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Theorem List for Metamath Proof Explorer - 39501-39600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempimgtpnf2 39501* Given a real valued function, the preimage of an open interval, unbounded above, with lower bound +∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = ∅)
 
Theoremsalpreimagelt 39502* If all the preimages of left-close, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iv) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐴 = 𝑆    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎𝐵} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)
 
Theorempimrecltpos 39503 The preimage of an unbounded below, open interval, with positive upper bound, for the reciprocal function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ≠ 0)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → {𝑥𝐴 ∣ (1 / 𝐵) < 𝐶} = ({𝑥𝐴 ∣ (1 / 𝐶) < 𝐵} ∪ {𝑥𝐴𝐵 < 0}))
 
Theoremsalpreimalegt 39504* If all the preimages of right-closed, unbounded below intervals, belong to a sigma-algebra, then all the preimages of left-open, unbounded above intervals, belong to the sigma-algebra. (ii) implies (iii) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐴 = 𝑆    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐶 < 𝐵} ∈ 𝑆)
 
Theorempimiooltgt 39505* The preimage of an open interval is the intersection of the preimage of an unbounded below open interval and an unbounded above open interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝐿 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ*)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)       (𝜑 → {𝑥𝐴𝐵 ∈ (𝐿(,)𝑅)} = ({𝑥𝐴𝐵 < 𝑅} ∩ {𝑥𝐴𝐿 < 𝐵}))
 
Theorempreimaicomnf 39506* Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐹 “ (-∞[,)𝐵)) = {𝑥𝐴 ∣ (𝐹𝑥) < 𝐵})
 
Theorempimltpnf2 39507* Given a real valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = 𝐴)
 
Theorempimgtmnf2 39508* Given a real valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} = 𝐴)
 
Theorempimdecfgtioc 39509* Given a non-increasing function, the preimage of an unbounded above, open interval, when the supremum of the preimage belongs to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴𝑅 < (𝐹𝑥)}    &   𝑆 = sup(𝑌, ℝ*, < )    &   (𝜑𝑆𝑌)    &   𝐼 = (-∞(,]𝑆)       (𝜑𝑌 = (𝐼𝐴))
 
Theorempimincfltioc 39510* Given a non decreasing function, the preimage of an unbounded below, open interval, when the supremum of the preimage belongs to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴 ∣ (𝐹𝑥) < 𝑅}    &   𝑆 = sup(𝑌, ℝ*, < )    &   (𝜑𝑆𝑌)    &   𝐼 = (-∞(,]𝑆)       (𝜑𝑌 = (𝐼𝐴))
 
Theorempimdecfgtioo 39511* Given a non decreasing function, the preimage of an unbounded below, open interval, when the supremum of the preimage does not belong to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴𝑅 < (𝐹𝑥)}    &   𝑆 = sup(𝑌, ℝ*, < )    &   (𝜑 → ¬ 𝑆𝑌)    &   𝐼 = (-∞(,)𝑆)       (𝜑𝑌 = (𝐼𝐴))
 
Theorempimincfltioo 39512* Given a non decreasing function, the preimage of an unbounded below, open interval, when the supremum of the preimage does not belong to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴 ∣ (𝐹𝑥) < 𝑅}    &   𝑆 = sup(𝑌, ℝ*, < )    &   (𝜑 → ¬ 𝑆𝑌)    &   𝐼 = (-∞(,)𝑆)       (𝜑𝑌 = (𝐼𝐴))
 
Theorempreimaioomnf 39513* Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐹 “ (-∞(,)𝐵)) = {𝑥𝐴 ∣ (𝐹𝑥) < 𝐵})
 
Theorempreimageiingt 39514* A preimage of a left-closed, unbounded above interval, expressed as an indexed intersection of preimages of open, unbounded above intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐶𝐵} = 𝑛 ∈ ℕ {𝑥𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵})
 
Theorempreimaleiinlt 39515* A preimage of a left-open, right-closed, unbounded below interval, expressed as an indexed intersection of preimages of open, unbound below intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵𝐶} = 𝑛 ∈ ℕ {𝑥𝐴𝐵 < (𝐶 + (1 / 𝑛))})
 
Theorempimgtmnf 39516* Given a real valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = 𝐴)
 
