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Theorem List for Metamath Proof Explorer - 23501-23600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiicmp 23501 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
II ∈ Comp

Theoremiiconn 23502 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
II ∈ Conn

Theoremcncfval 23503* The value of the continuous complex function operation is the set of continuous functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})

Theoremelcncf 23504* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))))

Theoremelcncf2 23505* Version of elcncf 23504 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑤𝑥)) < 𝑧 → (abs‘((𝐹𝑤) − (𝐹𝑥))) < 𝑦))))

Theoremcncfrss 23506 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)

Theoremcncfrss2 23507 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)

Theoremcncff 23508 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐹:𝐴𝐵)

Theoremcncfi 23509* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
((𝐹 ∈ (𝐴cn𝐵) ∧ 𝐶𝐴𝑅 ∈ ℝ+) → ∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑤𝐶)) < 𝑧 → (abs‘((𝐹𝑤) − (𝐹𝐶))) < 𝑅))

Theoremelcncf1di 23510* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑 → ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+))    &   (𝜑 → (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))       (𝜑 → ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵)))

Theoremelcncf1ii 23511* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
𝐹:𝐴𝐵    &   ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+)    &   (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))       ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵))

Theoremrescncf 23512 A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐶𝐴 → (𝐹 ∈ (𝐴cn𝐵) → (𝐹𝐶) ∈ (𝐶cn𝐵)))

Theoremcncffvrn 23513 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴cn𝐵)) → (𝐹 ∈ (𝐴cn𝐶) ↔ 𝐹:𝐴𝐶))

Theoremcncfss 23514 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐵𝐶𝐶 ⊆ ℂ) → (𝐴cn𝐵) ⊆ (𝐴cn𝐶))

Theoremclimcncf 23515 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹 ∈ (𝐴cn𝐵))    &   (𝜑𝐺:𝑍𝐴)    &   (𝜑𝐺𝐷)    &   (𝜑𝐷𝐴)       (𝜑 → (𝐹𝐺) ⇝ (𝐹𝐷))

Theoremabscncf 23516 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
abs ∈ (ℂ–cn→ℝ)

Theoremrecncf 23517 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℜ ∈ (ℂ–cn→ℝ)

Theoremimcncf 23518 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℑ ∈ (ℂ–cn→ℝ)

Theoremcjcncf 23519 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
∗ ∈ (ℂ–cn→ℂ)

Theoremmulc1cncf 23520* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))       (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremdivccncf 23521* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴))       ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremcncfco 23522 The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝜑𝐹 ∈ (𝐴cn𝐵))    &   (𝜑𝐺 ∈ (𝐵cn𝐶))       (𝜑 → (𝐺𝐹) ∈ (𝐴cn𝐶))

Theoremcncfcompt2 23523* Composition of continuous functions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑥𝜑    &   (𝜑 → (𝑥𝐴𝑅) ∈ (𝐴cn𝐵))    &   (𝜑 → (𝑦𝐶𝑆) ∈ (𝐶cn𝐸))    &   (𝜑𝐵𝐶)    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝑥𝐴𝑇) ∈ (𝐴cn𝐸))

Theoremcncfmet 23524 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) → (𝐴cn𝐵) = (𝐽 Cn 𝐾))

Theoremcncfcn 23525 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝐴)    &   𝐿 = (𝐽t 𝐵)       ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = (𝐾 Cn 𝐿))

Theoremcncfcn1 23526 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)

Theoremcncfmptc 23527* A constant function is a continuous function on . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
((𝐴𝑇𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥𝑆𝐴) ∈ (𝑆cn𝑇))

Theoremcncfmptid 23528* The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇))

Theoremcncfmpt1f 23529* Composition of continuous functions. cn analogue of cnmpt11f 22279. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝐹 ∈ (ℂ–cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝑋cn→ℂ))

Theoremcncfmpt2f 23530* Composition of continuous functions. cn analogue of cnmpt12f 22281. (Contributed by Mario Carneiro, 3-Sep-2014.)
𝐽 = (TopOpen‘ℂfld)    &   (𝜑𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋cn→ℂ))

Theoremcncfmpt2ss 23531* Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
𝐽 = (TopOpen‘ℂfld)    &   𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn𝑆))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn𝑆))    &   𝑆 ⊆ ℂ    &   ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)       (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋cn𝑆))

