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Theorem List for Metamath Proof Explorer - 43901-44000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfprodsubrecnncnvlem 43901* The sequence 𝑆 of finite products, where every factor is subtracted an "always smaller" amount, converges to the finite product of the factors. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   π‘† = (𝑛 ∈ β„• ↦ βˆπ‘˜ ∈ 𝐴 (𝐡 βˆ’ (1 / 𝑛)))    &   πΉ = (π‘₯ ∈ β„‚ ↦ βˆπ‘˜ ∈ 𝐴 (𝐡 βˆ’ π‘₯))    &   πΊ = (𝑛 ∈ β„• ↦ (1 / 𝑛))    β‡’   (πœ‘ β†’ 𝑆 ⇝ βˆπ‘˜ ∈ 𝐴 𝐡)
 
Theoremfprodsubrecnncnv 43902* The sequence 𝑆 of finite products, where every factor is subtracted an "always smaller" amount, converges to the finite product of the factors. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑋 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   π‘† = (𝑛 ∈ β„• ↦ βˆπ‘˜ ∈ 𝑋 (𝐴 βˆ’ (1 / 𝑛)))    β‡’   (πœ‘ β†’ 𝑆 ⇝ βˆπ‘˜ ∈ 𝑋 𝐴)
 
Theoremfprodaddrecnncnvlem 43903* The sequence 𝑆 of finite products, where every factor is added an "always smaller" amount, converges to the finite product of the factors. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   π‘† = (𝑛 ∈ β„• ↦ βˆπ‘˜ ∈ 𝐴 (𝐡 + (1 / 𝑛)))    &   πΉ = (π‘₯ ∈ β„‚ ↦ βˆπ‘˜ ∈ 𝐴 (𝐡 + π‘₯))    &   πΊ = (𝑛 ∈ β„• ↦ (1 / 𝑛))    β‡’   (πœ‘ β†’ 𝑆 ⇝ βˆπ‘˜ ∈ 𝐴 𝐡)
 
Theoremfprodaddrecnncnv 43904* The sequence 𝑆 of finite products, where every factor is added an "always smaller" amount, converges to the finite product of the factors. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑋 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   π‘† = (𝑛 ∈ β„• ↦ βˆπ‘˜ ∈ 𝑋 (𝐴 + (1 / 𝑛)))    β‡’   (πœ‘ β†’ 𝑆 ⇝ βˆπ‘˜ ∈ 𝑋 𝐴)
 
21.38.10  Derivatives
 
Theoremdvsinexp 43905* The derivative of sin^N . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ ((sinβ€˜π‘₯)↑𝑁))) = (π‘₯ ∈ β„‚ ↦ ((𝑁 Β· ((sinβ€˜π‘₯)↑(𝑁 βˆ’ 1))) Β· (cosβ€˜π‘₯))))
 
Theoremdvcosre 43906 The real derivative of the cosine. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(ℝ D (π‘₯ ∈ ℝ ↦ (cosβ€˜π‘₯))) = (π‘₯ ∈ ℝ ↦ -(sinβ€˜π‘₯))
 
Theoremdvsinax 43907* Derivative exercise: the derivative with respect to y of sin(Ay), given a constant 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ β„‚ β†’ (β„‚ D (𝑦 ∈ β„‚ ↦ (sinβ€˜(𝐴 Β· 𝑦)))) = (𝑦 ∈ β„‚ ↦ (𝐴 Β· (cosβ€˜(𝐴 Β· 𝑦)))))
 
Theoremdvsubf 43908 The subtraction rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ dom (𝑆 D 𝐹) = 𝑋)    &   (πœ‘ β†’ dom (𝑆 D 𝐺) = 𝑋)    β‡’   (πœ‘ β†’ (𝑆 D (𝐹 ∘f βˆ’ 𝐺)) = ((𝑆 D 𝐹) ∘f βˆ’ (𝑆 D 𝐺)))
 
Theoremdvmptconst 43909* Function-builder for derivative: derivative of a constant. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐴 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝐴 ↦ 𝐡)) = (π‘₯ ∈ 𝐴 ↦ 0))
 
Theoremdvcnre 43910 From complex differentiation to real differentiation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹:β„‚βŸΆβ„‚ ∧ ℝ βŠ† dom (β„‚ D 𝐹)) β†’ (ℝ D (𝐹 β†Ύ ℝ)) = ((β„‚ D 𝐹) β†Ύ ℝ))
 
Theoremdvmptidg 43911* Function-builder for derivative: derivative of the identity. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐴 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    β‡’   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝐴 ↦ π‘₯)) = (π‘₯ ∈ 𝐴 ↦ 1))
 
Theoremdvresntr 43912 Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 βŠ† β„‚)    &   (πœ‘ β†’ 𝑋 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   π½ = (𝐾 β†Ύt 𝑆)    &   πΎ = (TopOpenβ€˜β„‚fld)    &   (πœ‘ β†’ ((intβ€˜π½)β€˜π‘‹) = π‘Œ)    β‡’   (πœ‘ β†’ (𝑆 D 𝐹) = (𝑆 D (𝐹 β†Ύ π‘Œ)))
 
Theoremfperdvper 43913* The derivative of a periodic function is periodic. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹:β„βŸΆβ„‚)    &   (πœ‘ β†’ 𝑇 ∈ ℝ)    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (πΉβ€˜(π‘₯ + 𝑇)) = (πΉβ€˜π‘₯))    &   πΊ = (ℝ D 𝐹)    β‡’   ((πœ‘ ∧ π‘₯ ∈ dom 𝐺) β†’ ((π‘₯ + 𝑇) ∈ dom 𝐺 ∧ (πΊβ€˜(π‘₯ + 𝑇)) = (πΊβ€˜π‘₯)))
 
Theoremdvasinbx 43914* Derivative exercise: the derivative with respect to y of A x sin(By), given two constants 𝐴 and 𝐡. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (β„‚ D (𝑦 ∈ β„‚ ↦ (𝐴 Β· (sinβ€˜(𝐡 Β· 𝑦))))) = (𝑦 ∈ β„‚ ↦ ((𝐴 Β· 𝐡) Β· (cosβ€˜(𝐡 Β· 𝑦)))))
 
Theoremdvresioo 43915 Restriction of a derivative to an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 βŠ† ℝ ∧ 𝐹:π΄βŸΆβ„‚) β†’ (ℝ D (𝐹 β†Ύ (𝐡(,)𝐢))) = ((ℝ D 𝐹) β†Ύ (𝐡(,)𝐢)))
 
Theoremdvdivf 43916 The quotient rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆ(β„‚ βˆ– {0}))    &   (πœ‘ β†’ dom (𝑆 D 𝐹) = 𝑋)    &   (πœ‘ β†’ dom (𝑆 D 𝐺) = 𝑋)    β‡’   (πœ‘ β†’ (𝑆 D (𝐹 ∘f / 𝐺)) = ((((𝑆 D 𝐹) ∘f Β· 𝐺) ∘f βˆ’ ((𝑆 D 𝐺) ∘f Β· 𝐹)) ∘f / (𝐺 ∘f Β· 𝐺)))
 
