P. 250 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cncfmet 24901	Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
⊢ 𝐶 = ((abs ∘ − ) ↾ (𝐴 × 𝐴))    &   ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝐵 × 𝐵))    &   ⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐽 Cn 𝐾))
Theorem	cncfcn 24902	Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝐴)    &   ⊢ 𝐿 = (𝐽 ↾t 𝐵)    ⇒   ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐾 Cn 𝐿))
Theorem	cncfcn1 24903	Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
Theorem	cncfmptc 24904*	A constant function is a continuous function on ℂ. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝐴) ∈ (𝑆–cn→𝑇))
Theorem	cncfmptid 24905*	The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝑥) ∈ (𝑆–cn→𝑇))
Theorem	cncfmpt1f 24906*	Composition of continuous functions. –cn→ analogue of cnmpt11f 23654. (Contributed by Mario Carneiro, 3-Sep-2014.)
⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝑋–cn→ℂ))
Theorem	cncfmpt2f 24907*	Composition of continuous functions. –cn→ analogue of cnmpt12f 23656. (Contributed by Mario Carneiro, 3-Sep-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ))
Theorem	cncfmpt2ss 24908*	Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→𝑆))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→𝑆))    &   ⊢ 𝑆 ⊆ ℂ    &   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆))
Theorem	addccncf 24909*	Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 + 𝐴))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	idcncf 24910	The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 24905 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝑥)    ⇒   ⊢ 𝐹 ∈ (ℂ–cn→ℂ)
Theorem	sub1cncf 24911*	Subtracting a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 − 𝐴))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	sub2cncf 24912*	Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 − 𝑥))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	cdivcncf 24913*	Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.)
⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ ((ℂ ∖ {0})–cn→ℂ))
Theorem	negcncf 24914*	The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.) Avoid ax-mulf 11116. (Revised by GG, 16-Mar-2025.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥)    ⇒   ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ))
Theorem	negfcncf 24915*	The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ -(𝐹‘𝑥))    ⇒   ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐺 ∈ (𝐴–cn→ℂ))
Theorem	abscncfALT 24916	Absolute value is continuous. Alternate proof of abscncf 24893. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ abs ∈ (ℂ–cn→ℝ)
Theorem	cncfcnvcn 24917	Rewrite cmphaushmeo 23790 for functions on the complex numbers. (Contributed by Mario Carneiro, 19-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝑋)    ⇒   ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑌–cn→𝑋)))
Theorem	expcncf 24918*	The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 24864. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ))
Theorem	cnmptre 24919*	Lemma for iirevcn 24922 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.)
⊢ 𝑅 = (TopOpen‘ℂfld)    &   ⊢ 𝐽 = ((topGen‘ran (,)) ↾t 𝐴)    &   ⊢ 𝐾 = ((topGen‘ran (,)) ↾t 𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝐾))
Theorem	cnmpopc 24920*	Piecewise definition of a continuous function on a real interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
⊢ 𝑅 = (topGen‘ran (,))    &   ⊢ 𝑀 = (𝑅 ↾t (𝐴[,]𝐵))    &   ⊢ 𝑁 = (𝑅 ↾t (𝐵[,]𝐶))    &   ⊢ 𝑂 = (𝑅 ↾t (𝐴[,]𝐶))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐵 ∧ 𝑦 ∈ 𝑋)) → 𝐷 = 𝐸)    &   ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))    &   ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ 𝐸) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾))
Theorem	iirev 24921	Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑋 ∈ (0[,]1) → (1 − 𝑋) ∈ (0[,]1))
Theorem	iirevcn 24922	The reversion function is a continuous map of the unit interval. (Contributed by Mario Carneiro, 6-Jun-2014.)
⊢ (𝑥 ∈ (0[,]1) ↦ (1 − 𝑥)) ∈ (II Cn II)
Theorem	iihalf1 24923	Map the first half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑋 ∈ (0[,](1 / 2)) → (2 · 𝑋) ∈ (0[,]1))
Theorem	iihalf1cn 24924	The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11116. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = ((topGen‘ran (,)) ↾t (0[,](1 / 2)))    ⇒   ⊢ (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II)
Theorem	iihalf2 24925	Map the second half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑋 ∈ ((1 / 2)[,]1) → ((2 · 𝑋) − 1) ∈ (0[,]1))
Theorem	iihalf2cn 24926	The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11116. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1))    ⇒   ⊢ (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II)
Theorem	elii1 24927	Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
⊢ (𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ (0[,]1) ∧ 𝑋 ≤ (1 / 2)))
Theorem	elii2 24928	Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ∈ ((1 / 2)[,]1))
Theorem	iimulcl 24929	The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐴 ∈ (0[,]1) ∧ 𝐵 ∈ (0[,]1)) → (𝐴 · 𝐵) ∈ (0[,]1))
Theorem	iimulcn 24930*	Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.) Avoid ax-mulf 11116. (Revised by GG, 16-Mar-2025.)
⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥 · 𝑦)) ∈ ((II ×t II) Cn II)
Theorem	icoopnst 24931	A half-open interval starting at 𝐴 is open in the closed interval from 𝐴 to 𝐵. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) → (𝐴[,)𝐶) ∈ 𝐽))
Theorem	iocopnst 24932	A half-open interval ending at 𝐵 is open in the closed interval from 𝐴 to 𝐵. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶(,]𝐵) ∈ 𝐽))
Theorem	icchmeo 24933*	The natural bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.) Avoid ax-mulf 11116. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴)))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽 ↾t (𝐴[,]𝐵))))
Theorem	icopnfcnv 24934*	Define a bijection from [0, 1) to [0, +∞). (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))    ⇒   ⊢ (𝐹:(0[,)1)–1-1-onto→(0[,)+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦))))
Theorem	icopnfhmeo 24935*	The defined bijection from [0, 1) to [0, +∞) is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ∧ 𝐹 ∈ ((𝐽 ↾t (0[,)1))Homeo(𝐽 ↾t (0[,)+∞))))
Theorem	iccpnfcnv 24936*	Define a bijection from [0, 1] to [0, +∞]. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))    ⇒   ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
Theorem	iccpnfhmeo 24937	The defined bijection from [0, 1] to [0, +∞] is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))    &   ⊢ 𝐾 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    ⇒   ⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ∧ 𝐹 ∈ (IIHomeo𝐾))
Theorem	xrhmeo 24938*	The bijection from [-1, 1] to the extended reals is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))    &   ⊢ 𝐺 = (𝑦 ∈ (-1[,]1) ↦ if(0 ≤ 𝑦, (𝐹‘𝑦), -𝑒(𝐹‘-𝑦)))    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝐺 Isom < , < ((-1[,]1), ℝ*) ∧ 𝐺 ∈ ((𝐽 ↾t (-1[,]1))Homeo(ordTop‘ ≤ )))
Theorem	xrhmph 24939	The extended reals are homeomorphic to the interval [0, 1]. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ II ≃ (ordTop‘ ≤ )
Theorem	xrcmp 24940	The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 24797), this means that ℝ* is a compactification of ℝ. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (ordTop‘ ≤ ) ∈ Comp
Theorem	xrconn 24941	The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (ordTop‘ ≤ ) ∈ Conn
Theorem	icccvx 24942	A linear combination of two reals lies in the interval between them. Equivalently, a closed interval is a convex set. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝐶) + (𝑇 · 𝐷)) ∈ (𝐴[,]𝐵)))
Theorem	oprpiece1res1 24943*	Restriction to the first part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐴 ≤ 𝐵    &   ⊢ 𝑅 ∈ V    &   ⊢ 𝑆 ∈ V    &   ⊢ 𝐾 ∈ (𝐴[,]𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ 𝑅)    ⇒   ⊢ (𝐹 ↾ ((𝐴[,]𝐾) × 𝐶)) = 𝐺
Theorem	oprpiece1res2 24944*	Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐴 ≤ 𝐵    &   ⊢ 𝑅 ∈ V    &   ⊢ 𝑆 ∈ V    &   ⊢ 𝐾 ∈ (𝐴[,]𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))    &   ⊢ (𝑥 = 𝐾 → 𝑅 = 𝑃)    &   ⊢ (𝑥 = 𝐾 → 𝑆 = 𝑄)    &   ⊢ (𝑦 ∈ 𝐶 → 𝑃 = 𝑄)    &   ⊢ 𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ 𝑆)    ⇒   ⊢ (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = 𝐺
Theorem	cnrehmeo 24945*	The canonical bijection from (ℝ × ℝ) to ℂ described in cnref1o 12933 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if (ℝ × ℝ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.) Avoid ax-mulf 11116. (Revised by GG, 16-Mar-2025.)
