| Metamath
Proof Explorer Theorem List (p. 469 of 494) | < Previous Next > | |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | (1-30937) |
(30938-32460) |
(32461-49324) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | smfpimioompt 46801* | 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.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐿 ∈ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} ∈ (𝑆 ↾t 𝐴)) | ||
| Theorem | smfpimioo 46802 | 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.) |
| ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) ∈ (𝑆 ↾t 𝐷)) | ||
| Theorem | smfresal 46803* | Given a sigma-measurable function, the subsets of ℝ 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.) |
| ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)} ⇒ ⊢ (𝜑 → 𝑇 ∈ SAlg) | ||
| Theorem | smfrec 46804* | 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.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 0} ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (1 / 𝐵)) ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfres 46805 | The restriction of sigma-measurable function is sigma-measurable. Proposition 121E (h) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
| ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfmullem1 46806 | 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.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝑉 ∈ ℝ) & ⊢ (𝜑 → (𝑈 · 𝑉) < 𝐴) & ⊢ 𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))) & ⊢ 𝑌 = if(1 ≤ 𝑋, 1, 𝑋) & ⊢ (𝜑 → 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈)) & ⊢ (𝜑 → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌))) & ⊢ (𝜑 → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉)) & ⊢ (𝜑 → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌))) & ⊢ (𝜑 → 𝐻 ∈ (𝑃(,)𝑅)) & ⊢ (𝜑 → 𝐼 ∈ (𝑆(,)𝑍)) ⇒ ⊢ (𝜑 → (𝐻 · 𝐼) < 𝐴) | ||
| Theorem | smfmullem2 46807* | 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.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝐾 = {𝑞 ∈ (ℚ ↑m (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴} & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝑉 ∈ ℝ) & ⊢ (𝜑 → (𝑈 · 𝑉) < 𝐴) & ⊢ (𝜑 → 𝑃 ∈ ℚ) & ⊢ (𝜑 → 𝑅 ∈ ℚ) & ⊢ (𝜑 → 𝑆 ∈ ℚ) & ⊢ (𝜑 → 𝑍 ∈ ℚ) & ⊢ (𝜑 → 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈)) & ⊢ (𝜑 → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌))) & ⊢ (𝜑 → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉)) & ⊢ (𝜑 → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌))) & ⊢ 𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))) & ⊢ 𝑌 = if(1 ≤ 𝑋, 1, 𝑋) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3)))) | ||
| Theorem | smfmullem3 46808* | 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.) |
| ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ 𝐾 = {𝑞 ∈ (ℚ ↑m (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑅} & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝑉 ∈ ℝ) & ⊢ (𝜑 → (𝑈 · 𝑉) < 𝑅) & ⊢ 𝑋 = ((𝑅 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))) & ⊢ 𝑌 = if(1 ≤ 𝑋, 1, 𝑋) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3)))) | ||
| Theorem | smfmullem4 46809* | The multiplication of two sigma-measurable functions is measurable. Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ 𝐾 = {𝑞 ∈ (ℚ ↑m (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑅} & ⊢ 𝐸 = (𝑞 ∈ 𝐾 ↦ {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝐷 ∈ ((𝑞‘2)(,)(𝑞‘3)))}) ⇒ ⊢ (𝜑 → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 · 𝐷) < 𝑅} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) | ||
| Theorem | smfmul 46810* | The multiplication of two sigma-measurable functions is measurable. Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 · 𝐷)) ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfmulc1 46811* | 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.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfdiv 46812* | The fraction of two sigma-measurable functions is measurable. Proposition 121E (e) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) & ⊢ 𝐸 = {𝑥 ∈ 𝐶 ∣ 𝐷 ≠ 0} ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐸) ↦ (𝐵 / 𝐷)) ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfpimbor1lem1 46813* | 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.) |
| ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐺 ∈ 𝐽) & ⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)} ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝑇) | ||
| Theorem | smfpimbor1lem2 46814* | 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.) |
| ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ 𝑃 = (◡𝐹 “ 𝐸) & ⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)} ⇒ ⊢ (𝜑 → 𝑃 ∈ (𝑆 ↾t 𝐷)) | ||
| Theorem | smfpimbor1 46815 | 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.) |
| ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ 𝑃 = (◡𝐹 “ 𝐸) ⇒ ⊢ (𝜑 → 𝑃 ∈ (𝑆 ↾t 𝐷)) | ||
| Theorem | smf2id 46816* | 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) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (2 · 𝑥)) ∈ (SMblFn‘𝐵)) | ||
| Theorem | smfco 46817 | 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.) |
| ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐻 ∈ (SMblFn‘𝐵)) ⇒ ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfneg 46818* | The negative of a sigma-measurable function is measurable. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ (SMblFn‘𝑆)) | ||
| Theorem | smffmptf 46819 | A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 5-Jan-2025.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) | ||
| Theorem | smffmpt 46820* | A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) | ||
| Theorem | smflim2 46821* | 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 fewer distinct variable conditions than smflim 46792 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfpimcclem 46822* | Lemma for smfpimcc 46823 given the choice function 𝐶. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ 𝑍 ∈ 𝑉 & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) → (𝐶‘𝑦) ∈ 𝑦) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) ⇒ ⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) | ||
| Theorem | smfpimcc 46823* | Given a countable set of sigma-measurable functions, and a Borel set 𝐴 there exists a choice function ℎ 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.) |
| ⊢ Ⅎ𝑛𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) | ||
| Theorem | issmfle2d 46824* | A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,]𝑎)) ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smflimmpt 46825* | 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 indices 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 ∣ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfsuplem1 46826* | 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.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐻:𝑍⟶𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛))) ⇒ ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷)) | ||
| Theorem | smfsuplem2 46827* | 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.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷)) | ||
| Theorem | smfsuplem3 46828* | 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.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfsup 46829* | 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.) |
| ⊢ Ⅎ𝑛𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfsupmpt 46830* | 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.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfsupxr 46831* | 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.) |
| ⊢ Ⅎ𝑛𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfinflem 46832* | 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.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfinf 46833* | 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.) |
| ⊢ Ⅎ𝑛𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfinfmpt 46834* | 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.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smflimsuplem1 46835* | If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → 𝐾 ∈ 𝑍) ⇒ ⊢ (𝜑 → dom (𝐻‘𝐾) ⊆ dom (𝐹‘𝐾)) | ||
| Theorem | smflimsuplem2 46836* | 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.) |
| ⊢ Ⅎ𝑚𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → 𝑛 ∈ 𝑍) & ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → 𝑋 ∈ dom (𝐻‘𝑛)) | ||
| Theorem | smflimsuplem3 46837* | The limit of the (𝐻‘𝑛) functions is sigma-measurable. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) ⇒ ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)))) ∈ (SMblFn‘𝑆)) | ||
| Theorem | smflimsuplem4 46838* | If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝑥 ∈ ∩ 𝑛 ∈ (ℤ≥‘𝑁)dom (𝐻‘𝑛)) & ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) | ||