Theoremaddccncf 23532* 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))

Theoremidcncf 23533 The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 23528 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)

Theoremsub1cncf 23534* 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 ↦ (𝑥𝐴))       (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremsub2cncf 23535* 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 ↦ (𝐴𝑥))       (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremcdivcncf 23536* 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))

Theoremnegcncf 23537* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
𝐹 = (𝑥𝐴 ↦ -𝑥)       (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴cn→ℂ))

Theoremnegfcncf 23538* The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
𝐺 = (𝑥𝐴 ↦ -(𝐹𝑥))       (𝐹 ∈ (𝐴cn→ℂ) → 𝐺 ∈ (𝐴cn→ℂ))

TheoremabscncfALT 23539 Absolute value is continuous. Alternate proof of abscncf 23516. (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)

Theoremcncfcnvcn 23540 Rewrite cmphaushmeo 22415 for functions on the complex numbers. (Contributed by Mario Carneiro, 19-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝑋)       ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋cn𝑌)) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑌cn𝑋)))

Theoremexpcncf 23541* The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 23487. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥𝑁)) ∈ (ℂ–cn→ℂ))

Theoremcnmptre 23542* Lemma for iirevcn 23545 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.)
𝑅 = (TopOpen‘ℂfld)    &   𝐽 = ((topGen‘ran (,)) ↾t 𝐴)    &   𝐾 = ((topGen‘ran (,)) ↾t 𝐵)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐹𝐵)    &   (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅))       (𝜑 → (𝑥𝐴𝐹) ∈ (𝐽 Cn 𝐾))

Theoremcnmpopc 23543* 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 = (𝑅t (𝐴[,]𝐵))    &   𝑁 = (𝑅t (𝐵[,]𝐶))    &   𝑂 = (𝑅t (𝐴[,]𝐶))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 ∈ (𝐴[,]𝐶))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   ((𝜑 ∧ (𝑥 = 𝐵𝑦𝑋)) → 𝐷 = 𝐸)    &   (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))    &   (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋𝐸) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))       (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾))

Theoremiirev 23544 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ (0[,]1) → (1 − 𝑋) ∈ (0[,]1))

Theoremiirevcn 23545 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)

Theoremiihalf1 23546 Map the first half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ (0[,](1 / 2)) → (2 · 𝑋) ∈ (0[,]1))

Theoremiihalf1cn 23547 The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
𝐽 = ((topGen‘ran (,)) ↾t (0[,](1 / 2)))       (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II)

Theoremiihalf2 23548 Map the second half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ ((1 / 2)[,]1) → ((2 · 𝑋) − 1) ∈ (0[,]1))

Theoremiihalf2cn 23549 The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1))       (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II)

Theoremelii1 23550 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
(𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ (0[,]1) ∧ 𝑋 ≤ (1 / 2)))

Theoremelii2 23551 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ∈ ((1 / 2)[,]1))

Theoremiimulcl 23552 The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴 ∈ (0[,]1) ∧ 𝐵 ∈ (0[,]1)) → (𝐴 · 𝐵) ∈ (0[,]1))

Theoremiimulcn 23553* Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.)
(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥 · 𝑦)) ∈ ((II ×t II) Cn II)

Theoremicoopnst 23554 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))

Theoremiocopnst 23555 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))

Theoremicchmeo 23556* The natural bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)    &   𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴)))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽t (𝐴[,]𝐵))))

Theoremicopnfcnv 23557* 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))))

Theoremicopnfhmeo 23558* 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 ∈ ((𝐽t (0[,)1))Homeo(𝐽t (0[,)+∞))))

Theoremiccpnfcnv 23559* 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)))))

Theoremiccpnfhmeo 23560 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))

Theoremxrhmeo 23561* 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, (𝐹𝑦), -𝑒(𝐹‘-𝑦)))    &   𝐽 = (TopOpen‘ℂfld)       (𝐺 Isom < , < ((-1[,]1), ℝ*) ∧ 𝐺 ∈ ((𝐽t (-1[,]1))Homeo(ordTop‘ ≤ )))

Theoremxrhmph 23562 The extended reals are homeomorphic to the interval [0, 1]. (Contributed by Mario Carneiro, 9-Sep-2015.)
II ≃ (ordTop‘ ≤ )

