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Theorem List for Metamath Proof Explorer - 41301-41400   *Has distinct variable group(s)
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Statement

Theoremsmflimlem2 41301* 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 41302* 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 41303* 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 41304* 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 41305* 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 41306* 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 41307* 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 41308 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 41309* 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 41310* 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 41311* 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 41312 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 41313* 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 41314* 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 41315 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 41316* 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 41317* 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 41318 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 41319 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 41320* 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 41321* 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 41322* 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 41323* 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 41324* 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 41325* 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 41326* 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 41327* 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)    &   𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (𝐹𝑒) ∈ (𝑆t 𝐷)}       (𝜑𝑃 ∈ (𝑆t 𝐷))

Theoremsmfpimbor1 41328 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 41329* 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 41330 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))

Theoremsmfneg 41331* The negative of a sigma-measurable function is measurable. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))       (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ (SMblFn‘𝑆))

Theoremsmffmpt 41332* A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))       (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)

Theoremsmflim2 41333* The limit of a sequence 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 ). TODO this has less distinct variable restrictions than smflim and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-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))

Theoremsmfpimcclem 41334* Lemma for smfpimcc 41335 given the choice function 𝐶. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝜑    &   𝑍𝑉    &   (𝜑𝑆𝑊)    &   ((𝜑𝑦 ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})) → (𝐶𝑦) ∈ 𝑦)    &   𝐻 = (𝑛𝑍 ↦ (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))       (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))

Theoremsmfpimcc 41335* Given a countable set of sigma-measurable functions, and a Borel set 𝐴 there exists a choice function that, for each measurable function, chooses a measurable set that, when intersected with the function's domain, gives the preimage of 𝐴. This is a generalization of the observation at the beginning of the proof of Proposition 121F of [Fremlin1] p. 39 . The statement would also be provable for uncountable sets, but in most cases it will suffice to consider the countable case, and only the axiom of countable choice will be needed. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝐴𝐵)       (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))

Theoremissmfle2d 41336* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → (𝐹 “ (-∞(,]𝑎)) ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremsmflimmpt 41337* The limit of a sequence 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 ). 𝐴 can contain 𝑚 as a free variable, in other words it can be thought as an indexed collection 𝐴(𝑚). 𝐵 can be thought as a collection with two indexes 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑚𝜑    &   𝑥𝜑    &   𝑛𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑚𝑍) → 𝐴𝑉)    &   ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑊)    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍𝐵)))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfsuplem1 41338* The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 ((𝐹𝑛)‘𝑥) ≤ 𝑦}    &   𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐻:𝑍𝑆)    &   ((𝜑𝑛𝑍) → ((𝐹𝑛) “ (-∞(,]𝐴)) = ((𝐻𝑛) ∩ dom (𝐹𝑛)))       (𝜑 → (𝐺 “ (-∞(,]𝐴)) ∈ (𝑆t 𝐷))

Theoremsmfsuplem2 41339* The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 ((𝐹𝑛)‘𝑥) ≤ 𝑦}    &   𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐺 “ (-∞(,]𝐴)) ∈ (𝑆t 𝐷))

Theoremsmfsuplem3 41340* The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 ((𝐹𝑛)‘𝑥) ≤ 𝑦}    &   𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfsup 41341* The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 ((𝐹𝑛)‘𝑥) ≤ 𝑦}    &   𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfsupmpt 41342* The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝜑    &   𝑥𝜑    &   𝑦𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑛𝑍𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦}    &   𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍𝐵), ℝ, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfsupxr 41343* The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ sup(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ*, < ) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ*, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfinflem 41344* The infimum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (c) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)}    &   𝐺 = (𝑥𝐷 ↦ inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfinf 41345* The infimum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (c) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)}    &   𝐺 = (𝑥𝐷 ↦ inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfinfmpt 41346* The infimum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (c) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝜑    &   𝑥𝜑    &   𝑦𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑛𝑍𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵}    &   𝐺 = (𝑥𝐷 ↦ inf(ran (𝑛𝑍𝐵), ℝ, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmflimsuplem1 41347* If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)    &   𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))    &   (𝜑𝐾𝑍)       (𝜑 → dom (𝐻𝐾) ⊆ dom (𝐹𝐾))

