P. 369 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36801-36900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dnibndlem11 36801	Lemma for dnibnd 36804. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (1 / 2))
Theorem	dnibndlem12 36802*	Lemma for dnibnd 36804. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2))))    ⇒   ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴)))
Theorem	dnibndlem13 36803*	Lemma for dnibnd 36804. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2))))    ⇒   ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴)))
Theorem	dnibnd 36804*	The "distance to nearest integer" function is 1-Lipschitz continuous, i.e., is a short map. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴)))
Theorem	dnicn 36805	The "distance to nearest integer" function is continuous. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    ⇒   ⊢ 𝑇 ∈ (ℝ–cn→ℝ)
Theorem	knoppcnlem1 36806*	Lemma for knoppcn 36817. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴)‘𝑀) = ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))))
Theorem	knoppcnlem2 36807*	Lemma for knoppcn 36817. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))) ∈ ℝ)
Theorem	knoppcnlem3 36808*	Lemma for knoppcn 36817. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴)‘𝑀) ∈ ℝ)
Theorem	knoppcnlem4 36809*	Lemma for knoppcn 36817. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (abs‘((𝐹‘𝐴)‘𝑀)) ≤ ((𝑚 ∈ ℕ0 ↦ ((abs‘𝐶)↑𝑚))‘𝑀))
Theorem	knoppcnlem5 36810*	Lemma for knoppcn 36817. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ0⟶(ℂ ↑m ℝ))
Theorem	knoppcnlem6 36811*	Lemma for knoppcn 36817. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐶) < 1)    ⇒   ⊢ (𝜑 → seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))) ∈ dom (⇝𝑢‘ℝ))
Theorem	knoppcnlem7 36812*	Lemma for knoppcn 36817. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))‘𝑀) = (𝑤 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑤))‘𝑀)))
Theorem	knoppcnlem8 36813*	Lemma for knoppcn 36817. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))):ℕ0⟶(ℂ ↑m ℝ))
Theorem	knoppcnlem9 36814*	Lemma for knoppcn 36817. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐶) < 1)    ⇒   ⊢ (𝜑 → seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))(⇝𝑢‘ℝ)𝑊)
Theorem	knoppcnlem10 36815*	Lemma for knoppcn 36817. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) Avoid ax-mulf 11116. (Revised by GG, 19-Apr-2025.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑀)) ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂfld)))
Theorem	knoppcnlem11 36816*	Lemma for knoppcn 36817. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))):ℕ0⟶(ℝ–cn→ℂ))
Theorem	knoppcn 36817*	The continuous nowhere differentiable function 𝑊 ( Knopp, K. (1918). Math. Z. 2, 1-26 ) is, in fact, continuous. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐶) < 1)    ⇒   ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℂ))
Theorem	knoppcld 36818*	Closure theorem for Knopp's function. (Contributed by Asger C. Ipsen, 26-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐶) < 1)    ⇒   ⊢ (𝜑 → (𝑊‘𝐴) ∈ ℂ)
Theorem	unblimceq0lem 36819*	Lemma for unblimceq0 36820. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → ∀𝑏 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥))))    ⇒   ⊢ (𝜑 → ∀𝑐 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑦 ∈ 𝑆 (𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑦))))
Theorem	unblimceq0 36820*	If 𝐹 is unbounded near 𝐴 it has no limit at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → ∀𝑏 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥))))    ⇒   ⊢ (𝜑 → (𝐹 limℂ 𝐴) = ∅)
Theorem	unbdqndv1 36821*	If the difference quotient (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴)) is unbounded near 𝐴 then 𝐹 is not differentiable at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴)))    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → ∀𝑏 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑥 ∈ (𝑋 ∖ {𝐴})((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐺‘𝑥))))    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ dom (𝑆 D 𝐹))
Theorem	unbdqndv2lem1 36822	Lemma for unbdqndv2 36824. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐷 ≠ 0)    &   ⊢ (𝜑 → (2 · 𝐸) ≤ (abs‘((𝐴 − 𝐵) / 𝐷)))    ⇒   ⊢ (𝜑 → ((𝐸 · (abs‘𝐷)) ≤ (abs‘(𝐴 − 𝐶)) ∨ (𝐸 · (abs‘𝐷)) ≤ (abs‘(𝐵 − 𝐶))))
Theorem	unbdqndv2lem2 36823*	Lemma for unbdqndv2 36824. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴)))    &   ⊢ 𝑊 = if((𝐵 · (𝑉 − 𝑈)) ≤ (abs‘((𝐹‘𝑈) − (𝐹‘𝐴))), 𝑈, 𝑉)    &   ⊢ (𝜑 → 𝑋 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑈 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≤ 𝑉)    &   ⊢ (𝜑 → (𝑉 − 𝑈) < 𝐷)    &   ⊢ (𝜑 → (2 · 𝐵) ≤ ((abs‘((𝐹‘𝑉) − (𝐹‘𝑈))) / (𝑉 − 𝑈)))    ⇒   ⊢ (𝜑 → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊 − 𝐴)) < 𝐷 ∧ 𝐵 ≤ (abs‘(𝐺‘𝑊)))))
Theorem	unbdqndv2 36824*	Variant of unbdqndv1 36821 with the hypothesis that (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) is unbounded where 𝑥 ≤ 𝐴 and 𝐴 ≤ 𝑦. (Contributed by Asger C. Ipsen, 12-May-2021.)