Theoremdvdivbd 43917* A sufficient condition for the derivative to be bounded, for the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐢))    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐢 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝑅 ∈ ℝ)    &   (πœ‘ β†’ 𝑇 ∈ ℝ)    &   (πœ‘ β†’ 𝑄 ∈ ℝ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (absβ€˜πΆ) ≀ π‘ˆ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (absβ€˜π΅) ≀ 𝑅)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (absβ€˜π·) ≀ 𝑇)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (absβ€˜π΄) ≀ 𝑄)    &   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ 𝐡)) = (π‘₯ ∈ 𝑋 ↦ 𝐷))    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐷 ∈ β„‚)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 𝐸 ≀ (absβ€˜π΅))    &   πΉ = (𝑆 D (π‘₯ ∈ 𝑋 ↦ (𝐴 / 𝐡)))    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜(πΉβ€˜π‘₯)) ≀ 𝑏)
 
Theoremdvsubcncf 43918 A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ (𝑆 D 𝐹) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (𝑆 D 𝐺) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (𝑆 D (𝐹 ∘f βˆ’ 𝐺)) ∈ (𝑋–cnβ†’β„‚))
 
Theoremdvmulcncf 43919 A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ (𝑆 D 𝐹) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (𝑆 D 𝐺) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (𝑆 D (𝐹 ∘f Β· 𝐺)) ∈ (𝑋–cnβ†’β„‚))
 
Theoremdvcosax 43920* Derivative exercise: the derivative with respect to x of cos(Ax), given a constant 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ β„‚ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ (cosβ€˜(𝐴 Β· π‘₯)))) = (π‘₯ ∈ β„‚ ↦ (𝐴 Β· -(sinβ€˜(𝐴 Β· π‘₯)))))
 
Theoremdvdivcncf 43921 A sufficient condition for the derivative of a quotient to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆ(β„‚ βˆ– {0}))    &   (πœ‘ β†’ (𝑆 D 𝐹) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (𝑆 D 𝐺) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (𝑆 D (𝐹 ∘f / 𝐺)) ∈ (𝑋–cnβ†’β„‚))
 
Theoremdvbdfbdioolem1 43922* Given a function with bounded derivative, on an open interval, here is an absolute bound to the difference of the image of two points in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ 𝐾 ∈ ℝ)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝐾)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴(,)𝐡))    &   (πœ‘ β†’ 𝐷 ∈ (𝐢(,)𝐡))    β‡’   (πœ‘ β†’ ((absβ€˜((πΉβ€˜π·) βˆ’ (πΉβ€˜πΆ))) ≀ (𝐾 Β· (𝐷 βˆ’ 𝐢)) ∧ (absβ€˜((πΉβ€˜π·) βˆ’ (πΉβ€˜πΆ))) ≀ (𝐾 Β· (𝐡 βˆ’ 𝐴))))
 
Theoremdvbdfbdioolem2 43923* A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ 𝐾 ∈ ℝ)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝐾)    &   π‘€ = ((absβ€˜(πΉβ€˜((𝐴 + 𝐡) / 2))) + (𝐾 Β· (𝐡 βˆ’ 𝐴)))    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜(πΉβ€˜π‘₯)) ≀ 𝑀)
 
Theoremdvbdfbdioo 43924* A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ βˆƒπ‘Ž ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ π‘Ž)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜(πΉβ€˜π‘₯)) ≀ 𝑏)
 
Theoremioodvbdlimc1lem1 43925* If 𝐹 has bounded derivative on (𝐴(,)𝐡) then a sequence of points in its image converges to its lim sup. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 3-Oct-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴(,)𝐡)–cn→ℝ))    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝑦)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑅:(β„€β‰₯β€˜π‘€)⟢(𝐴(,)𝐡))    &   π‘† = (𝑗 ∈ (β„€β‰₯β€˜π‘€) ↦ (πΉβ€˜(π‘…β€˜π‘—)))    &   (πœ‘ β†’ 𝑅 ∈ dom ⇝ )    &   πΎ = inf({π‘˜ ∈ (β„€β‰₯β€˜π‘€) ∣ βˆ€π‘– ∈ (β„€β‰₯β€˜π‘˜)(absβ€˜((π‘…β€˜π‘–) βˆ’ (π‘…β€˜π‘˜))) < (π‘₯ / (sup(ran (𝑧 ∈ (𝐴(,)𝐡) ↦ (absβ€˜((ℝ D 𝐹)β€˜π‘§))), ℝ, < ) + 1))}, ℝ, < )    β‡’   (πœ‘ β†’ 𝑆 ⇝ (lim supβ€˜π‘†))
 
Theoremioodvbdlimc1lem2 43926* Limit at the lower bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 3-Oct-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝑦)    &   π‘Œ = sup(ran (π‘₯ ∈ (𝐴(,)𝐡) ↦ (absβ€˜((ℝ D 𝐹)β€˜π‘₯))), ℝ, < )    &   π‘€ = ((βŒŠβ€˜(1 / (𝐡 βˆ’ 𝐴))) + 1)    &   π‘† = (𝑗 ∈ (β„€β‰₯β€˜π‘€) ↦ (πΉβ€˜(𝐴 + (1 / 𝑗))))    &   π‘… = (𝑗 ∈ (β„€β‰₯β€˜π‘€) ↦ (𝐴 + (1 / 𝑗)))    &   π‘ = if(𝑀 ≀ ((βŒŠβ€˜(π‘Œ / (π‘₯ / 2))) + 1), ((βŒŠβ€˜(π‘Œ / (π‘₯ / 2))) + 1), 𝑀)    &   (πœ’ ↔ (((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘)) ∧ (absβ€˜((π‘†β€˜π‘—) βˆ’ (lim supβ€˜π‘†))) < (π‘₯ / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐡)) ∧ (absβ€˜(𝑧 βˆ’ 𝐴)) < (1 / 𝑗)))    β‡’   (πœ‘ β†’ (lim supβ€˜π‘†) ∈ (𝐹 limβ„‚ 𝐴))
 
Theoremioodvbdlimc1 43927* A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by AV, 3-Oct-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝑦)    β‡’   (πœ‘ β†’ (𝐹 limβ„‚ 𝐴) β‰  βˆ…)
 
Theoremioodvbdlimc2lem 43928* Limit at the upper bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 3-Oct-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝑦)    &   π‘Œ = sup(ran (π‘₯ ∈ (𝐴(,)𝐡) ↦ (absβ€˜((ℝ D 𝐹)β€˜π‘₯))), ℝ, < )    &   π‘€ = ((βŒŠβ€˜(1 / (𝐡 βˆ’ 𝐴))) + 1)    &   π‘† = (𝑗 ∈ (β„€β‰₯β€˜π‘€) ↦ (πΉβ€˜(𝐡 βˆ’ (1 / 𝑗))))    &   π‘… = (𝑗 ∈ (β„€β‰₯β€˜π‘€) ↦ (𝐡 βˆ’ (1 / 𝑗)))    &   π‘ = if(𝑀 ≀ ((βŒŠβ€˜(π‘Œ / (π‘₯ / 2))) + 1), ((βŒŠβ€˜(π‘Œ / (π‘₯ / 2))) + 1), 𝑀)    &   (πœ’ ↔ (((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘)) ∧ (absβ€˜((π‘†β€˜π‘—) βˆ’ (lim supβ€˜π‘†))) < (π‘₯ / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐡)) ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < (1 / 𝑗)))    β‡’   (πœ‘ β†’ (lim supβ€˜π‘†) ∈ (𝐹 limβ„‚ 𝐡))
 