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)
Theorem	cnheiborlem 24946*	Lemma for cnheibor 24947. (Contributed by Mario Carneiro, 14-Sep-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝑇 = (𝐽 ↾t 𝑋)    &   ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   ⊢ 𝑌 = (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))    ⇒   ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑇 ∈ Comp)
Theorem	cnheibor 24947*	Heine-Borel theorem for complex numbers. A subset of ℂ is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝑇 = (𝐽 ↾t 𝑋)    ⇒   ⊢ (𝑋 ⊆ ℂ → (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)))
Theorem	cnllycmp 24948	The topology on the complex numbers is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ 𝑛-Locally Comp
Theorem	rellycmp 24949	The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ (topGen‘ran (,)) ∈ 𝑛-Locally Comp
Theorem	bndth 24950*	The Boundedness Theorem. A continuous function from a compact topological space to the reals is bounded (above). (Boundedness below is obtained by applying this theorem to -𝐹.) (Contributed by Mario Carneiro, 12-Aug-2014.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)
Theorem	evth 24951*	The Extreme Value Theorem. A continuous function from a nonempty compact topological space to the reals attains its maximum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))
Theorem	evth2 24952*	The Extreme Value Theorem, minimum version. A continuous function from a nonempty compact topological space to the reals attains its minimum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦))
Theorem	lebnumlem1 24953*	Lemma for lebnum 24956. The function 𝐹 measures the sum of all of the distances to escape the sets of the cover. Since by assumption it is a cover, there is at least one set which covers a given point, and since it is open, the point is a positive distance from the edge of the set. Thus, the sum is a strictly positive number. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))    ⇒   ⊢ (𝜑 → 𝐹:𝑋⟶ℝ+)
Theorem	lebnumlem2 24954*	Lemma for lebnum 24956. As a finite sum of point-to-set distance functions, which are continuous by metdscn 24847, the function 𝐹 is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))    &   ⊢ 𝐾 = (topGen‘ran (,))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
Theorem	lebnumlem3 24955*	Lemma for lebnum 24956. By the previous lemmas, 𝐹 is continuous and positive on a compact set, so it has a positive minimum 𝑟. Then setting 𝑑 = 𝑟 / ♯(𝑈), since for each 𝑢 ∈ 𝑈 we have ball(𝑥, 𝑑) ⊆ 𝑢 iff 𝑑 ≤ 𝑑(𝑥, 𝑋 ∖ 𝑢), if ¬ ball(𝑥, 𝑑) ⊆ 𝑢 for all 𝑢 then summing over 𝑢 yields Σ𝑢 ∈ 𝑈𝑑(𝑥, 𝑋 ∖ 𝑢) = 𝐹(𝑥) < Σ𝑢 ∈ 𝑈𝑑 = 𝑟, in contradiction to the assumption that 𝑟 is the minimum of 𝐹. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) (Revised by AV, 30-Sep-2020.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))    &   ⊢ 𝐾 = (topGen‘ran (,))    ⇒   ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
Theorem	lebnum 24956*	The Lebesgue number lemma, or Lebesgue covering lemma. If 𝑋 is a compact metric space and 𝑈 is an open cover of 𝑋, then there exists a positive real number 𝑑 such that every ball of size 𝑑 (and every subset of a ball of size 𝑑, including every subset of diameter less than 𝑑) is a subset of some member of the cover. (Contributed by Mario Carneiro, 14-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
Theorem	xlebnum 24957*	Generalize lebnum 24956 to extended metrics. (Contributed by Mario Carneiro, 5-Sep-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
Theorem	lebnumii 24958*	Specialize the Lebesgue number lemma lebnum 24956 to the closed unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)
⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑢 ∈ 𝑈 (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢)
12.4.12  Path homotopy
Syntax	chtpy 24959	Extend class notation with the class of homotopies between two continuous functions.
class Htpy
Syntax	cphtpy 24960	Extend class notation with the class of path homotopies between two continuous functions.
class PHtpy
Syntax	cphtpc 24961	Extend class notation with the path homotopy relation.
class ≃ph
Definition	df-htpy 24962*	Define the function which takes topological spaces 𝑋, 𝑌 and two continuous functions 𝐹, 𝐺:𝑋⟶𝑌 and returns the class of homotopies from 𝐹 to 𝐺. (Contributed by Mario Carneiro, 22-Feb-2015.)