Theorempimrecltneg 39517 The preimage of an unbounded below, open interval, with negative upper bound, for the reciprocal function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ≠ 0)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 < 0)       (𝜑 → {𝑥𝐴 ∣ (1 / 𝐵) < 𝐶} = {𝑥𝐴𝐵 ∈ ((1 / 𝐶)(,)0)})
 
Theoremsalpreimagtge 39518* If all the preimages of left-open, unbounded above intervals, belong to a sigma-algebra, then all the preimages of left-closed, unbounded above intervals, belong to the sigma-algebra. (iii) implies (iv) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎 < 𝐵} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐶𝐵} ∈ 𝑆)
 
Theoremsalpreimaltle 39519* If all the preimages of right-open, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-closed, unbounded below intervals, belong to the sigma-algebra. (i) implies (ii) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵 < 𝑎} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵𝐶} ∈ 𝑆)
 
Theoremissmflem 39520* The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
 
Theoremissmf 39521* The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
 
Theoremsalpreimalelt 39522* If all the preimages of right-close, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (ii) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐴 = 𝑆    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)
 
Theoremsalpreimagtlt 39523* If all the preimages of lef-open, unbounded above intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iii) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐴 = 𝑆    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎 < 𝐵} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)
 
Theoremsmfpreimalt 39524* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))
 
Theoremsmff 39525 A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹       (𝜑𝐹:𝐷⟶ℝ)
 
Theoremsmfdmss 39526 The domain of a function measurable w.r.t. to a sigma-algebra, is a subset of the set underlying the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹       (𝜑𝐷 𝑆)
 
Theoremissmff 39527* The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
 
Theoremissmfd 39528* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))
 
Theoremissmfltle 39529* The definition of a measurable function w.r.t. a sigma-algebra, can be stated using less than or equal instead of less than. Proposition 121B (ii) of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))
 
Theoremsmfpreimaltf 39530* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))
 
Theoremissmfdf 39531* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))
 
Theoremsssmf 39532 The restriction of a sigma-measurable function, is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))       (𝜑 → (𝐹𝐵) ∈ (SMblFn‘𝑆))
 
Theoremmbfresmf 39533 A Real valued, measurable function is a sigma-measurable function (w.r.t. the Lebesgue measure on the Reals). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑 → ran 𝐹 ⊆ ℝ)    &   𝑆 = dom vol       (𝜑𝐹 ∈ (SMblFn‘𝑆))
 
Theoremcnfsmf 39534 A continuous function is measurable. Proposition 121D (b) of [Fremlin1] p. 36 is a special case of this theorem, where the topology on the domain is induced by the standard topology on n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐽 ∈ Top)    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐹 ∈ ((𝐽t dom 𝐹) Cn 𝐾))    &   𝑆 = (SalGen‘𝐽)       (𝜑𝐹 ∈ (SMblFn‘𝑆))
 
Theoremincsmflem 39535* A non decreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴 ∣ (𝐹𝑥) < 𝑅}    &   𝐶 = sup(𝑌, ℝ*, < )    &   𝐷 = (-∞(,)𝐶)    &   𝐸 = (-∞(,]𝐶)       (𝜑 → ∃𝑏𝐵 𝑌 = (𝑏𝐴))
 
Theoremincsmf 39536* A real valued, non-decreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       (𝜑𝐹 ∈ (SMblFn‘𝐵))
 
Theoremsmfsssmf 39537 If a function is measurable w.r.t. to a sigma-algebra, then it is measurable w.r.t. to a larger sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑅 ∈ SAlg)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝑅𝑆)    &   (𝜑𝐹 ∈ (SMblFn‘𝑅))       (𝜑𝐹 ∈ (SMblFn‘𝑆))
 
Theoremissmflelem 39538* The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all right closed intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (ii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))
 
Theoremissmfle 39539* The predicate "𝐹 is b measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all right closed intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (ii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))))
 
Theoremsmfpimltmpt 39540* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝑅 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < 𝑅} ∈ (𝑆t 𝐴))
 
Theoremsmfpimltxr 39541* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ*)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))
 
Theoremissmfdmpt 39542* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴 𝑆)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵 < 𝑎} ∈ (𝑆t 𝐴))       (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
 