Theoremioodvbdlimc2 43929* A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by AV, 3-Oct-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝑦)    β‡’   (πœ‘ β†’ (𝐹 limβ„‚ 𝐡) β‰  βˆ…)
 
Theoremdvdmsscn 43930 𝑋 is a subset of β„‚. This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑋 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    β‡’   (πœ‘ β†’ 𝑋 βŠ† β„‚)
 
Theoremdvmptmulf 43931* Function-builder for derivative, product rule. A version of dvmptmul 25247 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐡))    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐢 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐷 ∈ π‘Š)    &   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ 𝐢)) = (π‘₯ ∈ 𝑋 ↦ 𝐷))    β‡’   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐢))) = (π‘₯ ∈ 𝑋 ↦ ((𝐡 Β· 𝐢) + (𝐷 Β· 𝐴))))
 
Theoremdvnmptdivc 43932* Function-builder for iterated derivative, division rule for constant divisor. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑋 βŠ† 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ 𝑛 ∈ (0...𝑀)) β†’ ((𝑆 D𝑛 (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘›) = (π‘₯ ∈ 𝑋 ↦ 𝐡))    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 β‰  0)    &   (πœ‘ β†’ 𝑀 ∈ β„•0)    β‡’   ((πœ‘ ∧ 𝑛 ∈ (0...𝑀)) β†’ ((𝑆 D𝑛 (π‘₯ ∈ 𝑋 ↦ (𝐴 / 𝐢)))β€˜π‘›) = (π‘₯ ∈ 𝑋 ↦ (𝐡 / 𝐢)))
 
Theoremdvdsn1add 43933 If 𝐾 divides 𝑁 but 𝐾 does not divide 𝑀, then 𝐾 does not divide (𝑀 + 𝑁). (Contributed by Glauco Siliprandi, 5-Apr-2020.)
((𝐾 ∈ β„€ ∧ 𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ ((Β¬ 𝐾 βˆ₯ 𝑀 ∧ 𝐾 βˆ₯ 𝑁) β†’ Β¬ 𝐾 βˆ₯ (𝑀 + 𝑁)))
 
Theoremdvxpaek 43934* Derivative of the polynomial (π‘₯ + 𝐴)↑𝐾. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑋 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    &   (πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐾 ∈ β„•)    β‡’   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ ((π‘₯ + 𝐴)↑𝐾))) = (π‘₯ ∈ 𝑋 ↦ (𝐾 Β· ((π‘₯ + 𝐴)↑(𝐾 βˆ’ 1)))))
 
Theoremdvnmptconst 43935* The 𝑁-th derivative of a constant function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑋 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    &   (πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ ((𝑆 D𝑛 (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘) = (π‘₯ ∈ 𝑋 ↦ 0))
 
Theoremdvnxpaek 43936* The 𝑛-th derivative of the polynomial (π‘₯ + 𝐴)↑𝐾. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑋 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    &   (πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐾 ∈ β„•0)    &   πΉ = (π‘₯ ∈ 𝑋 ↦ ((π‘₯ + 𝐴)↑𝐾))    β‡’   ((πœ‘ ∧ 𝑁 ∈ β„•0) β†’ ((𝑆 D𝑛 𝐹)β€˜π‘) = (π‘₯ ∈ 𝑋 ↦ if(𝐾 < 𝑁, 0, (((!β€˜πΎ) / (!β€˜(𝐾 βˆ’ 𝑁))) Β· ((π‘₯ + 𝐴)↑(𝐾 βˆ’ 𝑁))))))
 
Theoremdvnmul 43937* Function-builder for the 𝑁-th derivative, product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑋 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   πΉ = (π‘₯ ∈ 𝑋 ↦ 𝐴)    &   πΊ = (π‘₯ ∈ 𝑋 ↦ 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ (0...𝑁)) β†’ ((𝑆 D𝑛 𝐹)β€˜π‘˜):π‘‹βŸΆβ„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (0...𝑁)) β†’ ((𝑆 D𝑛 𝐺)β€˜π‘˜):π‘‹βŸΆβ„‚)    &   πΆ = (π‘˜ ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)β€˜π‘˜))    &   π· = (π‘˜ ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)β€˜π‘˜))    β‡’   (πœ‘ β†’ ((𝑆 D𝑛 (π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡)))β€˜π‘) = (π‘₯ ∈ 𝑋 ↦ Ξ£π‘˜ ∈ (0...𝑁)((𝑁Cπ‘˜) Β· (((πΆβ€˜π‘˜)β€˜π‘₯) Β· ((π·β€˜(𝑁 βˆ’ π‘˜))β€˜π‘₯)))))
 
Theoremdvmptfprodlem 43938* Induction step for dvmptfprod 43939. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘₯πœ‘    &   β„²π‘–πœ‘    &   β„²π‘—πœ‘    &   β„²π‘–𝐹    &   β„²π‘—𝐺    &   ((πœ‘ ∧ 𝑖 ∈ 𝐼 ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐷 ∈ Fin)    &   (πœ‘ β†’ 𝐸 ∈ V)    &   (πœ‘ β†’ Β¬ 𝐸 ∈ 𝐷)    &   (πœ‘ β†’ (𝐷 βˆͺ {𝐸}) βŠ† 𝐼)    &   (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑗 ∈ 𝐷) β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ βˆπ‘– ∈ 𝐷 𝐴)) = (π‘₯ ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐷 (𝐢 Β· βˆπ‘– ∈ (𝐷 βˆ– {𝑗})𝐴)))    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐺 ∈ β„‚)    &   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ 𝐹)) = (π‘₯ ∈ 𝑋 ↦ 𝐺))    &   (𝑖 = 𝐸 β†’ 𝐴 = 𝐹)    &   (𝑗 = 𝐸 β†’ 𝐢 = 𝐺)    β‡’   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ βˆπ‘– ∈ (𝐷 βˆͺ {𝐸})𝐴)) = (π‘₯ ∈ 𝑋 ↦ Σ𝑗 ∈ (𝐷 βˆͺ {𝐸})(𝐢 Β· βˆπ‘– ∈ ((𝐷 βˆͺ {𝐸}) βˆ– {𝑗})𝐴)))
 
Theoremdvmptfprod 43939* Function-builder for derivative, finite product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘–πœ‘    &   β„²π‘—πœ‘    &   π½ = (𝐾 β†Ύt 𝑆)    &   πΎ = (TopOpenβ€˜β„‚fld)    &   (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑋 ∈ 𝐽)    &   (πœ‘ β†’ 𝐼 ∈ Fin)    &   ((πœ‘ ∧ 𝑖 ∈ 𝐼 ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ 𝑖 ∈ 𝐼 ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ 𝑖 ∈ 𝐼) β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐡))    &   (𝑖 = 𝑗 β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ βˆπ‘– ∈ 𝐼 𝐴)) = (π‘₯ ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐢 Β· βˆπ‘– ∈ (𝐼 βˆ– {𝑗})𝐴)))
 