Theoremxrcmp 23563 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 23421), this means that * is a compactification of . (Contributed by Mario Carneiro, 9-Sep-2015.)
(ordTop‘ ≤ ) ∈ Comp

Theoremxrconn 23564 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
(ordTop‘ ≤ ) ∈ Conn

Theoremicccvx 23565 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)))

Theoremoprpiece1res1 23566* 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    &   𝐴𝐵    &   𝑅 ∈ V    &   𝑆 ∈ V    &   𝐾 ∈ (𝐴[,]𝐵)    &   𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝐶 ↦ if(𝑥𝐾, 𝑅, 𝑆))    &   𝐺 = (𝑥 ∈ (𝐴[,]𝐾), 𝑦𝐶𝑅)       (𝐹 ↾ ((𝐴[,]𝐾) × 𝐶)) = 𝐺

Theoremoprpiece1res2 23567* 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    &   𝐴𝐵    &   𝑅 ∈ V    &   𝑆 ∈ V    &   𝐾 ∈ (𝐴[,]𝐵)    &   𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝐶 ↦ if(𝑥𝐾, 𝑅, 𝑆))    &   (𝑥 = 𝐾𝑅 = 𝑃)    &   (𝑥 = 𝐾𝑆 = 𝑄)    &   (𝑦𝐶𝑃 = 𝑄)    &   𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦𝐶𝑆)       (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = 𝐺

Theoremcnrehmeo 23568* The canonical bijection from (ℝ × ℝ) to described in cnref1o 12375 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.)
𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   𝐽 = (topGen‘ran (,))    &   𝐾 = (TopOpen‘ℂfld)       𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)

Theoremcnheiborlem 23569* Lemma for cnheibor 23570. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝑇 = (𝐽t 𝑋)    &   𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   𝑌 = (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))       ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑇 ∈ Comp)

Theoremcnheibor 23570* Heine-Borel theorem for complex numbers. A subset of is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝑇 = (𝐽t 𝑋)       (𝑋 ⊆ ℂ → (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)))

Theoremcnllycmp 23571 The topology on the complex numbers is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ 𝑛-Locally Comp

Theoremrellycmp 23572 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
(topGen‘ran (,)) ∈ 𝑛-Locally Comp

Theorembndth 23573* 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 (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥)

Theoremevth 23574* 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 (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑋 ≠ ∅)       (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))

Theoremevth2 23575* 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 (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑋 ≠ ∅)       (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑥) ≤ (𝐹𝑦))

Theoremlebnumlem1 23576* Lemma for lebnum 23579. 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)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑈 ∈ Fin)    &   (𝜑 → ¬ 𝑋𝑈)    &   𝐹 = (𝑦𝑋 ↦ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))       (𝜑𝐹:𝑋⟶ℝ+)

Theoremlebnumlem2 23577* Lemma for lebnum 23579. As a finite sum of point-to-set distance functions, which are continuous by metdscn 23471, the function 𝐹 is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑈 ∈ Fin)    &   (𝜑 → ¬ 𝑋𝑈)    &   𝐹 = (𝑦𝑋 ↦ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))    &   𝐾 = (topGen‘ran (,))       (𝜑𝐹 ∈ (𝐽 Cn 𝐾))

Theoremlebnumlem3 23578* Lemma for lebnum 23579. 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 𝑢𝑈 we have ball(𝑥, 𝑑) ⊆ 𝑢 iff 𝑑𝑑(𝑥, 𝑋𝑢), if ¬ ball(𝑥, 𝑑) ⊆ 𝑢 for all 𝑢 then summing over 𝑢 yields Σ𝑢𝑈𝑑(𝑥, 𝑋𝑢) = 𝐹(𝑥) < Σ𝑢𝑈𝑑 = 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)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑈 ∈ Fin)    &   (𝜑 → ¬ 𝑋𝑈)    &   𝐹 = (𝑦𝑋 ↦ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))    &   𝐾 = (topGen‘ran (,))       (𝜑 → ∃𝑑 ∈ ℝ+𝑥𝑋𝑢𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)

Theoremlebnum 23579* 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)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)       (𝜑 → ∃𝑑 ∈ ℝ+𝑥𝑋𝑢𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)