Theoremsmflimsuplem2 41348* The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑚𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))    &   (𝜑𝑛𝑍)    &   (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)    &   (𝜑𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))       (𝜑𝑋 ∈ dom (𝐻𝑛))

Theoremsmflimsuplem3 41349* The limit of the (𝐻𝑛) functions is sigma-measurable. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))       (𝜑 → (𝑥 ∈ {𝑥 𝑘𝑍 𝑛 ∈ (ℤ𝑘)dom (𝐻𝑛) ∣ (𝑛𝑍 ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛𝑍 ↦ ((𝐻𝑛)‘𝑥)))) ∈ (SMblFn‘𝑆))

Theoremsmflimsuplem4 41350* If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))    &   (𝜑𝑁𝑍)    &   (𝜑𝑥 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛))    &   (𝜑 → (𝑛𝑍 ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ )       (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ)

Theoremsmflimsuplem5 41351* 𝐻 converges to the superior limit of 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝜑    &   𝑚𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))    &   (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)    &   (𝜑𝑁𝑍)    &   (𝜑𝑋 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚))       (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑋))))

Theoremsmflimsuplem6 41352* The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝜑    &   𝑚𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))    &   (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)    &   (𝜑𝑁𝑍)    &   (𝜑𝑋 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚))       (𝜑 → (𝑛𝑍 ↦ ((𝐻𝑛)‘𝑋)) ∈ dom ⇝ )

Theoremsmflimsuplem7 41353* The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}    &   𝐸 = (𝑘𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑘)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑘𝑍 ↦ (𝑥 ∈ (𝐸𝑘) ↦ sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))       (𝜑𝐷 = {𝑥 𝑛𝑍 𝑘 ∈ (ℤ𝑛)dom (𝐻𝑘) ∣ (𝑘𝑍 ↦ ((𝐻𝑘)‘𝑥)) ∈ dom ⇝ })

Theoremsmflimsuplem8 41354* The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   𝐸 = (𝑘𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑘)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑘𝑍 ↦ (𝑥 ∈ (𝐸𝑘) ↦ sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmflimsup 41355* The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑚𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmflimsupmpt 41356* The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . 𝐴 can contain 𝑚 as a free variable, in other words it can be thought of as an indexed collection 𝐴(𝑚). 𝐵 can be thought of as a collection with two indexes 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑚𝜑    &   𝑥𝜑    &   𝑛𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑊)    &   ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍𝐵)))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfliminflem 41357* The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ (lim inf‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfliminf 41358* The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑚𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ (lim inf‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfliminfmpt 41359* The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of [Fremlin1] p. 39 . 𝐴 can contain 𝑚 as a free variable, in other words it can be thought of as an indexed collection 𝐴(𝑚). 𝐵 can be thought of as a collection with two indexes 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑚𝜑    &   𝑥𝜑    &   𝑛𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ (lim inf‘(𝑚𝑍𝐵)))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

20.33  Mathbox for Saveliy Skresanov

20.33.1  Ceva's theorem

Theoremsigarval 41360* 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 41361* 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 41362* 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 41363* 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 41364* 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 41365* 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)))

Theoremsigarms 41366* Signed area is additive (with respect to subtraction) 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(𝐵𝐶)) = ((𝐴𝐺𝐵) − (𝐴𝐺𝐶)))

Theoremsigarls 41367* Signed area is linear by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (𝐴𝐺(𝐵 · 𝐶)) = ((𝐴𝐺𝐵) · 𝐶))

Theoremsigarid 41368* Signed area of a flat parallelogram is zero. (Contributed by Saveliy Skresanov, 20-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))       (𝐴 ∈ ℂ → (𝐴𝐺𝐴) = 0)

Theoremsigarexp 41369* Expand the signed area formula by linearity. (Contributed by Saveliy Skresanov, 20-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐶)𝐺(𝐵𝐶)) = (((𝐴𝐺𝐵) − (𝐴𝐺𝐶)) − (𝐶𝐺𝐵)))

