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Theorem List for Metamath Proof Explorer - 43801-43900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsmfco 43801 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 43802* 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 43803* 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 43804* 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 43777 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 43805* Lemma for smfpimcc 43806 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 43806* 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 43807* 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 43808* 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.)
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 43809* 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 43810* 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 43811* 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 43812* 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 43813* 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 43814* 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 43815* 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 43816* 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 43817* 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 43818* 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 43819* 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 43820* 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 43821* 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 43822* 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 43823* 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 43824* 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 43825* 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 43826* 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 43827* 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.)
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 43828* 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 43829* 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 43830* 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.)
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))

20.38  Mathbox for Saveliy Skresanov

20.38.1  Ceva's theorem

Theoremsigarval 43831* 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 43832* 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 43833* 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 43834* 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 43835* 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 43836* 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 43837* 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 43838* 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 43839* 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 43840* 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 43841* 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 43842* 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 43843* 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 43844* 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 43845* 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 43846* 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 43847* 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 43848 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 43849* 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 43850* 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 43849 three times and then using cevathlem1 43848 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 43845. 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)))

20.38.2  Simple groups

Theoremsimpcntrab 43851 The center of a simple group is trivial or the group is abelian. (Contributed by SS, 3-Jan-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntr‘𝐺)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑 → (𝑍 = { 0 } ∨ 𝐺 ∈ Abel))

20.39  Mathbox for Jarvin Udandy

TheoremhirstL-ax3 43852 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 43853 Recover ax-3 8 from hirstL-ax3 43852. (Contributed by Jarvin Udandy, 3-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))

Theoremaibandbiaiffaiffb 43854 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 43855 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 43856 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 43857 Given a is equivalent to , there exists a proof for a. (Contributed by Jarvin Udandy, 30-Aug-2016.)
(𝜑 ↔ ⊤)       𝜑

Theoremaisfina 43858 Given a is equivalent to , there exists a proof for not a. (Contributed by Jarvin Udandy, 30-Aug-2016.)
(𝜑 ↔ ⊥)        ¬ 𝜑

Theorembothtbothsame 43859 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 43860 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 43861 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 43862 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 43863 Given a is exclusive to b, there exists a proof for (not (a if-and-only-if b)); df-xor 1504 is a closed form of this. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜓)        ¬ (𝜑𝜓)

Theoremaiffnbandciffatnotciffb 43864 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 43865 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 43866 Given a implies b, (not b), there exists a proof for (not a). (Contributed by Jarvin Udandy, 1-Sep-2016.)
(𝜑𝜓)    &    ¬ 𝜓        ¬ 𝜑

Theoremaibnbaif 43867 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 43868 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 43869 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 43870 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 43871 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 43872 If a implies b, then a implies not not b. (Contributed by Jarvin Udandy, 28-Aug-2016.)
((𝜑𝜓) → (𝜑 → ¬ ¬ 𝜓))

Theorembisaiaisb 43873 Application of bicom1 with a, b swapped. (Contributed by Jarvin Udandy, 31-Aug-2016.)
((𝜓𝜑) → (𝜑𝜓))

Theorematbiffatnnbalt 43874 If a implies b, then a implies not not b. (Contributed by Jarvin Udandy, 29-Aug-2016.)
((𝜑𝜓) → (𝜑 → ¬ ¬ 𝜓))

Theoremabnotbtaxb 43875 Assuming a, not b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)
𝜑    &    ¬ 𝜓       (𝜑𝜓)

Theoremabnotataxb 43876 Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)
¬ 𝜑    &   𝜓       (𝜑𝜓)

Theoremconimpf 43877 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 28-Aug-2016.)
𝜑    &    ¬ 𝜓    &   (𝜑𝜓)       (𝜑 ↔ ⊥)

Theoremconimpfalt 43878 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 29-Aug-2016.)
𝜑    &    ¬ 𝜓    &   (𝜑𝜓)       (𝜑 ↔ ⊥)

Theoremaistbisfiaxb 43879 Given a is equivalent to T., Given b is equivalent to F. there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.)
(𝜑 ↔ ⊤)    &   (𝜓 ↔ ⊥)       (𝜑𝜓)

Theoremaisfbistiaxb 43880 Given a is equivalent to F., Given b is equivalent to T., there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)       (𝜑𝜓)

Theoremaifftbifffaibif 43881 Given a is equivalent to T., Given b is equivalent to F., there exists a proof for that a implies b is false. (Contributed by Jarvin Udandy, 7-Sep-2020.)
(𝜑 ↔ ⊤)    &   (𝜓 ↔ ⊥)       ((𝜑𝜓) ↔ ⊥)

Theoremaifftbifffaibifff 43882 Given a is equivalent to T., Given b is equivalent to F., there exists a proof for that a iff b is false. (Contributed by Jarvin Udandy, 7-Sep-2020.)
(𝜑 ↔ ⊤)    &   (𝜓 ↔ ⊥)       ((𝜑𝜓) ↔ ⊥)