Theoremdvnprodlem1 43940* 𝐷 is bijective. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝐢 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ β„•0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑑 ∈ 𝑠 (π‘β€˜π‘‘) = 𝑛}))    &   (πœ‘ β†’ 𝐽 ∈ β„•0)    &   π· = (𝑐 ∈ ((πΆβ€˜(𝑅 βˆͺ {𝑍}))β€˜π½) ↦ ⟨(𝐽 βˆ’ (π‘β€˜π‘)), (𝑐 β†Ύ 𝑅)⟩)    &   (πœ‘ β†’ 𝑇 ∈ Fin)    &   (πœ‘ β†’ 𝑍 ∈ 𝑇)    &   (πœ‘ β†’ Β¬ 𝑍 ∈ 𝑅)    &   (πœ‘ β†’ (𝑅 βˆͺ {𝑍}) βŠ† 𝑇)    β‡’   (πœ‘ β†’ 𝐷:((πΆβ€˜(𝑅 βˆͺ {𝑍}))β€˜π½)–1-1-ontoβ†’βˆͺ π‘˜ ∈ (0...𝐽)({π‘˜} Γ— ((πΆβ€˜π‘…)β€˜π‘˜)))
 
Theoremdvnprodlem2 43941* Induction step for dvnprodlem2 43941. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑋 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    &   (πœ‘ β†’ 𝑇 ∈ Fin)    &   ((πœ‘ ∧ 𝑑 ∈ 𝑇) β†’ (π»β€˜π‘‘):π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   ((πœ‘ ∧ 𝑑 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) β†’ ((𝑆 D𝑛 (π»β€˜π‘‘))β€˜π‘—):π‘‹βŸΆβ„‚)    &   πΆ = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ β„•0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑑 ∈ 𝑠 (π‘β€˜π‘‘) = 𝑛}))    &   (πœ‘ β†’ 𝑅 βŠ† 𝑇)    &   (πœ‘ β†’ 𝑍 ∈ (𝑇 βˆ– 𝑅))    &   (πœ‘ β†’ βˆ€π‘˜ ∈ (0...𝑁)((𝑆 D𝑛 (π‘₯ ∈ 𝑋 ↦ βˆπ‘‘ ∈ 𝑅 ((π»β€˜π‘‘)β€˜π‘₯)))β€˜π‘˜) = (π‘₯ ∈ 𝑋 ↦ Σ𝑐 ∈ ((πΆβ€˜π‘…)β€˜π‘˜)(((!β€˜π‘˜) / βˆπ‘‘ ∈ 𝑅 (!β€˜(π‘β€˜π‘‘))) Β· βˆπ‘‘ ∈ 𝑅 (((𝑆 D𝑛 (π»β€˜π‘‘))β€˜(π‘β€˜π‘‘))β€˜π‘₯))))    &   (πœ‘ β†’ 𝐽 ∈ (0...𝑁))    &   π· = (𝑐 ∈ ((πΆβ€˜(𝑅 βˆͺ {𝑍}))β€˜π½) ↦ ⟨(𝐽 βˆ’ (π‘β€˜π‘)), (𝑐 β†Ύ 𝑅)⟩)    β‡’   (πœ‘ β†’ ((𝑆 D𝑛 (π‘₯ ∈ 𝑋 ↦ βˆπ‘‘ ∈ (𝑅 βˆͺ {𝑍})((π»β€˜π‘‘)β€˜π‘₯)))β€˜π½) = (π‘₯ ∈ 𝑋 ↦ Σ𝑐 ∈ ((πΆβ€˜(𝑅 βˆͺ {𝑍}))β€˜π½)(((!β€˜π½) / βˆπ‘‘ ∈ (𝑅 βˆͺ {𝑍})(!β€˜(π‘β€˜π‘‘))) Β· βˆπ‘‘ ∈ (𝑅 βˆͺ {𝑍})(((𝑆 D𝑛 (π»β€˜π‘‘))β€˜(π‘β€˜π‘‘))β€˜π‘₯))))
 
Theoremdvnprodlem3 43942* The multinomial formula for the π‘˜-th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑋 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    &   (πœ‘ β†’ 𝑇 ∈ Fin)    &   ((πœ‘ ∧ 𝑑 ∈ 𝑇) β†’ (π»β€˜π‘‘):π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   ((πœ‘ ∧ 𝑑 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) β†’ ((𝑆 D𝑛 (π»β€˜π‘‘))β€˜π‘—):π‘‹βŸΆβ„‚)    &   πΉ = (π‘₯ ∈ 𝑋 ↦ βˆπ‘‘ ∈ 𝑇 ((π»β€˜π‘‘)β€˜π‘₯))    &   π· = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ β„•0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑑 ∈ 𝑠 (π‘β€˜π‘‘) = 𝑛}))    &   πΆ = (𝑛 ∈ β„•0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑇) ∣ Σ𝑑 ∈ 𝑇 (π‘β€˜π‘‘) = 𝑛})    β‡’   (πœ‘ β†’ ((𝑆 D𝑛 𝐹)β€˜π‘) = (π‘₯ ∈ 𝑋 ↦ Σ𝑐 ∈ (πΆβ€˜π‘)(((!β€˜π‘) / βˆπ‘‘ ∈ 𝑇 (!β€˜(π‘β€˜π‘‘))) Β· βˆπ‘‘ ∈ 𝑇 (((𝑆 D𝑛 (π»β€˜π‘‘))β€˜(π‘β€˜π‘‘))β€˜π‘₯))))
 
Theoremdvnprod 43943* The multinomial formula for the 𝑁-th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑋 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    &   (πœ‘ β†’ 𝑇 ∈ Fin)    &   ((πœ‘ ∧ 𝑑 ∈ 𝑇) β†’ (π»β€˜π‘‘):π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   ((πœ‘ ∧ 𝑑 ∈ 𝑇 ∧ π‘˜ ∈ (0...𝑁)) β†’ ((𝑆 D𝑛 (π»β€˜π‘‘))β€˜π‘˜):π‘‹βŸΆβ„‚)    &   πΉ = (π‘₯ ∈ 𝑋 ↦ βˆπ‘‘ ∈ 𝑇 ((π»β€˜π‘‘)β€˜π‘₯))    &   πΆ = (𝑛 ∈ β„•0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑇) ∣ Σ𝑑 ∈ 𝑇 (π‘β€˜π‘‘) = 𝑛})    β‡’   (πœ‘ β†’ ((𝑆 D𝑛 𝐹)β€˜π‘) = (π‘₯ ∈ 𝑋 ↦ Σ𝑐 ∈ (πΆβ€˜π‘)(((!β€˜π‘) / βˆπ‘‘ ∈ 𝑇 (!β€˜(π‘β€˜π‘‘))) Β· βˆπ‘‘ ∈ 𝑇 (((𝑆 D𝑛 (π»β€˜π‘‘))β€˜(π‘β€˜π‘‘))β€˜π‘₯))))
 