Theoremsigarperm 41370* Signed area (𝐴𝐶)𝐺(𝐵𝐶) acts as a double area of a triangle 𝐴𝐵𝐶. Here we prove that cyclically permuting the vertices doesn't change the area. (Contributed by Saveliy Skresanov, 20-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐶)𝐺(𝐵𝐶)) = ((𝐵𝐴)𝐺(𝐶𝐴)))

Theoremsigardiv 41371* If signed area between vectors 𝐵𝐴 and 𝐶𝐴 is zero, then those vectors lie on the same line. (Contributed by Saveliy Skresanov, 22-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))    &   (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))    &   (𝜑 → ¬ 𝐶 = 𝐴)    &   (𝜑 → ((𝐵𝐴)𝐺(𝐶𝐴)) = 0)       (𝜑 → ((𝐵𝐴) / (𝐶𝐴)) ∈ ℝ)

Theoremsigarimcd 41372* Signed area takes value in complex numbers. Deduction version. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))    &   (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))       (𝜑 → (𝐴𝐺𝐵) ∈ ℂ)

Theoremsigariz 41373* If signed area is zero, the signed area with swapped arguments is also zero. Deduction version. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))    &   (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))    &   (𝜑 → (𝐴𝐺𝐵) = 0)       (𝜑 → (𝐵𝐺𝐴) = 0)

Theoremsigarcol 41374* Given three points 𝐴, 𝐵 and 𝐶 such that ¬ 𝐴 = 𝐵, the point 𝐶 lies on the line going through 𝐴 and 𝐵 iff the corresponding signed area is zero. That justifies the usage of signed area as a collinearity indicator. (Contributed by Saveliy Skresanov, 22-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))    &   (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))    &   (𝜑 → ¬ 𝐴 = 𝐵)       (𝜑 → (((𝐴𝐶)𝐺(𝐵𝐶)) = 0 ↔ ∃𝑡 ∈ ℝ 𝐶 = (𝐵 + (𝑡 · (𝐴𝐵)))))

Theoremsharhght 41375* Let 𝐴𝐵𝐶 be a triangle, and let 𝐷 lie on the line 𝐴𝐵. Then (doubled) areas of triangles 𝐴𝐷𝐶 and 𝐶𝐷𝐵 relate as lengths of corresponding bases 𝐴𝐷 and 𝐷𝐵. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))    &   (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))    &   (𝜑 → (𝐷 ∈ ℂ ∧ ((𝐴𝐷)𝐺(𝐵𝐷)) = 0))       (𝜑 → (((𝐶𝐴)𝐺(𝐷𝐴)) · (𝐵𝐷)) = (((𝐶𝐵)𝐺(𝐷𝐵)) · (𝐴𝐷)))

Theoremsigaradd 41376* Subtracting (double) area of 𝐴𝐷𝐶 from 𝐴𝐵𝐶 yields the (double) area of 𝐷𝐵𝐶. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))    &   (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))    &   (𝜑 → (𝐷 ∈ ℂ ∧ ((𝐴𝐷)𝐺(𝐵𝐷)) = 0))       (𝜑 → (((𝐵𝐶)𝐺(𝐴𝐶)) − ((𝐷𝐶)𝐺(𝐴𝐶))) = ((𝐵𝐶)𝐺(𝐷𝐶)))

Theoremcevathlem1 41377 Ceva's theorem first lemma. Multiplies three identities and divides by the common factors. (Contributed by Saveliy Skresanov, 24-Sep-2017.)
(𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))    &   (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ))    &   (𝜑 → (𝐺 ∈ ℂ ∧ 𝐻 ∈ ℂ ∧ 𝐾 ∈ ℂ))    &   (𝜑 → (𝐴 ≠ 0 ∧ 𝐸 ≠ 0 ∧ 𝐶 ≠ 0))    &   (𝜑 → ((𝐴 · 𝐵) = (𝐶 · 𝐷) ∧ (𝐸 · 𝐹) = (𝐴 · 𝐺) ∧ (𝐶 · 𝐻) = (𝐸 · 𝐾)))       (𝜑 → ((𝐵 · 𝐹) · 𝐻) = ((𝐷 · 𝐺) · 𝐾))