| Theorem | smflimsuplem5 46839* | 𝐻 converges to the superior limit of 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑁)dom (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋)))) | ||
| Theorem | smflimsuplem6 46840* | 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.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑁)dom (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ dom ⇝ ) | ||
| Theorem | smflimsuplem7 46841* | 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.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐸 = (𝑘 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑘 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑘) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) ⇒ ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) | ||
| Theorem | smflimsuplem8 46842* | 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.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ 𝐸 = (𝑘 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑘 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑘) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smflimsup 46843* | 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.) |
| ⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smflimsupmpt 46844* | 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 indices 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfliminflem 46845* | 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.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfliminf 46846* | 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.) |
| ⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | smfliminfmpt 46847* | 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 indices 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| Theorem | adddmmbl 46848 | If two functions have domains in the sigma-algebra, the domain of their addition also belongs to the sigma-algebra. This is the first statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their addition. (Contributed by Glauco Siliprandi, 30-Dec-2024.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 + 𝐷)) ∈ 𝑆) | ||
| Theorem | adddmmbl2 46849 | If two functions have domains in the sigma-algebra, the domain of their addition also belongs to the sigma-algebra. This is the first statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their addition. (Contributed by Glauco Siliprandi, 30-Dec-2024.) |
| ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐺 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → dom 𝐹 ∈ 𝑆) & ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) & ⊢ 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) ⇒ ⊢ (𝜑 → dom 𝐻 ∈ 𝑆) | ||
| Theorem | muldmmbl 46850 | If two functions have domains in the sigma-algebra, the domain of their multiplication also belongs to the sigma-algebra. This is the second statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their multiplication. (Contributed by Glauco Siliprandi, 30-Dec-2024.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) ∈ 𝑆) | ||
| Theorem | muldmmbl2 46851 | If two functions have domains in the sigma-algebra, the domain of their multiplication also belongs to the sigma-algebra. This is the second statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their multiplication. (Contributed by Glauco Siliprandi, 30-Dec-2024.) |
| ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐺 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → dom 𝐹 ∈ 𝑆) & ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) & ⊢ 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) · (𝐺‘𝑥))) ⇒ ⊢ (𝜑 → dom 𝐻 ∈ 𝑆) | ||
| Theorem | smfdmmblpimne 46852* | If a measurable function w.r.t. to a sigma-algebra has domain in the sigma-algebra, the set of elements that are not mapped to a given real, is in the sigma-algebra (Contributed by Glauco Siliprandi, 5-Jan-2025.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶} ⇒ ⊢ (𝜑 → 𝐷 ∈ 𝑆) | ||
| Theorem | smfdivdmmbl 46853 | If a functions and a sigma-measurable function have domains in the sigma-algebra, the domain of the division of the two functions is in the sigma-algebra. This is the third statement of Proposition 121H of [Fremlin1] p. 39 . Note: While the theorem in the book assumes both functions are sigma-measurable, this assumption is unnecessary for the part concerning their division, for the function at the numerator (it is needed only for the function at the denominator). (Contributed by Glauco Siliprandi, 5-Jan-2025.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ (SMblFn‘𝑆)) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ 𝐷 ≠ 0} ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐸) ∈ 𝑆) | ||