⊢ Htpy = (𝑥 ∈ Top, 𝑦 ∈ Top ↦ (𝑓 ∈ (𝑥 Cn 𝑦), 𝑔 ∈ (𝑥 Cn 𝑦) ↦ {ℎ ∈ ((𝑥 ×t II) Cn 𝑦) ∣ ∀𝑠 ∈ ∪ 𝑥((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}))
Definition	df-phtpy 24963*	Define the class of path homotopies between two paths 𝐹, 𝐺:II⟶𝑋; these are homotopies (in the sense of df-htpy 24962) which also preserve both endpoints of the paths throughout the homotopy. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ PHtpy = (𝑥 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑥), 𝑔 ∈ (II Cn 𝑥) ↦ {ℎ ∈ (𝑓(II Htpy 𝑥)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))}))
Theorem	ishtpy 24964*	Membership in the class of homotopies between two continuous functions. (Contributed by Mario Carneiro, 22-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)))))
Theorem	htpycn 24965	A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×t II) Cn 𝐾))
Theorem	htpyi 24966	A homotopy evaluated at its endpoints. (Contributed by Mario Carneiro, 22-Feb-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴)))
Theorem	ishtpyd 24967*	Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐻 ∈ ((𝐽 ×t II) Cn 𝐾))    &   ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐻0) = (𝐹‘𝑠))    &   ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐻1) = (𝐺‘𝑠))    ⇒   ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
Theorem	htpycom 24968*	Given a homotopy from 𝐹 to 𝐺, produce a homotopy from 𝐺 to 𝐹. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    &   ⊢ 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦)))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))    ⇒   ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐹))
Theorem	htpyid 24969*	A homotopy from a function to itself. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑥))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝐹(𝐽 Htpy 𝐾)𝐹))
Theorem	htpyco1 24970*	Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐿))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐾 Htpy 𝐿)𝐺))    ⇒   ⊢ (𝜑 → 𝑁 ∈ ((𝐹 ∘ 𝑃)(𝐽 Htpy 𝐿)(𝐺 ∘ 𝑃)))
Theorem	htpyco2 24971	Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑃 ∈ (𝐾 Cn 𝐿))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))    ⇒   ⊢ (𝜑 → (𝑃 ∘ 𝐻) ∈ ((𝑃 ∘ 𝐹)(𝐽 Htpy 𝐿)(𝑃 ∘ 𝐺)))
Theorem	htpycc 24972*	Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))    &   ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐻))    ⇒   ⊢ (𝜑 → 𝑁 ∈ (𝐹(𝐽 Htpy 𝐾)𝐻))
Theorem	isphtpy 24973*	Membership in the class of path homotopies between two continuous functions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))))
Theorem	phtpyhtpy 24974	A path homotopy is a homotopy. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺))
Theorem	phtpycn 24975	A path homotopy is a continuous function. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ ((II ×t II) Cn 𝐽))
Theorem	phtpyi 24976	Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1)))
Theorem	phtpy01 24977	Two path-homotopic paths have the same start and end point. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))    ⇒   ⊢ (𝜑 → ((𝐹‘0) = (𝐺‘0) ∧ (𝐹‘1) = (𝐺‘1)))
Theorem	isphtpyd 24978*	Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺))    &   ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘0))    &   ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘1))    ⇒   ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))
Theorem	isphtpy2d 24979*	Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐻 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹‘𝑠))    &   ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐺‘𝑠))    &   ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘0))    &   ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘1))    ⇒   ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))
Theorem	phtpycom 24980*	Given a homotopy from 𝐹 to 𝐺, produce a homotopy from 𝐺 to 𝐹. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦)))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐹))
Theorem	phtpyid 24981*	A homotopy from a path to itself. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
⊢ 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑥))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝐹(PHtpy‘𝐽)𝐹))
Theorem	phtpyco2 24982	Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))    ⇒   ⊢ (𝜑 → (𝑃 ∘ 𝐻) ∈ ((𝑃 ∘ 𝐹)(PHtpy‘𝐾)(𝑃 ∘ 𝐺)))
Theorem	phtpycc 24983*	Concatenate two path homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)
⊢ 𝑀 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), (𝑥𝐾(2 · 𝑦)), (𝑥𝐿((2 · 𝑦) − 1))))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ (𝐹(PHtpy‘𝐽)𝐺))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐺(PHtpy‘𝐽)𝐻))    ⇒   ⊢ (𝜑 → 𝑀 ∈ (𝐹(PHtpy‘𝐽)𝐻))
Definition	df-phtpc 24984*	Define the function which takes a topology and returns the path homotopy relation on that topology. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
⊢ ≃ph = (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
Theorem	phtpcrel 24985	The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.)