Theoremcevathlem2 41378* Ceva's theorem second lemma. Relate (doubled) areas of triangles 𝐶𝐴𝑂 and 𝐴𝐵𝑂 with of segments 𝐵𝐷 and 𝐷𝐶. (Contributed by Saveliy Skresanov, 24-Sep-2017.)
𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))    &   (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))    &   (𝜑 → (𝐹 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))    &   (𝜑𝑂 ∈ ℂ)    &   (𝜑 → (((𝐴𝑂)𝐺(𝐷𝑂)) = 0 ∧ ((𝐵𝑂)𝐺(𝐸𝑂)) = 0 ∧ ((𝐶𝑂)𝐺(𝐹𝑂)) = 0))    &   (𝜑 → (((𝐴𝐹)𝐺(𝐵𝐹)) = 0 ∧ ((𝐵𝐷)𝐺(𝐶𝐷)) = 0 ∧ ((𝐶𝐸)𝐺(𝐴𝐸)) = 0))    &   (𝜑 → (((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0))       (𝜑 → (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐵𝐷)) = (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐷𝐶)))

Theoremcevath 41379* Ceva's theorem. Let 𝐴𝐵𝐶 be a triangle and let points 𝐹, 𝐷 and 𝐸 lie on sides 𝐴𝐵, 𝐵𝐶, 𝐶𝐴 correspondingly. Suppose that cevians 𝐴𝐷, 𝐵𝐸 and 𝐶𝐹 intersect at one point 𝑂. Then triangle's sides are partitioned into segments and their lengths satisfy a certain identity. Here we obtain a bit stronger version by using complex numbers themselves instead of their absolute values.

The proof goes by applying cevathlem2 41378 three times and then using cevathlem1 41377 to multiply obtained identities and prove the theorem.

In the theorem statement we are using function 𝐺 as a collinearity indicator. For justification of that use, see sigarcol 41374. This is Metamath 100 proof #61. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))    &   (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))    &   (𝜑 → (𝐹 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))    &   (𝜑𝑂 ∈ ℂ)    &   (𝜑 → (((𝐴𝑂)𝐺(𝐷𝑂)) = 0 ∧ ((𝐵𝑂)𝐺(𝐸𝑂)) = 0 ∧ ((𝐶𝑂)𝐺(𝐹𝑂)) = 0))    &   (𝜑 → (((𝐴𝐹)𝐺(𝐵𝐹)) = 0 ∧ ((𝐵𝐷)𝐺(𝐶𝐷)) = 0 ∧ ((𝐶𝐸)𝐺(𝐴𝐸)) = 0))    &   (𝜑 → (((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0))       (𝜑 → (((𝐴𝐹) · (𝐶𝐸)) · (𝐵𝐷)) = (((𝐹𝐵) · (𝐸𝐴)) · (𝐷𝐶)))

20.34  Mathbox for Jarvin Udandy

TheoremhirstL-ax3 41380 The third axiom of a system called "L" but proven to be a theorem since set.mm uses a different third axiom. This is named hirst after Holly P. Hirst and Jeffry L. Hirst. Axiom A3 of [Mendelson] p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015.) (Proof modification is discouraged.)
((¬ 𝜑 → ¬ 𝜓) → ((¬ 𝜑𝜓) → 𝜑))

Theoremax3h 41381 Recovery of ax-3 8 from hirstL-ax3 41380. (Contributed by Jarvin Udandy, 3-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))

Theoremaibandbiaiffaiffb 41382 A closed form showing (a implies b and b implies a) same-as (a same-as b). (Contributed by Jarvin Udandy, 3-Sep-2016.)
(((𝜑𝜓) ∧ (𝜓𝜑)) ↔ (𝜑𝜓))