Theorematnaiana 43883 Given a, it is not the case a implies a self contradiction. (Contributed by Jarvin Udandy, 7-Sep-2020.)
𝜑        ¬ (𝜑 → (𝜑 ∧ ¬ 𝜑))

Theoremainaiaandna 43884 Given a, a implies it is not the case a implies a self contradiction. (Contributed by Jarvin Udandy, 7-Sep-2020.)
𝜑       (𝜑 → ¬ (𝜑 → (𝜑 ∧ ¬ 𝜑)))

Theoremabcdta 43885 Given (((a and b) and c) and d), there exists a proof for a. (Contributed by Jarvin Udandy, 3-Sep-2016.)
(((𝜑𝜓) ∧ 𝜒) ∧ 𝜃)       𝜑

Theoremabcdtb 43886 Given (((a and b) and c) and d), there exists a proof for b. (Contributed by Jarvin Udandy, 3-Sep-2016.)
(((𝜑𝜓) ∧ 𝜒) ∧ 𝜃)       𝜓

Theoremabcdtc 43887 Given (((a and b) and c) and d), there exists a proof for c. (Contributed by Jarvin Udandy, 3-Sep-2016.)
(((𝜑𝜓) ∧ 𝜒) ∧ 𝜃)       𝜒

Theoremabcdtd 43888 Given (((a and b) and c) and d), there exists a proof for d. (Contributed by Jarvin Udandy, 3-Sep-2016.)
(((𝜑𝜓) ∧ 𝜒) ∧ 𝜃)       𝜃

Theoremabciffcbatnabciffncba 43889 Operands in a biconditional expression converted negated. Additionally biconditional converted to show antecedent implies sequent. Closed form. (Contributed by Jarvin Udandy, 7-Sep-2020.)
(¬ ((𝜑𝜓) ∧ 𝜒) → ¬ ((𝜒𝜓) ∧ 𝜑))

Theoremabciffcbatnabciffncbai 43890 Operands in a biconditional expression converted negated. Additionally biconditional converted to show antecedent implies sequent. (Contributed by Jarvin Udandy, 7-Sep-2020.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜒𝜓) ∧ 𝜑))       (¬ ((𝜑𝜓) ∧ 𝜒) → ¬ ((𝜒𝜓) ∧ 𝜑))

Theoremnabctnabc 43891 not ( a -> ( b /\ c ) ) we can show: not a implies ( b /\ c ). (Contributed by Jarvin Udandy, 7-Sep-2020.)
¬ (𝜑 → (𝜓𝜒))       𝜑 → (𝜓𝜒))

Theoremjabtaib 43892 For when pm3.4 lacks a pm3.4i. (Contributed by Jarvin Udandy, 9-Sep-2020.)
(𝜑𝜓)       (𝜑𝜓)

Theoremonenotinotbothi 43893 From one negated implication it is not the case its nonnegated form and a random others are both true. (Contributed by Jarvin Udandy, 11-Sep-2020.)
¬ (𝜑𝜓)        ¬ ((𝜑𝜓) ∧ (𝜒𝜃))

Theoremtwonotinotbothi 43894 From these two negated implications it is not the case their nonnegated forms are both true. (Contributed by Jarvin Udandy, 11-Sep-2020.)
¬ (𝜑𝜓)    &    ¬ (𝜒𝜃)        ¬ ((𝜑𝜓) ∧ (𝜒𝜃))

Theoremclifte 43895 show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)
(𝜑 ∧ ¬ 𝜒)    &   𝜃       (𝜃 ↔ ((𝜑 ∧ ¬ 𝜒) ∨ (𝜓𝜒)))

Theoremcliftet 43896 show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)
(𝜑𝜒)    &   𝜃       (𝜃 ↔ ((𝜑𝜒) ∨ (𝜓 ∧ ¬ 𝜒)))

Theoremclifteta 43897 show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)
((𝜑 ∧ ¬ 𝜒) ∨ (𝜓𝜒))    &   𝜃       (𝜃 ↔ ((𝜑 ∧ ¬ 𝜒) ∨ (𝜓𝜒)))

Theoremcliftetb 43898 show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)
((𝜑𝜒) ∨ (𝜓 ∧ ¬ 𝜒))    &   𝜃       (𝜃 ↔ ((𝜑𝜒) ∨ (𝜓 ∧ ¬ 𝜒)))

Theoremconfun 43899 Given the hypotheses there exists a proof for (c implies ( d iff a ) ). (Contributed by Jarvin Udandy, 6-Sep-2020.)
𝜑    &   (𝜒𝜓)    &   (𝜒𝜃)    &   (𝜑 → (𝜑𝜓))       (𝜒 → (𝜃𝜑))

Theoremconfun2 43900 Confun simplified to two propositions. (Contributed by Jarvin Udandy, 6-Sep-2020.)
(𝜓𝜑)    &   (𝜓 → ¬ (𝜓 → (𝜓 ∧ ¬ 𝜓)))    &   ((𝜓𝜑) → ((𝜓𝜑) → 𝜑))       (𝜓 → (¬ (𝜓 → (𝜓 ∧ ¬ 𝜓)) ↔ (𝜓𝜑)))

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45725
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