⊢ Rel ( ≃ph‘𝐽)
Theorem	isphtpc 24986	The relation "is path homotopic to". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Sep-2015.)
⊢ (𝐹( ≃ph‘𝐽)𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
Theorem	phtpcer 24987	Path homotopy is an equivalence relation. Proposition 1.2 of [Hatcher] p. 26. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 6-Jul-2015.) (Proof shortened by AV, 1-May-2021.)
⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽)
Theorem	phtpc01 24988	Path homotopic paths have the same endpoints. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝐹( ≃ph‘𝐽)𝐺 → ((𝐹‘0) = (𝐺‘0) ∧ (𝐹‘1) = (𝐺‘1)))
Theorem	reparphti 24989*	Lemma for reparpht 24990. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.) Avoid ax-mulf 11116. (Revised by GG, 16-Mar-2025.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn II))    &   ⊢ (𝜑 → (𝐺‘0) = 0)    &   ⊢ (𝜑 → (𝐺‘1) = 1)    &   ⊢ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))))    ⇒   ⊢ (𝜑 → 𝐻 ∈ ((𝐹 ∘ 𝐺)(PHtpy‘𝐽)𝐹))
Theorem	reparpht 24990	Reparametrization lemma. The reparametrization of a path by any continuous map 𝐺:II⟶II with 𝐺(0) = 0 and 𝐺(1) = 1 is path-homotopic to the original path. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn II))    &   ⊢ (𝜑 → (𝐺‘0) = 0)    &   ⊢ (𝜑 → (𝐺‘1) = 1)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺)( ≃ph‘𝐽)𝐹)
Theorem	phtpcco2 24991	Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ (𝜑 → 𝐹( ≃ph‘𝐽)𝐺)    &   ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑃 ∘ 𝐹)( ≃ph‘𝐾)(𝑃 ∘ 𝐺))
12.4.13  The fundamental group
Syntax	cpco 24992	Extend class notation with the concatenation operation for paths in a topological space.
class *𝑝
Syntax	comi 24993	Extend class notation with the loop space.
class Ω1
Syntax	comn 24994	Extend class notation with the higher loop spaces.
class Ω𝑛
Syntax	cpi1 24995	Extend class notation with the fundamental group.
class π1
Syntax	cpin 24996	Extend class notation with the higher homotopy groups.
class πn
Definition	df-pco 24997*	Define the concatenation of two paths in a topological space 𝐽. For simplicity of definition, we define it on all paths, not just those whose endpoints line up. Definition of [Hatcher] p. 26. Hatcher denotes path concatenation with a square dot; other authors, such as Munkres, use a star. (Contributed by Jeff Madsen, 15-Jun-2010.)
⊢ *𝑝 = (𝑗 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑗), 𝑔 ∈ (II Cn 𝑗) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1))))))
Definition	df-om1 24998*	Define the loop space of a topological space, with a magma structure on it given by concatenation of loops. This structure is not a group, but the operation is compatible with homotopy, which allows the homotopy groups to be defined based on this operation. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ Ω1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝‘𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ↑ko II)⟩})
Definition	df-omn 24999*	Define the n-th iterated loop space of a topological space. Unlike Ω1 this is actually a pointed topological space, which is to say a tuple of a topological space (a member of TopSp, not Top) and a point in the space. Higher loop spaces select the constant loop at the point from the lower loop space for the distinguished point. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ Ω𝑛 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ seq0(((𝑥 ∈ V, 𝑝 ∈ V ↦ ⟨((TopOpen‘(1st ‘𝑥)) Ω1 (2nd ‘𝑥)), ((0[,]1) × {(2nd ‘𝑥)})⟩) ∘ 1st ), ⟨{⟨(Base‘ndx), ∪ 𝑗⟩, ⟨(TopSet‘ndx), 𝑗⟩}, 𝑦⟩))
Definition	df-pi1 25000*	Define the fundamental group, whose operation is given by concatenation of homotopy classes of loops. Definition of [Hatcher] p. 26. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)))