| Theorem | smfpimne 46854* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of reals that are different from a value in the extended reals is in the subspace of sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 5-Jan-2025.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≠ 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
| Theorem | smfpimne2 46855* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of reals that are different from a value is in the subspace sigma-algebra induced by its domain. Notice that 𝐴 is not assumed to be an extended real. (Contributed by Glauco Siliprandi, 5-Jan-2025.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≠ 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
| Theorem | smfdivdmmbl2 46856 | If a functions and a sigma-measurable function have domains in the sigma-algebra, the domain of the division of the two functions is in the sigma-algebra. This is the third statement of Proposition 121H of [Fremlin1] p. 39 . Note: While the theorem in the book assumes both functions are sigma-measurable, this assumption is unnecessary for the part concerning their division, for the function at the numerator. It is required only for the function at the denominator. (Contributed by Glauco Siliprandi, 5-Jan-2025.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐺 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑉) & ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) & ⊢ 𝐷 = {𝑥 ∈ dom 𝐺 ∣ (𝐺‘𝑥) ≠ 0} & ⊢ 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ 𝐷) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥))) ⇒ ⊢ (𝜑 → dom 𝐻 ∈ 𝑆) | ||
| Theorem | fsupdm 46857* | The domain of the sup function is defined in Proposition 121F (b) of [Fremlin1], p. 38. Note that this definition of the sup function is quite general, as it does not require the original functions to be sigma-measurable, and it could be applied to uncountable sets of functions. The equality proved here is part of the proof of the fourth statement of Proposition 121H in [Fremlin1], p. 39. (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ*) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚})) ⇒ ⊢ (𝜑 → 𝐷 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) | ||
| Theorem | fsupdm2 46858* | The domain of the sup function is defined in Proposition 121F (b) of [Fremlin1], p. 38. Note that this definition of the sup function is quite general, as it does not require the original functions to be sigma-measurable, and it could be applied to uncountable sets of functions. The equality proved here is part of the proof of the fourth statement of Proposition 121H in [Fremlin1], p. 39. (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ*) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚})) ⇒ ⊢ (𝜑 → dom 𝐺 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) | ||
| Theorem | smfsupdmmbllem 46859* | If a countable set of sigma-measurable functions have domains in the sigma-algebra, then their supremum function has the domain in the sigma-algebra. This is the fourth statement of Proposition 121H of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚})) & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) | ||
| Theorem | smfsupdmmbl 46860* | If a countable set of sigma-measurable functions have domains in the sigma-algebra, then their supremum function has the domain in the sigma-algebra. This is the fourth statement of Proposition 121H of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) | ||
| Theorem | finfdm 46861* | The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 46833. Note that this definition of the inf function is quite general, as it does not require the original functions to be sigma-measurable, and it could be applied to uncountable sets of functions. The equality proved here is part of the proof of the fifth statement of Proposition 121H in [Fremlin1], p. 39. (Contributed by Glauco Siliprandi, 1-Feb-2025.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ*) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})) ⇒ ⊢ (𝜑 → 𝐷 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) | ||