Theoremsmfconst 39543* A constant function is measurable. Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴 𝑆)    &   (𝜑𝐵 ∈ ℝ)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑𝐹 ∈ (SMblFn‘𝑆))
 
Theoremsssmfmpt 39544* The restriction of a sigma-measurable function, is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝐶𝐴)       (𝜑 → (𝑥𝐶𝐵) ∈ (SMblFn‘𝑆))
 
Theoremcnfrrnsmf 39545 A function, continuous from the standard topology on the space of n-dimensional reals, and the standard topology on the reals, is Borel measurable. Proposition 121D (b) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋 ∈ Fin)    &   𝐽 = (TopOpen‘(ℝ^‘𝑋))    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐹 ∈ ((𝐽t dom 𝐹) Cn 𝐾))    &   𝐵 = (SalGen‘𝐽)       (𝜑𝐹 ∈ (SMblFn‘𝐵))
 
Theoremsmfid 39546* The identity function is Borel sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → (𝑥𝐴𝑥) ∈ (SMblFn‘𝐵))
 
Theorembormflebmf 39547 A Borel measurable function is Lebesgue measurable. Proposition 121D (a) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋 ∈ Fin)    &   𝐵 = (SalGen‘(TopOpen‘(ℝ^‘𝑋)))    &   𝐿 = dom (voln‘𝑋)    &   (𝜑𝐹 ∈ (SMblFn‘𝐵))       (𝜑𝐹 ∈ (SMblFn‘𝐿))
 
Theoremsmfpreimale 39548* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an closed interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))
 
Theoremissmfgtlem 39549* The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-open intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   (𝜑 → ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))
 
Theoremissmfgt 39550* The predicate "𝐹 is b measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-open intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷))))
 
Theoremissmfled 39551* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))
 
Theoremsmfpimltxrmpt 39552* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝑅 ∈ ℝ*)       (𝜑 → {𝑥𝐴𝐵 < 𝑅} ∈ (𝑆t 𝐴))
 
Theoremsmfmbfcex 39553* A constant function, with non-lebesgue-measurable domain is a sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals) but it is not a measurable functions ( w.r.t. to df-mbf 23069). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑆 = dom vol    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑 → ¬ 𝑋𝑆)    &   𝐹 = (𝑥𝑋 ↦ 0)       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ∧ ¬ 𝐹 ∈ MblFn))
 
Theoremissmfgtd 39554* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))
 
Theoremsmfpreimagt 39555* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷𝐴 < (𝐹𝑥)} ∈ (𝑆t 𝐷))
 
Theoremsmfaddlem1 39556* Given the sum of two functions, the preimage of an unbounded below, open interval, expressed as the countable union of intersections of preimages of both functions. Proposition 121E (b) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐶) → 𝐷 ∈ ℝ)    &   (𝜑𝑅 ∈ ℝ)    &   𝐾 = (𝑝 ∈ ℚ ↦ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅})       (𝜑 → {𝑥 ∈ (𝐴𝐶) ∣ (𝐵 + 𝐷) < 𝑅} = 𝑝 ∈ ℚ 𝑞 ∈ (𝐾𝑝){𝑥 ∈ (𝐴𝐶) ∣ (𝐵 < 𝑝𝐷 < 𝑞)})
 
Theoremsmfaddlem2 39557* The sum of two sigma-measurable functions is measurable. Proposition 121E (b) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐶) → 𝐷 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → (𝑥𝐶𝐷) ∈ (SMblFn‘𝑆))    &   (𝜑𝑅 ∈ ℝ)    &   𝐾 = (𝑝 ∈ ℚ ↦ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅})       (𝜑 → {𝑥 ∈ (𝐴𝐶) ∣ (𝐵 + 𝐷) < 𝑅} ∈ (𝑆t (𝐴𝐶)))
 
Theoremsmfadd 39558* The sum of two sigma-measurable functions is measurable. Proposition 121E (b) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐶) → 𝐷 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → (𝑥𝐶𝐷) ∈ (SMblFn‘𝑆))       (𝜑 → (𝑥 ∈ (𝐴𝐶) ↦ (𝐵 + 𝐷)) ∈ (SMblFn‘𝑆))
 