21.38.11  Integrals
 
Theoremitgsin0pilem1 43944* Calculation of the integral for sine on the (0,Ο€) interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐢 = (𝑑 ∈ (0[,]Ο€) ↦ -(cosβ€˜π‘‘))    β‡’   βˆ«(0(,)Ο€)(sinβ€˜π‘₯) dπ‘₯ = 2
 
Theoremibliccsinexp 43945* sin^n on a closed interval is integrable. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝑁 ∈ β„•0) β†’ (π‘₯ ∈ (𝐴[,]𝐡) ↦ ((sinβ€˜π‘₯)↑𝑁)) ∈ 𝐿1)
 
Theoremitgsin0pi 43946 Calculation of the integral for sine on the (0,Ο€) interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
∫(0(,)Ο€)(sinβ€˜π‘₯) dπ‘₯ = 2
 
Theoremiblioosinexp 43947* sin^n on an open integral is integrable. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝑁 ∈ β„•0) β†’ (π‘₯ ∈ (𝐴(,)𝐡) ↦ ((sinβ€˜π‘₯)↑𝑁)) ∈ 𝐿1)
 
Theoremitgsinexplem1 43948* Integration by parts is applied to integrate sin^(N+1). (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐹 = (π‘₯ ∈ β„‚ ↦ ((sinβ€˜π‘₯)↑𝑁))    &   πΊ = (π‘₯ ∈ β„‚ ↦ -(cosβ€˜π‘₯))    &   π» = (π‘₯ ∈ β„‚ ↦ ((𝑁 Β· ((sinβ€˜π‘₯)↑(𝑁 βˆ’ 1))) Β· (cosβ€˜π‘₯)))    &   πΌ = (π‘₯ ∈ β„‚ ↦ (((sinβ€˜π‘₯)↑𝑁) Β· (sinβ€˜π‘₯)))    &   πΏ = (π‘₯ ∈ β„‚ ↦ (((𝑁 Β· ((sinβ€˜π‘₯)↑(𝑁 βˆ’ 1))) Β· (cosβ€˜π‘₯)) Β· -(cosβ€˜π‘₯)))    &   π‘€ = (π‘₯ ∈ β„‚ ↦ (((cosβ€˜π‘₯)↑2) Β· ((sinβ€˜π‘₯)↑(𝑁 βˆ’ 1))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ ∫(0(,)Ο€)(((sinβ€˜π‘₯)↑𝑁) Β· (sinβ€˜π‘₯)) dπ‘₯ = (𝑁 Β· ∫(0(,)Ο€)(((cosβ€˜π‘₯)↑2) Β· ((sinβ€˜π‘₯)↑(𝑁 βˆ’ 1))) dπ‘₯))
 
Theoremitgsinexp 43949* A recursive formula for the integral of sin^N on the interval (0,Ο€) . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐼 = (𝑛 ∈ β„•0 ↦ ∫(0(,)Ο€)((sinβ€˜π‘₯)↑𝑛) dπ‘₯)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜2))    β‡’   (πœ‘ β†’ (πΌβ€˜π‘) = (((𝑁 βˆ’ 1) / 𝑁) Β· (πΌβ€˜(𝑁 βˆ’ 2))))
 
Theoremiblconstmpt 43950* A constant function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ dom vol ∧ (volβ€˜π΄) ∈ ℝ ∧ 𝐡 ∈ β„‚) β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1)
 
Theoremitgeq1d 43951* Equality theorem for an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ ∫𝐴𝐢 dπ‘₯ = ∫𝐡𝐢 dπ‘₯)
 
Theoremmbfres2cn 43952 Measurability of a piecewise function: if 𝐹 is measurable on subsets 𝐡 and 𝐢 of its domain, and these pieces make up all of 𝐴, then 𝐹 is measurable on the whole domain. Similar to mbfres2 24931 but here the theorem is extended to complex-valued functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ (𝐹 β†Ύ 𝐡) ∈ MblFn)    &   (πœ‘ β†’ (𝐹 β†Ύ 𝐢) ∈ MblFn)    &   (πœ‘ β†’ (𝐡 βˆͺ 𝐢) = 𝐴)    β‡’   (πœ‘ β†’ 𝐹 ∈ MblFn)
 
Theoremvol0 43953 The measure of the empty set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(volβ€˜βˆ…) = 0
 
Theoremditgeqiooicc 43954* A function 𝐹 on an open interval, has the same directed integral as its extension 𝐺 on the closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐺 = (π‘₯ ∈ (𝐴[,]𝐡) ↦ if(π‘₯ = 𝐴, 𝑅, if(π‘₯ = 𝐡, 𝐿, (πΉβ€˜π‘₯))))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    β‡’   (πœ‘ β†’ ⨜[𝐴 β†’ 𝐡](πΉβ€˜π‘₯) dπ‘₯ = ⨜[𝐴 β†’ 𝐡](πΊβ€˜π‘₯) dπ‘₯)
 
Theoremvolge0 43955 The volume of a set is always nonnegative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ dom vol β†’ 0 ≀ (volβ€˜π΄))
 
Theoremcnbdibl 43956* A continuous bounded function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ dom vol)    &   (πœ‘ β†’ (volβ€˜π΄) ∈ ℝ)    &   (πœ‘ β†’ 𝐹 ∈ (𝐴–cnβ†’β„‚))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ dom 𝐹(absβ€˜(πΉβ€˜π‘¦)) ≀ π‘₯)    β‡’   (πœ‘ β†’ 𝐹 ∈ 𝐿1)
 
Theoremsnmbl 43957 A singleton is measurable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ β†’ {𝐴} ∈ dom vol)
 
Theoremditgeq3d 43958* Equality theorem for the directed integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ≀ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴(,)𝐡)) β†’ 𝐷 = 𝐸)    β‡’   (πœ‘ β†’ ⨜[𝐴 β†’ 𝐡]𝐷 dπ‘₯ = ⨜[𝐴 β†’ 𝐡]𝐸 dπ‘₯)
 
Theoremiblempty 43959 The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
βˆ… ∈ 𝐿1
 
Theoremiblsplit 43960* The union of two integrable functions is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ (vol*β€˜(𝐴 ∩ 𝐡)) = 0)    &   (πœ‘ β†’ π‘ˆ = (𝐴 βˆͺ 𝐡))    &   ((πœ‘ ∧ π‘₯ ∈ π‘ˆ) β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐢) ∈ 𝐿1)    &   (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ 𝐢) ∈ 𝐿1)    β‡’   (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↦ 𝐢) ∈ 𝐿1)
 
Theoremvolsn 43961 A singleton has 0 Lebesgue measure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ β†’ (volβ€˜{𝐴}) = 0)
 
Theoremitgvol0 43962* If the domani is negligible, the function is integrable and the integral is 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ (vol*β€˜π΄) = 0)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1 ∧ ∫𝐴𝐡 dπ‘₯ = 0))
 