| Theorem | finfdm2 46862* | The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 46833. Note that this definition of the inf function is quite general, as it does not require the original functions to be sigma-measurable, and it could be applied to uncountable sets of functions. The equality proved here is part of the proof of the fifth statement of Proposition 121H in [Fremlin1], p. 39. (Contributed by Glauco Siliprandi, 1-Feb-2025.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ*) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})) ⇒ ⊢ (𝜑 → dom 𝐺 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) | ||
| Theorem | smfinfdmmbllem 46863* | If a countable set of sigma-measurable functions have domains in the sigma-algebra, then their infimum function has the domain in the sigma-algebra. This is the fifth statement of Proposition 121H of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 1-Feb-2025.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})) ⇒ ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) | ||
| Theorem | smfinfdmmbl 46864* | If a countable set of sigma-measurable functions have domains in the sigma-algebra, then their infimum function has the domain in the sigma-algebra. This is the fifth statement of Proposition 121H of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 1-Feb-2025.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) | ||
| Theorem | sigarval 46865* | 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))) | ||
| Theorem | sigarim 46866* | 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) | ||
| Theorem | sigarac 46867* | 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)) | ||
| Theorem | sigaraf 46868* | 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))) | ||
| Theorem | sigarmf 46869* | 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) → ((𝐴 − 𝐶)𝐺𝐵) = ((𝐴𝐺𝐵) − (𝐶𝐺𝐵))) | ||
| Theorem | sigaras 46870* | 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))) | ||
| Theorem | sigarms 46871* | 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(𝐵 − 𝐶)) = ((𝐴𝐺𝐵) − (𝐴𝐺𝐶))) | ||
| Theorem | sigarls 46872* | 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)) | ||
| Theorem | sigarid 46873* | 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) | ||
| Theorem | sigarexp 46874* | 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) → ((𝐴 − 𝐶)𝐺(𝐵 − 𝐶)) = (((𝐴𝐺𝐵) − (𝐴𝐺𝐶)) − (𝐶𝐺𝐵))) | ||
| Theorem | sigarperm 46875* | Signed area (𝐴 − 𝐶)𝐺(𝐵 − 𝐶) 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) → ((𝐴 − 𝐶)𝐺(𝐵 − 𝐶)) = ((𝐵 − 𝐴)𝐺(𝐶 − 𝐴))) | ||
| Theorem | sigardiv 46876* | If signed area between vectors 𝐵 − 𝐴 and 𝐶 − 𝐴 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 → ((𝐵 − 𝐴)𝐺(𝐶 − 𝐴)) = 0) ⇒ ⊢ (𝜑 → ((𝐵 − 𝐴) / (𝐶 − 𝐴)) ∈ ℝ) | ||
| Theorem | sigarimcd 46877* | 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) | ||
| Theorem | sigariz 46878* | 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) | ||
| Theorem | sigarcol 46879* | 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 → (((𝐴 − 𝐶)𝐺(𝐵 − 𝐶)) = 0 ↔ ∃𝑡 ∈ ℝ 𝐶 = (𝐵 + (𝑡 · (𝐴 − 𝐵))))) | ||
| Theorem | sharhght 46880* | 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 ∧ ((𝐴 − 𝐷)𝐺(𝐵 − 𝐷)) = 0)) ⇒ ⊢ (𝜑 → (((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) · (𝐵 − 𝐷)) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · (𝐴 − 𝐷))) | ||
| Theorem | sigaradd 46881* | 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 ∧ ((𝐴 − 𝐷)𝐺(𝐵 − 𝐷)) = 0)) ⇒ ⊢ (𝜑 → (((𝐵 − 𝐶)𝐺(𝐴 − 𝐶)) − ((𝐷 − 𝐶)𝐺(𝐴 − 𝐶))) = ((𝐵 − 𝐶)𝐺(𝐷 − 𝐶))) | ||
| Theorem | cevathlem1 46882 | 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)) | ||
| Theorem | cevathlem2 46883* | 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)) & ⊢ (𝜑 → 𝑂 ∈ ℂ) & ⊢ (𝜑 → (((𝐴 − 𝑂)𝐺(𝐷 − 𝑂)) = 0 ∧ ((𝐵 − 𝑂)𝐺(𝐸 − 𝑂)) = 0 ∧ ((𝐶 − 𝑂)𝐺(𝐹 − 𝑂)) = 0)) & ⊢ (𝜑 → (((𝐴 − 𝐹)𝐺(𝐵 − 𝐹)) = 0 ∧ ((𝐵 − 𝐷)𝐺(𝐶 − 𝐷)) = 0 ∧ ((𝐶 − 𝐸)𝐺(𝐴 − 𝐸)) = 0)) & ⊢ (𝜑 → (((𝐴 − 𝑂)𝐺(𝐵 − 𝑂)) ≠ 0 ∧ ((𝐵 − 𝑂)𝐺(𝐶 − 𝑂)) ≠ 0 ∧ ((𝐶 − 𝑂)𝐺(𝐴 − 𝑂)) ≠ 0)) ⇒ ⊢ (𝜑 → (((𝐶 − 𝑂)𝐺(𝐴 − 𝑂)) · (𝐵 − 𝐷)) = (((𝐴 − 𝑂)𝐺(𝐵 − 𝑂)) · (𝐷 − 𝐶))) | ||
| Theorem | cevath 46884* |
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 46883 three times and then using cevathlem1 46882 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 46879. 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)) & ⊢ (𝜑 → 𝑂 ∈ ℂ) & ⊢ (𝜑 → (((𝐴 − 𝑂)𝐺(𝐷 − 𝑂)) = 0 ∧ ((𝐵 − 𝑂)𝐺(𝐸 − 𝑂)) = 0 ∧ ((𝐶 − 𝑂)𝐺(𝐹 − 𝑂)) = 0)) & ⊢ (𝜑 → (((𝐴 − 𝐹)𝐺(𝐵 − 𝐹)) = 0 ∧ ((𝐵 − 𝐷)𝐺(𝐶 − 𝐷)) = 0 ∧ ((𝐶 − 𝐸)𝐺(𝐴 − 𝐸)) = 0)) & ⊢ (𝜑 → (((𝐴 − 𝑂)𝐺(𝐵 − 𝑂)) ≠ 0 ∧ ((𝐵 − 𝑂)𝐺(𝐶 − 𝑂)) ≠ 0 ∧ ((𝐶 − 𝑂)𝐺(𝐴 − 𝑂)) ≠ 0)) ⇒ ⊢ (𝜑 → (((𝐴 − 𝐹) · (𝐶 − 𝐸)) · (𝐵 − 𝐷)) = (((𝐹 − 𝐵) · (𝐸 − 𝐴)) · (𝐷 − 𝐶))) | ||