Theoremaibandbiaiaiffb 41383 A closed form showing (a implies b and b implies a) implies (a same-as b). (Contributed by Jarvin Udandy, 3-Sep-2016.)
(((𝜑𝜓) ∧ (𝜓𝜑)) → (𝜑𝜓))

Theoremnotatnand 41384 Do not use. Use intnanr instead. Given not a, there exists a proof for not (a and b). (Contributed by Jarvin Udandy, 31-Aug-2016.)
¬ 𝜑        ¬ (𝜑𝜓)

Theoremaistia 41385 Given a is equivalent to , there exists a proof for a. (Contributed by Jarvin Udandy, 30-Aug-2016.)
(𝜑 ↔ ⊤)       𝜑

Theoremaisfina 41386 Given a is equivalent to , there exists a proof for not a. (Contributed by Jarvin Udandy, 30-Aug-2016.)
(𝜑 ↔ ⊥)        ¬ 𝜑

Theorembothtbothsame 41387 Given both a, b are equivalent to , there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)
(𝜑 ↔ ⊤)    &   (𝜓 ↔ ⊤)       (𝜑𝜓)

Theorembothfbothsame 41388 Given both a, b are equivalent to , there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊥)       (𝜑𝜓)

Theoremaiffbbtat 41389 Given a is equivalent to b, b is equivalent to there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)
(𝜑𝜓)    &   (𝜓 ↔ ⊤)       (𝜑 ↔ ⊤)

Theoremaisbbisfaisf 41390 Given a is equivalent to b, b is equivalent to there exists a proof for a is equivalent to F. (Contributed by Jarvin Udandy, 30-Aug-2016.)
(𝜑𝜓)    &   (𝜓 ↔ ⊥)       (𝜑 ↔ ⊥)

Theoremaxorbtnotaiffb 41391 Given a is exclusive to b, there exists a proof for (not (a if-and-only-if b)); df-xor 1505 is a closed form of this. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜓)        ¬ (𝜑𝜓)

Theoremaiffnbandciffatnotciffb 41392 Given a is equivalent to (not b), c is equivalent to a, there exists a proof for ( not ( c iff b ) ). (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜒𝜑)        ¬ (𝜒𝜓)

Theoremaxorbciffatcxorb 41393 Given a is equivalent to (not b), c is equivalent to a. there exists a proof for ( c xor b ) . (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜓)    &   (𝜒𝜑)       (𝜒𝜓)

Theoremaibnbna 41394 Given a implies b, (not b), there exists a proof for (not a). (Contributed by Jarvin Udandy, 1-Sep-2016.)
(𝜑𝜓)    &    ¬ 𝜓        ¬ 𝜑

Theoremaibnbaif 41395 Given a implies b, not b, there exists a proof for a is F. (Contributed by Jarvin Udandy, 1-Sep-2016.)
(𝜑𝜓)    &    ¬ 𝜓       (𝜑 ↔ ⊥)

Theoremaiffbtbat 41396 Given a is equivalent to b, T. is equivalent to b. there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)
(𝜑𝜓)    &   (⊤ ↔ 𝜓)       (𝜑 ↔ ⊤)

Theoremastbstanbst 41397 Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for a and b is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)
(𝜑 ↔ ⊤)    &   (𝜓 ↔ ⊤)       ((𝜑𝜓) ↔ ⊤)

Theoremaistbistaandb 41398 Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for (a and b). (Contributed by Jarvin Udandy, 9-Sep-2016.)
(𝜑 ↔ ⊤)    &   (𝜓 ↔ ⊤)       (𝜑𝜓)

Theoremaisbnaxb 41399 Given a is equivalent to b, there exists a proof for (not (a xor b)). (Contributed by Jarvin Udandy, 28-Aug-2016.)
(𝜑𝜓)        ¬ (𝜑𝜓)

Theorematbiffatnnb 41400 If a implies b, then a implies not not b. (Contributed by Jarvin Udandy, 28-Aug-2016.)
((𝜑𝜓) → (𝜑 → ¬ ¬ 𝜓))

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