Theoremitgcoscmulx 43963* Exercise: the integral of π‘₯ ↦ cosπ‘Žπ‘₯ on an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ≀ 𝐢)    &   (πœ‘ β†’ 𝐴 β‰  0)    β‡’   (πœ‘ β†’ ∫(𝐡(,)𝐢)(cosβ€˜(𝐴 Β· π‘₯)) dπ‘₯ = (((sinβ€˜(𝐴 Β· 𝐢)) βˆ’ (sinβ€˜(𝐴 Β· 𝐡))) / 𝐴))
 
Theoremiblsplitf 43964* A version of iblsplit 43960 using bound-variable hypotheses instead of distinct variable conditions". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ (vol*β€˜(𝐴 ∩ 𝐡)) = 0)    &   (πœ‘ β†’ π‘ˆ = (𝐴 βˆͺ 𝐡))    &   ((πœ‘ ∧ π‘₯ ∈ π‘ˆ) β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐢) ∈ 𝐿1)    &   (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ 𝐢) ∈ 𝐿1)    β‡’   (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↦ 𝐢) ∈ 𝐿1)
 
Theoremibliooicc 43965* If a function is integrable on an open interval, it is integrable on the corresponding closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (π‘₯ ∈ (𝐴(,)𝐡) ↦ 𝐢) ∈ 𝐿1)    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ (π‘₯ ∈ (𝐴[,]𝐡) ↦ 𝐢) ∈ 𝐿1)
 
Theoremvolioc 43966 The measure of a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐴 ≀ 𝐡) β†’ (volβ€˜(𝐴(,]𝐡)) = (𝐡 βˆ’ 𝐴))
 
Theoremiblspltprt 43967* If a function is integrable on any interval of a partition, then it is integrable on the whole interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
β„²π‘‘πœ‘    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑀 + 1)))    &   ((πœ‘ ∧ 𝑖 ∈ (𝑀...𝑁)) β†’ (π‘ƒβ€˜π‘–) ∈ ℝ)    &   ((πœ‘ ∧ 𝑖 ∈ (𝑀..^𝑁)) β†’ (π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1)))    &   ((πœ‘ ∧ 𝑑 ∈ ((π‘ƒβ€˜π‘€)[,](π‘ƒβ€˜π‘))) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ 𝑖 ∈ (𝑀..^𝑁)) β†’ (𝑑 ∈ ((π‘ƒβ€˜π‘–)[,](π‘ƒβ€˜(𝑖 + 1))) ↦ 𝐴) ∈ 𝐿1)    β‡’   (πœ‘ β†’ (𝑑 ∈ ((π‘ƒβ€˜π‘€)[,](π‘ƒβ€˜π‘)) ↦ 𝐴) ∈ 𝐿1)
 
Theoremitgsincmulx 43968* Exercise: the integral of π‘₯ ↦ sinπ‘Žπ‘₯ on an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐴 β‰  0)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ≀ 𝐢)    β‡’   (πœ‘ β†’ ∫(𝐡(,)𝐢)(sinβ€˜(𝐴 Β· π‘₯)) dπ‘₯ = (((cosβ€˜(𝐴 Β· 𝐡)) βˆ’ (cosβ€˜(𝐴 Β· 𝐢))) / 𝐴))
 
Theoremitgsubsticclem 43969* lemma for itgsubsticc 43970. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑒 ∈ (𝐾[,]𝐿) ↦ 𝐢)    &   πΊ = (𝑒 ∈ ℝ ↦ if(𝑒 ∈ (𝐾[,]𝐿), (πΉβ€˜π‘’), if(𝑒 < 𝐾, (πΉβ€˜πΎ), (πΉβ€˜πΏ))))    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   (πœ‘ β†’ π‘Œ ∈ ℝ)    &   (πœ‘ β†’ 𝑋 ≀ π‘Œ)    &   (πœ‘ β†’ (π‘₯ ∈ (𝑋[,]π‘Œ) ↦ 𝐴) ∈ ((𝑋[,]π‘Œ)–cnβ†’(𝐾[,]𝐿)))    &   (πœ‘ β†’ (π‘₯ ∈ (𝑋(,)π‘Œ) ↦ 𝐡) ∈ (((𝑋(,)π‘Œ)–cnβ†’β„‚) ∩ 𝐿1))    &   (πœ‘ β†’ 𝐹 ∈ ((𝐾[,]𝐿)–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐾 ∈ ℝ)    &   (πœ‘ β†’ 𝐿 ∈ ℝ)    &   (πœ‘ β†’ 𝐾 ≀ 𝐿)    &   (πœ‘ β†’ (ℝ D (π‘₯ ∈ (𝑋[,]π‘Œ) ↦ 𝐴)) = (π‘₯ ∈ (𝑋(,)π‘Œ) ↦ 𝐡))    &   (𝑒 = 𝐴 β†’ 𝐢 = 𝐸)    &   (π‘₯ = 𝑋 β†’ 𝐴 = 𝐾)    &   (π‘₯ = π‘Œ β†’ 𝐴 = 𝐿)    β‡’   (πœ‘ β†’ ⨜[𝐾 β†’ 𝐿]𝐢 d𝑒 = ⨜[𝑋 β†’ π‘Œ](𝐸 Β· 𝐡) dπ‘₯)
 
Theoremitgsubsticc 43970* Integration by u-substitution. The main difference with respect to itgsubst 25335 is that here we consider the range of 𝐴(π‘₯) to be in the closed interval (𝐾[,]𝐿). If 𝐴(π‘₯) is a continuous, differentiable function from [𝑋, π‘Œ] to (𝑍, π‘Š), whose derivative is continuous and integrable, and 𝐢(𝑒) is a continuous function on (𝑍, π‘Š), then the integral of 𝐢(𝑒) from 𝐾 = 𝐴(𝑋) to 𝐿 = 𝐴(π‘Œ) is equal to the integral of 𝐢(𝐴(π‘₯)) D 𝐴(π‘₯) from 𝑋 to π‘Œ. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑋 ∈ ℝ)    &   (πœ‘ β†’ π‘Œ ∈ ℝ)    &   (πœ‘ β†’ 𝑋 ≀ π‘Œ)    &   (πœ‘ β†’ (π‘₯ ∈ (𝑋[,]π‘Œ) ↦ 𝐴) ∈ ((𝑋[,]π‘Œ)–cnβ†’(𝐾[,]𝐿)))    &   (πœ‘ β†’ (𝑒 ∈ (𝐾[,]𝐿) ↦ 𝐢) ∈ ((𝐾[,]𝐿)–cnβ†’β„‚))    &   (πœ‘ β†’ (π‘₯ ∈ (𝑋(,)π‘Œ) ↦ 𝐡) ∈ (((𝑋(,)π‘Œ)–cnβ†’β„‚) ∩ 𝐿1))    &   (πœ‘ β†’ (ℝ D (π‘₯ ∈ (𝑋[,]π‘Œ) ↦ 𝐴)) = (π‘₯ ∈ (𝑋(,)π‘Œ) ↦ 𝐡))    &   (𝑒 = 𝐴 β†’ 𝐢 = 𝐸)    &   (π‘₯ = 𝑋 β†’ 𝐴 = 𝐾)    &   (π‘₯ = π‘Œ β†’ 𝐴 = 𝐿)    &   (πœ‘ β†’ 𝐾 ∈ ℝ)    &   (πœ‘ β†’ 𝐿 ∈ ℝ)    β‡’   (πœ‘ β†’ ⨜[𝐾 β†’ 𝐿]𝐢 d𝑒 = ⨜[𝑋 β†’ π‘Œ](𝐸 Β· 𝐡) dπ‘₯)
 