Theoremxlebnum 23580* Generalize lebnum 23579 to extended metrics. (Contributed by Mario Carneiro, 5-Sep-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)       (𝜑 → ∃𝑑 ∈ ℝ+𝑥𝑋𝑢𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)

Theoremlebnumii 23581* Specialize the Lebesgue number lemma lebnum 23579 to the closed unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)
((𝑈 ⊆ II ∧ (0[,]1) = 𝑈) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑢𝑈 (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢)

12.4.12  Path homotopy

Syntaxchtpy 23582 Extend class notation with the class of homotopies between two continuous functions.
class Htpy

Syntaxcphtpy 23583 Extend class notation with the class of path homotopies between two continuous functions.
class PHtpy

Syntaxcphtpc 23584 Extend class notation with the path homotopy relation.
class ph

Definitiondf-htpy 23585* Define the function which takes topological spaces 𝑋, 𝑌 and two continuous functions 𝐹, 𝐺:𝑋𝑌 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) ↦ { ∈ ((𝑥 ×t II) Cn 𝑦) ∣ ∀𝑠 𝑥((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}))

Definitiondf-phtpy 23586* Define the class of path homotopies between two paths 𝐹, 𝐺:II⟶𝑋; these are homotopies (in the sense of df-htpy 23585) 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) ↦ { ∈ (𝑓(II Htpy 𝑥)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}))

Theoremishtpy 23587* Membership in the class of homotopies between two continuous functions. (Contributed by Mario Carneiro, 22-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))))

Theoremhtpycn 23588 A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×t II) Cn 𝐾))

Theoremhtpyi 23589 A homotopy evaluated at its endpoints. (Contributed by Mario Carneiro, 22-Feb-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))       ((𝜑𝐴𝑋) → ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴)))

Theoremishtpyd 23590* Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐻 ∈ ((𝐽 ×t II) Cn 𝐾))    &   ((𝜑𝑠𝑋) → (𝑠𝐻0) = (𝐹𝑠))    &   ((𝜑𝑠𝑋) → (𝑠𝐻1) = (𝐺𝑠))       (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))

Theoremhtpycom 23591* Given a homotopy from 𝐹 to 𝐺, produce a homotopy from 𝐺 to 𝐹. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   𝑀 = (𝑥𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦)))    &   (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))       (𝜑𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐹))

Theoremhtpyid 23592* A homotopy from a function to itself. (Contributed by Mario Carneiro, 23-Feb-2015.)
𝐺 = (𝑥𝑋, 𝑦 ∈ (0[,]1) ↦ (𝐹𝑥))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐺 ∈ (𝐹(𝐽 Htpy 𝐾)𝐹))

Theoremhtpyco1 23593* Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝑁 = (𝑥𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃𝑥)𝐻𝑦))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑃 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐹 ∈ (𝐾 Cn 𝐿))    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐿))    &   (𝜑𝐻 ∈ (𝐹(𝐾 Htpy 𝐿)𝐺))       (𝜑𝑁 ∈ ((𝐹𝑃)(𝐽 Htpy 𝐿)(𝐺𝑃)))

Theoremhtpyco2 23594 Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
(𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑃 ∈ (𝐾 Cn 𝐿))    &   (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))       (𝜑 → (𝑃𝐻) ∈ ((𝑃𝐹)(𝐽 Htpy 𝐿)(𝑃𝐺)))

Theoremhtpycc 23595* Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
𝑁 = (𝑥𝑋, 𝑦 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐻 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐿 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))    &   (𝜑𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐻))       (𝜑𝑁 ∈ (𝐹(𝐽 Htpy 𝐾)𝐻))

Theoremisphtpy 23596* 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.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))       (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))))

Theoremphtpyhtpy 23597 A path homotopy is a homotopy. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))       (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺))

Theoremphtpycn 23598 A path homotopy is a continuous function. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))       (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ ((II ×t II) Cn 𝐽))

Theoremphtpyi 23599 Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))       ((𝜑𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1)))

Theoremphtpy01 23600 Two path-homotopic paths have the same start and end point. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))       (𝜑 → ((𝐹‘0) = (𝐺‘0) ∧ (𝐹‘1) = (𝐺‘1)))

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