Theoremdecsmflem 39559* A non-increasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴𝑅 < (𝐹𝑥)}    &   𝐶 = sup(𝑌, ℝ*, < )    &   𝐷 = (-∞(,)𝐶)    &   𝐸 = (-∞(,]𝐶)       (𝜑 → ∃𝑏𝐵 𝑌 = (𝑏𝐴))
 
Theoremdecsmf 39560* A real valued, non-increasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       (𝜑𝐹 ∈ (SMblFn‘𝐵))
 
Theoremsmfpreimagtf 39561* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷𝐴 < (𝐹𝑥)} ∈ (𝑆t 𝐷))
 
Theoremissmfgelem 39562* The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-closed intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iv) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   (𝜑 → ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))
 
Theoremissmfge 39563* The predicate "𝐹 is b measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-closed intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iv) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))))
 
Theoremsmflimlem1 39564* Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that (𝐷𝐼) is in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})    &   𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))    &   𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)    &   ((𝜑𝑟 ∈ ran 𝑃) → (𝐶𝑟) ∈ 𝑟)       (𝜑 → (𝐷𝐼) ∈ (𝑆t 𝐷))
 
Theoremsmflimlem2 39565* Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves one-side of the double inclusion for the proof that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})    &   𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))    &   𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)    &   ((𝜑𝑟 ∈ ran 𝑃) → (𝐶𝑟) ∈ 𝑟)       (𝜑 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ⊆ (𝐷𝐼))
 
Theoremsmflimlem3 39566* The limit of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   (𝜑𝐴 ∈ ℝ)    &   𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})    &   𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))    &   𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)    &   ((𝜑𝑦 ∈ ran 𝑃) → (𝐶𝑦) ∈ 𝑦)    &   (𝜑𝑋 ∈ (𝐷𝐼))    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑌 ∈ ℝ+)    &   (𝜑 → (1 / 𝐾) < 𝑌)       (𝜑 → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌)))
 
Theoremsmflimlem4 39567* Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves one-side of the double inclusion for the proof that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})    &   𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))    &   𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)    &   ((𝜑𝑟 ∈ ran 𝑃) → (𝐶𝑟) ∈ 𝑟)       (𝜑 → (𝐷𝐼) ⊆ {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴})
 
Theoremsmflimlem5 39568* Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})    &   𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))    &   𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)    &   ((𝜑𝑟 ∈ ran 𝑃) → (𝐶𝑟) ∈ 𝑟)       (𝜑 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))
 
Theoremsmflimlem6 39569* Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})       (𝜑 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))
 
Theoremsmflim 39570* The limit of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . Notice that every function in the sequence can have a different (partial) domain, and the domain of convergence can be decidedly irregular (Remark 121G of [Fremlin1] p. 39 ). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑚𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))       (𝜑𝐺 ∈ (SMblFn‘𝑆))
 
Theoremnsssmfmbflem 39571* The sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals) are not a subset of the measurable functions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑆 = dom vol    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑 → ¬ 𝑋𝑆)    &   𝐹 = (𝑥𝑋 ↦ 0)       (𝜑 → ∃𝑓(𝑓 ∈ (SMblFn‘𝑆) ∧ ¬ 𝑓 ∈ MblFn))
 
Theoremnsssmfmbf 39572 The sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals) are not a subset of the measurable functions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑆 = dom vol        ¬ (SMblFn‘𝑆) ⊆ MblFn
 
Theoremsmfpimgtxr 39573* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ*)       (𝜑 → {𝑥𝐷𝐴 < (𝐹𝑥)} ∈ (𝑆t 𝐷))
 
Theoremsmfpimgtmpt 39574* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝐿 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐿 < 𝐵} ∈ (𝑆t 𝐴))
 
Theoremsmfpreimage 39575* Given a function measurable w.r.t. to a sigma-algebra, the preimage of a closed interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷𝐴 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))
 
Theoremmbfpsssmf 39576 Real valued, measurable functions are a proper subset of sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑆 = dom vol       (MblFn ∩ (ℝ ↑pm ℝ)) ⊊ (SMblFn‘𝑆)
 
Theoremsmfpimgtxrmpt 39577* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝐿 ∈ ℝ*)       (𝜑 → {𝑥𝐴𝐿 < 𝐵} ∈ (𝑆t 𝐴))
 
Theoremsmfpimioompt 39578* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝐿 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ*)       (𝜑 → {𝑥𝐴𝐵 ∈ (𝐿(,)𝑅)} ∈ (𝑆t 𝐴))
 
Theoremsmfpimioo 39579 Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐹 “ (𝐴(,)𝐵)) ∈ (𝑆t 𝐷))
 
Theoremsmfresal 39580* Given a sigma-measurable function, the subsets of whose preimage is in the sigma-algebra induced by the function's domain, form a sigma-algebra. First part of the proof of Proposition 121E (f) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (𝐹𝑒) ∈ (𝑆t 𝐷)}       (𝜑𝑇 ∈ SAlg)
 
Theoremsmfrec 39581* The reciprocal of a sigma-measurable functions is sigma-measurable. First part of Proposition 121E (e) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   𝐶 = {𝑥𝐴𝐵 ≠ 0}       (𝜑 → (𝑥𝐶 ↦ (1 / 𝐵)) ∈ (SMblFn‘𝑆))
 
Theoremsmfres 39582 The restriction of sigma-measurable function is sigma-measurable. Proposition 121E (h) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   (𝜑𝐴𝑉)       (𝜑 → (𝐹𝐴) ∈ (SMblFn‘𝑆))
 
Theoremsmfmullem1 39583 The multiplication of two sigma-measurable functions is measurable: this is the step (i) of the proof of Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝑉 ∈ ℝ)    &   (𝜑 → (𝑈 · 𝑉) < 𝐴)    &   𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉))))    &   𝑌 = if(1 ≤ 𝑋, 1, 𝑋)    &   (𝜑𝑃 ∈ ((𝑈𝑌)(,)𝑈))    &   (𝜑𝑅 ∈ (𝑈(,)(𝑈 + 𝑌)))    &   (𝜑𝑆 ∈ ((𝑉𝑌)(,)𝑉))    &   (𝜑𝑍 ∈ (𝑉(,)(𝑉 + 𝑌)))    &   (𝜑𝐻 ∈ (𝑃(,)𝑅))    &   (𝜑𝐼 ∈ (𝑆(,)𝑍))       (𝜑 → (𝐻 · 𝐼) < 𝐴)
 
Theoremsmfmullem2 39584* The multiplication of two sigma-measurable functions is measurable: this is the step (i) of the proof of Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   𝐾 = {𝑞 ∈ (ℚ ↑𝑚 (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴}    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝑉 ∈ ℝ)    &   (𝜑 → (𝑈 · 𝑉) < 𝐴)    &   (𝜑𝑃 ∈ ℚ)    &   (𝜑𝑅 ∈ ℚ)    &   (𝜑𝑆 ∈ ℚ)    &   (𝜑𝑍 ∈ ℚ)    &   (𝜑𝑃 ∈ ((𝑈𝑌)(,)𝑈))    &   (𝜑𝑅 ∈ (𝑈(,)(𝑈 + 𝑌)))    &   (𝜑𝑆 ∈ ((𝑉𝑌)(,)𝑉))    &   (𝜑𝑍 ∈ (𝑉(,)(𝑉 + 𝑌)))    &   𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉))))    &   𝑌 = if(1 ≤ 𝑋, 1, 𝑋)       (𝜑 → ∃𝑞𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3))))
 
Theoremsmfmullem3 39585* The multiplication of two sigma-measurable functions is measurable: this is the step (i) of the proof of Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑅 ∈ ℝ)    &   𝐾 = {𝑞 ∈ (ℚ ↑𝑚 (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑅}    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝑉 ∈ ℝ)    &   (𝜑 → (𝑈 · 𝑉) < 𝑅)    &   𝑋 = ((𝑅 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉))))    &   𝑌 = if(1 ≤ 𝑋, 1, 𝑋)       (𝜑 → ∃𝑞𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3))))
 
