P. 456 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45501-45600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	smfmul 45501*	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 45502*	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 45503*	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 45504*	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 45505*	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 45506	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 45507*	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 45508	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 45509*	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 45510	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 45511*	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 45512*	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 45483 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 45513*	Lemma for smfpimcc 45514 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 45514*	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 45515*	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 45516*	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 45517*	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 45518*	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 45519*	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 45520*	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 45521*	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 45522*	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 45523*	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 45524*	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 45525*	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 45526*	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 45527*	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 45528*	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 45529*	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 45530*	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 45531*	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 45532*	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 45533*	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 45534*	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 45535*	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 45536*	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 45537*	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 45538*	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 45539	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 45540	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 45541	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 45542	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 45543*	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 45544	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 45545*	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 45546*	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 45547	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 45548*	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 45549*	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 45550*	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 45551*	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 45552*	The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 45524. 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 45553*	The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 45524. 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 45554*	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 45555*	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)
21.41  Mathbox for Saveliy Skresanov
21.41.1  Ceva's theorem
Theorem	sigarval 45556*	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 45557*	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 45558*	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 45559*	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 45560*	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 45561*	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 45562*	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 45563*	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 45564*	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 45565*	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 45566*	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 45567*	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 45568*	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 45569*	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 45570*	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 45571*	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 45572*	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 45573	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 45574*	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 45575*	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 45574 three times and then using cevathlem1 45573 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 45570. 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)))
21.41.2  Simple groups
Theorem	simpcntrab 45576	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))
21.42  Mathbox for Ender Ting
21.42.1  Increasing sequences and subsequences
Theorem	et-ltneverrefl 45577	Less-than class is never reflexive. (Contributed by Ender Ting, 22-Nov-2024.) Prefer to specify theorem domain and then apply ltnri 11322. (New usage is discouraged.)
⊢ ¬ 𝐴 < 𝐴
Theorem	et-equeucl 45578	Alternative proof that equality is left-Euclidean, using ax7 2019 directly instead of utility theorems; done for practice. (Contributed by Ender Ting, 21-Dec-2024.)
⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦))
Theorem	et-sqrtnegnre 45579	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	natlocalincr 45580*	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 45581*	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 45582	Extend class notation to include the set of strictly increasing sequences.
class UpWord 𝑆
Definition	df-upword 45583*	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 45584	Empty set is an increasing sequence for every range. (Contributed by Ender Ting, 19-Nov-2024.)
⊢ ∅ ∈ UpWord 𝑆
Theorem	upwordisword 45585	Any increasing sequence is a sequence. (Contributed by Ender Ting, 19-Nov-2024.)
⊢ (𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ Word 𝑆)
Theorem	singoutnword 45586	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 45587	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 45588	Singleton is an increasing sequence for any compatible range. (Contributed by Ender Ting, 21-Nov-2024.)
⊢ 𝐴 ∈ 𝑆    ⇒   ⊢ ⟨“𝐴”⟩ ∈ UpWord 𝑆
Theorem	upwordsseti 45589	Strictly increasing sequences with a set given for range form a set. (Contributed by Ender Ting, 21-Nov-2024.)
⊢ 𝑆 ∈ V    ⇒   ⊢ UpWord 𝑆 ∈ V
Theorem	tworepnotupword 45590	Concatenation of identical singletons is never an increasing sequence. (Contributed by Ender Ting, 22-Nov-2024.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ¬ (⟨“𝐴”⟩ ++ ⟨“𝐴”⟩) ∈ UpWord 𝑆
Theorem	upwrdfi 45591*	There is a finite number of strictly increasing sequences of a given length over finite alphabet. Trivially holds for invalid lengths where there're zero matching sequences. (Contributed by Ender Ting, 5-Jan-2024.)
⊢ (𝑆 ∈ Fin → {𝑎 ∈ UpWord 𝑆 ∣ (♯‘𝑎) = 𝑇} ∈ Fin)
21.43  Mathbox for Jarvin Udandy
Theorem	hirstL-ax3 45592	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) → ((¬ 𝜑 → 𝜓) → 𝜑))
Theorem	ax3h 45593	Recover ax-3 8 from hirstL-ax3 45592. (Contributed by Jarvin Udandy, 3-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))
Theorem	aibandbiaiffaiffb 45594	A closed form showing (a implies b and b implies a) same-as (a same-as b). (Contributed by Jarvin Udandy, 3-Sep-2016.)
⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (𝜑 ↔ 𝜓))
Theorem	aibandbiaiaiffb 45595	A closed form showing (a implies b and b implies a) implies (a same-as b). (Contributed by Jarvin Udandy, 3-Sep-2016.)
⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))
Theorem	notatnand 45596	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    ⇒   ⊢ ¬ (𝜑 ∧ 𝜓)
Theorem	aistia 45597	Given a is equivalent to ⊤, there exists a proof for a. (Contributed by Jarvin Udandy, 30-Aug-2016.)
⊢ (𝜑 ↔ ⊤)    ⇒   ⊢ 𝜑
Theorem	aisfina 45598	Given a is equivalent to ⊥, there exists a proof for not a. (Contributed by Jarvin Udandy, 30-Aug-2016.)
⊢ (𝜑 ↔ ⊥)    ⇒   ⊢ ¬ 𝜑
Theorem	bothtbothsame 45599	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 ↔ ⊤)    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	bothfbothsame 45600	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 ↔ ⊥)    ⇒   ⊢ (𝜑 ↔ 𝜓)