Theoremitgioocnicc 43971* The integral of a piecewise continuous function 𝐹 on an open interval is equal to the integral of the continuous function 𝐺, in the corresponding closed interval. 𝐺 is equal to 𝐹 on the open interval, but it is continuous at the two boundaries, also. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹:dom πΉβŸΆβ„‚)    &   (πœ‘ β†’ (𝐹 β†Ύ (𝐴(,)𝐡)) ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† dom 𝐹)    &   (πœ‘ β†’ 𝑅 ∈ ((𝐹 β†Ύ (𝐴(,)𝐡)) limβ„‚ 𝐴))    &   (πœ‘ β†’ 𝐿 ∈ ((𝐹 β†Ύ (𝐴(,)𝐡)) limβ„‚ 𝐡))    &   πΊ = (π‘₯ ∈ (𝐴[,]𝐡) ↦ if(π‘₯ = 𝐴, 𝑅, if(π‘₯ = 𝐡, 𝐿, (πΉβ€˜π‘₯))))    β‡’   (πœ‘ β†’ (𝐺 ∈ 𝐿1 ∧ ∫(𝐴[,]𝐡)(πΊβ€˜π‘₯) dπ‘₯ = ∫(𝐴[,]𝐡)(πΉβ€˜π‘₯) dπ‘₯))
 
Theoremiblcncfioo 43972 A continuous function 𝐹 on an open interval (𝐴(,)𝐡) with a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐡 is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐿 ∈ (𝐹 limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝑅 ∈ (𝐹 limβ„‚ 𝐴))    β‡’   (πœ‘ β†’ 𝐹 ∈ 𝐿1)
 
Theoremitgspltprt 43973* The ∫ integral splits on a given partition 𝑃. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑀 + 1)))    &   (πœ‘ β†’ 𝑃:(𝑀...𝑁)βŸΆβ„)    &   ((πœ‘ ∧ 𝑖 ∈ (𝑀..^𝑁)) β†’ (π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1)))    &   ((πœ‘ ∧ 𝑑 ∈ ((π‘ƒβ€˜π‘€)[,](π‘ƒβ€˜π‘))) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ 𝑖 ∈ (𝑀..^𝑁)) β†’ (𝑑 ∈ ((π‘ƒβ€˜π‘–)[,](π‘ƒβ€˜(𝑖 + 1))) ↦ 𝐴) ∈ 𝐿1)    β‡’   (πœ‘ β†’ ∫((π‘ƒβ€˜π‘€)[,](π‘ƒβ€˜π‘))𝐴 d𝑑 = Σ𝑖 ∈ (𝑀..^𝑁)∫((π‘ƒβ€˜π‘–)[,](π‘ƒβ€˜(𝑖 + 1)))𝐴 d𝑑)
 
Theoremitgiccshift 43974* The integral of a function, 𝐹 stays the same if its closed interval domain is shifted by 𝑇. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴[,]𝐡)–cnβ†’β„‚))    &   (πœ‘ β†’ 𝑇 ∈ ℝ+)    &   πΊ = (π‘₯ ∈ ((𝐴 + 𝑇)[,](𝐡 + 𝑇)) ↦ (πΉβ€˜(π‘₯ βˆ’ 𝑇)))    β‡’   (πœ‘ β†’ ∫((𝐴 + 𝑇)[,](𝐡 + 𝑇))(πΊβ€˜π‘₯) dπ‘₯ = ∫(𝐴[,]𝐡)(πΉβ€˜π‘₯) dπ‘₯)
 
Theoremitgperiod 43975* The integral of a periodic function, with period 𝑇 stays the same if the domain of integration is shifted. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   (πœ‘ β†’ 𝑇 ∈ ℝ+)    &   (πœ‘ β†’ 𝐹:β„βŸΆβ„‚)    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜(π‘₯ + 𝑇)) = (πΉβ€˜π‘₯))    &   (πœ‘ β†’ (𝐹 β†Ύ (𝐴[,]𝐡)) ∈ ((𝐴[,]𝐡)–cnβ†’β„‚))    β‡’   (πœ‘ β†’ ∫((𝐴 + 𝑇)[,](𝐡 + 𝑇))(πΉβ€˜π‘₯) dπ‘₯ = ∫(𝐴[,]𝐡)(πΉβ€˜π‘₯) dπ‘₯)
 
Theoremitgsbtaddcnst 43976* Integral substitution, adding a constant to the function's argument. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴[,]𝐡)–cnβ†’β„‚))    β‡’   (πœ‘ β†’ ⨜[(𝐴 βˆ’ 𝑋) β†’ (𝐡 βˆ’ 𝑋)](πΉβ€˜(𝑋 + 𝑠)) d𝑠 = ⨜[𝐴 β†’ 𝐡](πΉβ€˜π‘‘) d𝑑)
 
Theoremvolico 43977 The measure of left-closed, right-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (volβ€˜(𝐴[,)𝐡)) = if(𝐴 < 𝐡, (𝐡 βˆ’ 𝐴), 0))
 
Theoremsublevolico 43978 The Lebesgue measure of a left-closed, right-open interval is greater than or equal to the difference of the two bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (𝐡 βˆ’ 𝐴) ≀ (volβ€˜(𝐴[,)𝐡)))
 
Theoremdmvolss 43979 Lebesgue measurable sets are subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
dom vol βŠ† 𝒫 ℝ
 
Theoremismbl3 43980* The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl2 24813, but here +𝑒 is used, and the precondition (vol*β€˜π‘₯) ∈ ℝ can be dropped. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐴 ∈ dom vol ↔ (𝐴 βŠ† ℝ ∧ βˆ€π‘₯ ∈ 𝒫 ℝ((vol*β€˜(π‘₯ ∩ 𝐴)) +𝑒 (vol*β€˜(π‘₯ βˆ– 𝐴))) ≀ (vol*β€˜π‘₯)))
 
Theoremvolioof 43981 The function that assigns the Lebesgue measure to open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(vol ∘ (,)):(ℝ* Γ— ℝ*)⟢(0[,]+∞)
 
Theoremovolsplit 43982 The Lebesgue outer measure function is finitely sub-additive: application to a set split in two parts, using addition for extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 βŠ† ℝ)    β‡’   (πœ‘ β†’ (vol*β€˜π΄) ≀ ((vol*β€˜(𝐴 ∩ 𝐡)) +𝑒 (vol*β€˜(𝐴 βˆ– 𝐡))))
 
Theoremfvvolioof 43983 The function value of the Lebesgue measure of an open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐹:𝐴⟢(ℝ* Γ— ℝ*))    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    β‡’   (πœ‘ β†’ (((vol ∘ (,)) ∘ 𝐹)β€˜π‘‹) = (volβ€˜((1st β€˜(πΉβ€˜π‘‹))(,)(2nd β€˜(πΉβ€˜π‘‹)))))
 
Theoremvolioore 43984 The measure of an open interval. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (volβ€˜(𝐴(,)𝐡)) = if(𝐴 ≀ 𝐡, (𝐡 βˆ’ 𝐴), 0))
 
Theoremfvvolicof 43985 The function value of the Lebesgue measure of a left-closed right-open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐹:𝐴⟢(ℝ* Γ— ℝ*))    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    β‡’   (πœ‘ β†’ (((vol ∘ [,)) ∘ 𝐹)β€˜π‘‹) = (volβ€˜((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹)))))
 
Theoremvoliooico 43986 An open interval and a left-closed, right-open interval with the same real bounds, have the same Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (volβ€˜(𝐴(,)𝐡)) = (volβ€˜(𝐴[,)𝐡)))
 
Theoremismbl4 43987* The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl 24812, but here +𝑒 is used, and the precondition (vol*β€˜π‘₯) ∈ ℝ can be dropped. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐴 ∈ dom vol ↔ (𝐴 βŠ† ℝ ∧ βˆ€π‘₯ ∈ 𝒫 ℝ(vol*β€˜π‘₯) = ((vol*β€˜(π‘₯ ∩ 𝐴)) +𝑒 (vol*β€˜(π‘₯ βˆ– 𝐴)))))
 
Theoremvolioofmpt 43988* ((vol ∘ (,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
β„²π‘₯𝐹    &   (πœ‘ β†’ 𝐹:𝐴⟢(ℝ* Γ— ℝ*))    β‡’   (πœ‘ β†’ ((vol ∘ (,)) ∘ 𝐹) = (π‘₯ ∈ 𝐴 ↦ (volβ€˜((1st β€˜(πΉβ€˜π‘₯))(,)(2nd β€˜(πΉβ€˜π‘₯))))))
 
Theoremvolicoff 43989 ((vol ∘ [,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐹:𝐴⟢(ℝ Γ— ℝ*))    β‡’   (πœ‘ β†’ ((vol ∘ [,)) ∘ 𝐹):𝐴⟢(0[,]+∞))
 
Theoremvoliooicof 43990 The Lebesgue measure of open intervals is the same as the Lebesgue measure of left-closed right-open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐹:𝐴⟢(ℝ Γ— ℝ))    β‡’   (πœ‘ β†’ ((vol ∘ (,)) ∘ 𝐹) = ((vol ∘ [,)) ∘ 𝐹))
 
Theoremvolicofmpt 43991* ((vol ∘ [,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
β„²π‘₯𝐹    &   (πœ‘ β†’ 𝐹:𝐴⟢(ℝ Γ— ℝ*))    β‡’   (πœ‘ β†’ ((vol ∘ [,)) ∘ 𝐹) = (π‘₯ ∈ 𝐴 ↦ (volβ€˜((1st β€˜(πΉβ€˜π‘₯))[,)(2nd β€˜(πΉβ€˜π‘₯))))))
 
Theoremvolicc 43992 The Lebesgue measure of a closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐴 ≀ 𝐡) β†’ (volβ€˜(𝐴[,]𝐡)) = (𝐡 βˆ’ 𝐴))
 
Theoremvoliccico 43993 A closed interval and a left-closed, right-open interval with the same real bounds, have the same Lebesgue measure. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (volβ€˜(𝐴[,]𝐡)) = (volβ€˜(𝐴[,)𝐡)))
 
Theoremmbfdmssre 43994 The domain of a measurable function is a subset of the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ MblFn β†’ dom 𝐹 βŠ† ℝ)
 
21.38.12  Stone Weierstrass theorem - real version
 
Theoremstoweidlem1 43995 Lemma for stoweid 44057. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90; the key step uses Bernoulli's inequality bernneq 14057. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐾 ∈ β„•)    &   (πœ‘ β†’ 𝐷 ∈ ℝ+)    &   (πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 0 ≀ 𝐴)    &   (πœ‘ β†’ 𝐴 ≀ 1)    &   (πœ‘ β†’ 𝐷 ≀ 𝐴)    β‡’   (πœ‘ β†’ ((1 βˆ’ (𝐴↑𝑁))↑(𝐾↑𝑁)) ≀ (1 / ((𝐾 Β· 𝐷)↑𝑁)))
 
Theoremstoweidlem2 43996* lemma for stoweid 44057: here we prove that the subalgebra of continuous functions, which contains constant functions, is closed under scaling. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
β„²π‘‘πœ‘    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ 𝑓:π‘‡βŸΆβ„)    &   (πœ‘ β†’ 𝐸 ∈ ℝ)    &   (πœ‘ β†’ 𝐹 ∈ 𝐴)    β‡’   (πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ (𝐸 Β· (πΉβ€˜π‘‘))) ∈ 𝐴)
 
Theoremstoweidlem3 43997* Lemma for stoweid 44057: if 𝐴 is positive and all 𝑀 terms of a finite product are larger than 𝐴, then the finite product is larger than 𝐴↑𝑀. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Ⅎ𝑖𝐹    &   β„²π‘–πœ‘    &   π‘‹ = seq1( Β· , 𝐹)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑀)βŸΆβ„)    &   ((πœ‘ ∧ 𝑖 ∈ (1...𝑀)) β†’ 𝐴 < (πΉβ€˜π‘–))    &   (πœ‘ β†’ 𝐴 ∈ ℝ+)    β‡’   (πœ‘ β†’ (𝐴↑𝑀) < (π‘‹β€˜π‘€))
 
Theoremstoweidlem4 43998* Lemma for stoweid 44057: a class variable replaces a setvar variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴)    β‡’   ((πœ‘ ∧ 𝐡 ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ 𝐡) ∈ 𝐴)
 
Theoremstoweidlem5 43999* There exists a Ξ΄ as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 0 < Ξ΄ < 1 , p >= Ξ΄ on 𝑇 βˆ– π‘ˆ. Here 𝐷 is used to represent Ξ΄ in the paper and 𝑄 to represent 𝑇 βˆ– π‘ˆ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
β„²π‘‘πœ‘    &   π· = if(𝐢 ≀ (1 / 2), 𝐢, (1 / 2))    &   (πœ‘ β†’ 𝑃:π‘‡βŸΆβ„)    &   (πœ‘ β†’ 𝑄 βŠ† 𝑇)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘‘ ∈ 𝑄 𝐢 ≀ (π‘ƒβ€˜π‘‘))    β‡’   (πœ‘ β†’ βˆƒπ‘‘(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ βˆ€π‘‘ ∈ 𝑄 𝑑 ≀ (π‘ƒβ€˜π‘‘)))
 
Theoremstoweidlem6 44000* Lemma for stoweid 44057: two class variables replace two setvar variables, for multiplication of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Ⅎ𝑑 𝑓 = 𝐹    &   β„²π‘‘ 𝑔 = 𝐺    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    β‡’   ((πœ‘ ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((πΉβ€˜π‘‘) Β· (πΊβ€˜π‘‘))) ∈ 𝐴)
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