Theoremsmfmullem4 39586* The multiplication of two sigma-measurable functions is measurable. Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐶) → 𝐷 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → (𝑥𝐶𝐷) ∈ (SMblFn‘𝑆))    &   (𝜑𝑅 ∈ ℝ)    &   𝐾 = {𝑞 ∈ (ℚ ↑𝑚 (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑅}    &   𝐸 = (𝑞𝐾 ↦ {𝑥 ∈ (𝐴𝐶) ∣ (𝐵 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝐷 ∈ ((𝑞‘2)(,)(𝑞‘3)))})       (𝜑 → {𝑥 ∈ (𝐴𝐶) ∣ (𝐵 · 𝐷) < 𝑅} ∈ (𝑆t (𝐴𝐶)))
 
Theoremsmfmul 39587* The multiplication of two sigma-measurable functions is measurable. Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐶) → 𝐷 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → (𝑥𝐶𝐷) ∈ (SMblFn‘𝑆))       (𝜑 → (𝑥 ∈ (𝐴𝐶) ↦ (𝐵 · 𝐷)) ∈ (SMblFn‘𝑆))
 
Theoremsmfmulc1 39588* A sigma-measurable function multiplied by a constant, is sigma-measurable. Proposition 121E (c) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))       (𝜑 → (𝑥𝐴 ↦ (𝐶 · 𝐵)) ∈ (SMblFn‘𝑆))
 
Theoremsmfdiv 39589* The fraction of two sigma-measurable functions is measurable. Proposition 121E (e) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑𝐶𝑊)    &   ((𝜑𝑥𝐶) → 𝐷 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → (𝑥𝐶𝐷) ∈ (SMblFn‘𝑆))    &   𝐸 = {𝑥𝐶𝐷 ≠ 0}       (𝜑 → (𝑥 ∈ (𝐴𝐸) ↦ (𝐵 / 𝐷)) ∈ (SMblFn‘𝑆))
 
Theoremsmfpimbor1lem1 39590* Every open set belongs to 𝑇. This is the second step in the proof of Proposition 121E (f) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   𝐽 = (topGen‘ran (,))    &   (𝜑𝐺𝐽)    &   𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (𝐹𝑒) ∈ (𝑆t 𝐷)}       (𝜑𝐺𝑇)
 
Theoremsmfpimbor1lem2 39591* Given D sigma-measurable function, the preimage of D Borel set belongs to the subspace sigma-algebra induced by the domain of the function. Proposition 121E (f) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝐸𝐵)    &   𝑃 = (𝐹𝐸)    &   𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (𝐹𝑒) ∈ (𝑆t 𝐷)}       (𝜑𝑃 ∈ (𝑆t 𝐷))
 
Theoremsmfpimbor1 39592 Given a sigma-measurable function, the preimage of a Borel set belongs to the subspace sigma-algebra induced by the domain of the function. Proposition 121E (f) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝐸𝐵)    &   𝑃 = (𝐹𝐸)       (𝜑𝑃 ∈ (𝑆t 𝐷))
 
Theoremsmf2id 39593* Twice the identity function is Borel sigma-measurable (just an example, to test previous general theorems). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → (𝑥𝐴 ↦ (2 · 𝑥)) ∈ (SMblFn‘𝐵))
 
Theoremsmfco 39594 The composition of a Borel sigma-measurable function with a sigma-measurable function, is sigma-measurable. Proposition 121E (g) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝐻 ∈ (SMblFn‘𝐵))       (𝜑 → (𝐻𝐹) ∈ (SMblFn‘𝑆))
 
20.32  Mathbox for Saveliy Skresanov
 
20.32.1  Ceva's theorem
 
Theoremsigarval 39595* Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵)))
 
Theoremsigarim 39596* Signed area takes value in reals. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) ∈ ℝ)
 
Theoremsigarac 39597* Signed area is anticommutative. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = -(𝐵𝐺𝐴))
 
Theoremsigaraf 39598* Signed area is additive by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶)𝐺𝐵) = ((𝐴𝐺𝐵) + (𝐶𝐺𝐵)))
 
Theoremsigarmf 39599* Signed area is additive (with respect to subtraction) by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐶)𝐺𝐵) = ((𝐴𝐺𝐵) − (𝐶𝐺𝐵)))
 
Theoremsigaras 39600* Signed area is additive by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴𝐺(𝐵 + 𝐶)) = ((𝐴𝐺𝐵) + (𝐴𝐺𝐶)))
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