| Theorem | simpcntrab 46885 | The center of a simple group is trivial or the group is abelian. (Contributed by SS, 3-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntr‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ SimpGrp) ⇒ ⊢ (𝜑 → (𝑍 = { 0 } ∨ 𝐺 ∈ Abel)) | ||
| Theorem | et-ltneverrefl 46886 | Less-than class is never reflexive. (Contributed by Ender Ting, 22-Nov-2024.) Prefer to specify theorem domain and then apply ltnri 11370. (New usage is discouraged.) |
| ⊢ ¬ 𝐴 < 𝐴 | ||
| Theorem | et-equeucl 46887 | Alternative proof that equality is left-Euclidean, using ax7 2015 directly instead of utility theorems; done for practice. (Contributed by Ender Ting, 21-Dec-2024.) |
| ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) | ||
| Theorem | et-sqrtnegnre 46888 | The square root of a negative number is not a real number. (Contributed by Ender Ting, 5-Jan-2025.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → ¬ (√‘𝐴) ∈ ℝ) | ||
| Theorem | ormklocald 46889* | If elements of a certain sequence are ordered with respect to a certain relation, then its consecutive elements satisfy that relation (so-called "local monotonicity"). (Contributed by Ender Ting, 30-Apr-2025.) |
| ⊢ (𝜑 → 𝑅 Or 𝑆) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^(𝑇 + 1))(𝐵‘𝑘) ∈ 𝑆) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ (1..^(𝑇 + 1))(𝑘 < 𝑡 → (𝐵‘𝑘)𝑅(𝐵‘𝑡))) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘)𝑅(𝐵‘(𝑘 + 1))) | ||
| Theorem | ormkglobd 46890* | If all adjacent elements of a certain sequence are ordered according to a relation which is a total order on S, then any element is so related to anything to right of it (so-called "global monotonicity"). Deduction form. (Contributed by Ender Ting, 30-Apr-2025.) |
| ⊢ (𝜑 → 𝑅 Or 𝑆) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^(𝑇 + 1))(𝐵‘𝑘) ∈ 𝑆) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘)𝑅(𝐵‘(𝑘 + 1))) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ (1..^(𝑇 + 1))(𝑘 < 𝑡 → (𝐵‘𝑘)𝑅(𝐵‘𝑡))) | ||
| Theorem | natlocalincr 46891* | Global monotonicity on half-open range implies local monotonicity. Inference form. (Contributed by Ender Ting, 22-Nov-2024.) |
| ⊢ ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ (1..^(𝑇 + 1))(𝑘 < 𝑡 → (𝐵‘𝑘) < (𝐵‘𝑡)) ⇒ ⊢ ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘) < (𝐵‘(𝑘 + 1)) | ||
| Theorem | natglobalincr 46892* | Local monotonicity on half-open integer range implies global monotonicity. Inference form. (Contributed by Ender Ting, 23-Nov-2024.) |
| ⊢ ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘) < (𝐵‘(𝑘 + 1)) & ⊢ 𝑇 ∈ ℤ ⇒ ⊢ ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ ((𝑘 + 1)...𝑇)(𝐵‘𝑘) < (𝐵‘𝑡) | ||
| Syntax | cupword 46893 | Extend class notation to include the set of strictly increasing sequences. |
| class UpWord 𝑆 | ||
| Definition | df-upword 46894* | Strictly increasing sequence is a sequence, adjacent elements of which increase. (Contributed by Ender Ting, 19-Nov-2024.) |
| ⊢ UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))} | ||
| Theorem | upwordnul 46895 | Empty set is an increasing sequence for every range. (Contributed by Ender Ting, 19-Nov-2024.) |
| ⊢ ∅ ∈ UpWord 𝑆 | ||
| Theorem | upwordisword 46896 | Any increasing sequence is a sequence. (Contributed by Ender Ting, 19-Nov-2024.) |
| ⊢ (𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ Word 𝑆) | ||
| Theorem | singoutnword 46897 | Singleton with character out of range 𝑉 is not a word for that range. (Contributed by Ender Ting, 21-Nov-2024.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (¬ 𝐴 ∈ 𝑉 → ¬ 〈“𝐴”〉 ∈ Word 𝑉) | ||
| Theorem | singoutnupword 46898 | Singleton with character out of range 𝑆 is not an increasing sequence for that range. (Contributed by Ender Ting, 22-Nov-2024.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (¬ 𝐴 ∈ 𝑆 → ¬ 〈“𝐴”〉 ∈ UpWord 𝑆) | ||
| Theorem | upwordsing 46899 | Singleton is an increasing sequence for any compatible range. (Contributed by Ender Ting, 21-Nov-2024.) |
| ⊢ 𝐴 ∈ 𝑆 ⇒ ⊢ 〈“𝐴”〉 ∈ UpWord 𝑆 | ||
| Theorem | upwordsseti 46900 | Strictly increasing sequences with a set given for range form a set. (Contributed by Ender Ting, 21-Nov-2024.) |
| ⊢ 𝑆 ∈ V ⇒ ⊢ UpWord 𝑆 ∈ V | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |