Syntax	cacycgr 35126	Extend class notation with acyclic graphs.
class AcyclicGraph
Definition	df-acycgr 35127*	Define the class of all acyclic graphs. A graph is called acyclic if it has no (non-trivial) cycles. (Contributed by BTernaryTau, 11-Oct-2023.)
⊢ AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)}
Theorem	dfacycgr1 35128*	An alternate definition of the class of all acyclic graphs that requires all cycles to be trivial. (Contributed by BTernaryTau, 11-Oct-2023.)
⊢ AcyclicGraph = {𝑔 ∣ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)}
Theorem	isacycgr 35129*	The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)))
Theorem	isacycgr1 35130*	The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)))
Theorem	acycgrcycl 35131	Any cycle in an acyclic graph is trivial (i.e. has one vertex and no edges). (Contributed by BTernaryTau, 12-Oct-2023.)
⊢ ((𝐺 ∈ AcyclicGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → 𝐹 = ∅)
Theorem	acycgr0v 35132	A null graph (with no vertices) is an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → 𝐺 ∈ AcyclicGraph)
Theorem	acycgr1v 35133	A multigraph with one vertex is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ (♯‘𝑉) = 1) → 𝐺 ∈ AcyclicGraph)
Theorem	acycgr2v 35134	A simple graph with two vertices is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 2) → 𝐺 ∈ AcyclicGraph)
Theorem	prclisacycgr 35135*	A proper class (representing a null graph, see vtxvalprc 29076) has the property of an acyclic graph (see also acycgr0v 35132). (Contributed by BTernaryTau, 11-Oct-2023.) (New usage is discouraged.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (¬ 𝐺 ∈ V → ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))
Theorem	acycgrislfgr 35136*	An acyclic hypergraph is a loop-free hypergraph. (Contributed by BTernaryTau, 15-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ AcyclicGraph ∧ 𝐺 ∈ UHGraph) → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
Theorem	upgracycumgr 35137	An acyclic pseudograph is a multigraph. (Contributed by BTernaryTau, 15-Oct-2023.)
⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ UMGraph)
Theorem	umgracycusgr 35138	An acyclic multigraph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.)
⊢ ((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ USGraph)
Theorem	upgracycusgr 35139	An acyclic pseudograph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.)
⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ USGraph)
Theorem	cusgracyclt3v 35140	A complete simple graph is acyclic if and only if it has fewer than three vertices. (Contributed by BTernaryTau, 20-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ (♯‘𝑉) < 3))
Theorem	pthacycspth 35141	A path in an acyclic graph is a simple path. (Contributed by BTernaryTau, 21-Oct-2023.)
⊢ ((𝐺 ∈ AcyclicGraph ∧ 𝐹(Paths‘𝐺)𝑃) → 𝐹(SPaths‘𝐺)𝑃)
Theorem	acycgrsubgr 35142	The subgraph of an acyclic graph is also acyclic. (Contributed by BTernaryTau, 23-Oct-2023.)
⊢ ((𝐺 ∈ AcyclicGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ AcyclicGraph)
21.6  Mathbox for Mario Carneiro
21.6.1  Predicate calculus with all distinct variables
Axiom	ax-7d 35143*	Distinct variable version of ax-11 2154. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Axiom	ax-8d 35144*	Distinct variable version of ax-7 2004. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
Axiom	ax-9d1 35145	Distinct variable version of ax-6 1964, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥
Axiom	ax-9d2 35146*	Distinct variable version of ax-6 1964, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Axiom	ax-10d 35147*	Distinct variable version of axc11n 2428. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Axiom	ax-11d 35148*	Distinct variable version of ax-12 2174. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
21.6.2  Miscellaneous stuff
Theorem	quartfull 35149	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)))))))))
21.6.3  Derangements and the Subfactorial
Theorem	deranglem 35150*	Lemma for derangements. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ (𝐴 ∈ Fin → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ 𝜑)} ∈ Fin)
Theorem	derangval 35151*	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 35152*	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 35153*	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 35154*	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 35155*	One half of derangen 35156. (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 35156*	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 35157*	The subfactorial is defined as the number of derangements (see derangval 35151) 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 35158*	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 35159*	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 35160*	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 35161*	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 35162*	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 35163*	Lemma for subfacp1 35170. 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 35164*	Lemma for subfacp1 35170. 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 35165*	Lemma for subfacp1 35170. 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 35166*	Lemma for subfacp1 35170. In subfacp1lem6 35169 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 35167*	Lemma for subfacp1 35170. 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 35168*	Lemma for subfacp1 35170. In subfacp1lem6 35169 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 35169*	Lemma for subfacp1 35170. 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 35166), while the subset with (𝑓‘𝑀) ≠ 1 has size 𝑆(𝑁) (by subfacp1lem5 35168). 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 35170*	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 35161, subfac1 35162. (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 35171*	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 35172*	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 35173*	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 35174*	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))))
21.6.4  The Erdős-Szekeres theorem
Theorem	erdszelem1 35175*	Lemma for erdsze 35186. (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 35176*	Lemma for erdsze 35186. (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 35177*	Lemma for erdsze 35186. (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 35178*	Lemma for erdsze 35186. (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 35179*	Lemma for erdsze 35186. (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 35180*	Lemma for erdsze 35186. (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 35181*	Lemma for erdsze 35186. (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 35182*	Lemma for erdsze 35186. (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 35183*	Lemma for erdsze 35186. (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 35184*	Lemma for erdsze 35186. (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 35185*	Lemma for erdsze 35186. (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 35186*	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 35187*	Lemma for erdsze2 35189. (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 35188*	Lemma for erdsze2 35189. (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 35189*	Generalize the statement of the Erdős-Szekeres theorem erdsze 35186 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)))))
21.6.5  The Kuratowski closure-complement theorem
Theorem	kur14lem1 35190	Lemma for kur14 35200. (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 35191	Lemma for kur14 35200. 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 35192	Lemma for kur14 35200. 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 35193	Lemma for kur14 35200. 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 35194	Lemma for kur14 35200. 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 35195	Lemma for kur14 35200. 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 35196	Lemma for kur14 35200: 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 35195. (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 35197	Lemma for kur14 35200. 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 35198*	Lemma for kur14 35200. 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 35199*	Lemma for kur14 35200. 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 35200*	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))