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Theorem List for Metamath Proof Explorer - 22501-22600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelcncf2 22501* Version of elcncf 22500 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 22502 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)

Theoremcncfrss2 22503 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)

Theoremcncff 22504 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 22505* 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 22506* 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 22507* 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 22508 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 22509 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 22510 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐵𝐶𝐶 ⊆ ℂ) → (𝐴cn𝐵) ⊆ (𝐴cn𝐶))

Theoremclimcncf 22511 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 22512 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
abs ∈ (ℂ–cn→ℝ)

Theoremrecncf 22513 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℜ ∈ (ℂ–cn→ℝ)

Theoremimcncf 22514 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℑ ∈ (ℂ–cn→ℝ)

Theoremcjcncf 22515 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
∗ ∈ (ℂ–cn→ℂ)

Theoremmulc1cncf 22516* 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 22517* 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 22518 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))

Theoremcncfmet 22519 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 22520 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 22521 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 22522* 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 22523* 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 22524* Composition of continuous functions. cn analogue of cnmpt11f 21277. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝐹 ∈ (ℂ–cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝑋cn→ℂ))

Theoremcncfmpt2f 22525* Composition of continuous functions. cn analogue of cnmpt12f 21279. (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 22526* 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 22527* 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))

Theoremcdivcncf 22528* 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 22529* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
𝐹 = (𝑥𝐴 ↦ -𝑥)       (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴cn→ℂ))

Theoremnegfcncf 22530* 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 22531 Absolute value is continuous. Alternate proof of abscncf 22512. (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 22532 Rewrite cmphaushmeo 21413 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 22533* The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 22483. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥𝑁)) ∈ (ℂ–cn→ℂ))

Theoremcnmptre 22534* Lemma for iirevcn 22537 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))

Theoremcnmpt2pc 22535* 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 22536 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ (0[,]1) → (1 − 𝑋) ∈ (0[,]1))

Theoremiirevcn 22537 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 22538 Map the first half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ (0[,](1 / 2)) → (2 · 𝑋) ∈ (0[,]1))

Theoremiihalf1cn 22539 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 22540 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 22541 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 22542 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 22543 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 22544 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 22545* 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 22546 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 22547 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 22548* 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 22549* 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 22550* 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 22551* 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 22552 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 22553* 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)    &   𝐾 = (ordTop‘ ≤ )       (𝐺 Isom < , < ((-1[,]1), ℝ*) ∧ 𝐺 ∈ ((𝐽t (-1[,]1))Homeo(ordTop‘ ≤ )))

Theoremxrhmph 22554 The extended reals are homeomorphic to the interval [0, 1]. (Contributed by Mario Carneiro, 9-Sep-2015.)
II ≃ (ordTop‘ ≤ )

Theoremxrcmp 22555 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 22417), this means that * is a compactification of . (Contributed by Mario Carneiro, 9-Sep-2015.)
(ordTop‘ ≤ ) ∈ Comp

Theoremxrcon 22556 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
(ordTop‘ ≤ ) ∈ Con

Theoremicccvx 22557 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 22558* 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 22559* 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 22560* The canonical bijection from (ℝ × ℝ) to described in cnref1o 11703 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 22561* Lemma for cnheibor 22562. (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 22562* 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 22563 The topology on the complex numbers is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ 𝑛-Locally Comp

Theoremrellycmp 22564 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
(topGen‘ran (,)) ∈ 𝑛-Locally Comp

Theorembndth 22565* 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 22566* 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 22567* 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 22568* Lemma for lebnum 22571. 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 22569* Lemma for lebnum 22571. As a finite sum of point-to-set distance functions, which are continuous by metdscn 22467, 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 22570* Lemma for lebnum 22571. 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 22571* 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 22572* Generalize lebnum 22571 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 22573* Specialize the Lebesgue number lemma lebnum 22571 to the 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 22574 Extend class notation with the class of homotopies between two continuous functions.
class Htpy

Syntaxcphtpy 22575 Extend class notation with the class of path homotopies between two continuous functions.
class PHtpy

Syntaxcphtpc 22576 Extend class notation with the path homotopy relation.
class ph

Definitiondf-htpy 22577* 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 22578* Define the class of path homotopies between two paths 𝐹, 𝐺:II⟶𝑋; these are homotopies (in the sense of df-htpy 22577) 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 22579* 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 22580 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 22581 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 22582* 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 22583* 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 22584* 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 22585* 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 22586 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 22587* 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 22588* 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 22589 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 22590 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 22591 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 22592 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)))

Theoremisphtpyd 22593* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺))    &   ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘0))    &   ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘1))       (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))

Theoremisphtpy2d 22594* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝐻 ∈ ((II ×t II) Cn 𝐽))    &   ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹𝑠))    &   ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐺𝑠))    &   ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘0))    &   ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘1))       (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))

Theoremphtpycom 22595* 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.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦)))    &   (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))       (𝜑𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐹))

Theoremphtpyid 22596* 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) ↦ (𝐹𝑥))    &   (𝜑𝐹 ∈ (II Cn 𝐽))       (𝜑𝐺 ∈ (𝐹(PHtpy‘𝐽)𝐹))

Theoremphtpyco2 22597 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))       (𝜑 → (𝑃𝐻) ∈ ((𝑃𝐹)(PHtpy‘𝐾)(𝑃𝐺)))

Theoremphtpycc 22598* 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))))    &   (𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝐻 ∈ (II Cn 𝐽))    &   (𝜑𝐾 ∈ (𝐹(PHtpy‘𝐽)𝐺))    &   (𝜑𝐿 ∈ (𝐺(PHtpy‘𝐽)𝐻))       (𝜑𝑀 ∈ (𝐹(PHtpy‘𝐽)𝐻))

Definitiondf-phtpc 22599* 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) ≠ ∅)})

Theoremphtpcrel 22600 The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.)
Rel ( ≃ph𝐽)

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