⊢ (𝜑 → 𝑋 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → ∀𝑏 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑥 ≤ 𝐴 ∧ 𝐴 ≤ 𝑦) ∧ ((𝑦 − 𝑥) < 𝑑 ∧ 𝑥 ≠ 𝑦) ∧ 𝑏 ≤ ((abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) / (𝑦 − 𝑥))))    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ dom (ℝ D 𝐹))
Theorem	knoppndvlem1 36825	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    ⇒   ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ)
Theorem	knoppndvlem2 36826	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ ℤ)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐽 < 𝐼)    ⇒   ⊢ (𝜑 → (((2 · 𝑁)↑𝐼) · ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)) ∈ ℤ)
Theorem	knoppndvlem3 36827	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 15-Jun-2021.)
⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    ⇒   ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
Theorem	knoppndvlem4 36828*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → seq0( + , (𝐹‘𝐴)) ⇝ (𝑊‘𝐴))
Theorem	knoppndvlem5 36829*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) ∈ ℝ)
Theorem	knoppndvlem6 36830*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝑊‘𝐴) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖))
Theorem	knoppndvlem7 36831*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = ((𝐶↑𝐽) · (𝑇‘(𝑀 / 2))))
Theorem	knoppndvlem8 36832*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 2 ∥ 𝑀)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = 0)
Theorem	knoppndvlem9 36833*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝑀)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = ((𝐶↑𝐽) / 2))
Theorem	knoppndvlem10 36834*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))
Theorem	knoppndvlem11 36835*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 28-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖))) ≤ ((abs‘(𝐵 − 𝐴)) · Σ𝑖 ∈ (0...(𝐽 − 1))(((2 · 𝑁) · (abs‘𝐶))↑𝑖)))
Theorem	knoppndvlem12 36836	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 29-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    ⇒   ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)))
Theorem	knoppndvlem13 36837	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    ⇒   ⊢ (𝜑 → 𝐶 ≠ 0)
Theorem	knoppndvlem14 36838*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 7-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    ⇒   ⊢ (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖))) ≤ ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))
Theorem	knoppndvlem15 36839*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 6-Jul-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    ⇒   ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ (abs‘((𝑊‘𝐵) − (𝑊‘𝐴))))
Theorem	knoppndvlem16 36840	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 19-Jul-2021.)
⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐵 − 𝐴) = (((2 · 𝑁)↑-𝐽) / 2))
Theorem	knoppndvlem17 36841*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 12-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    ⇒   ⊢ (𝜑 → ((((2 · 𝑁) · (abs‘𝐶))↑𝐽) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ ((abs‘((𝑊‘𝐵) − (𝑊‘𝐴))) / (𝐵 − 𝐴)))
Theorem	knoppndvlem18 36842*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 14-Aug-2021.)
⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐺 ∈ ℝ+)    &   ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ ℕ0 ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · 𝐺)))
Theorem	knoppndvlem19 36843*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 17-Aug-2021.)
⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑚)    &   ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑚 + 1))    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐻 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ∃𝑚 ∈ ℤ (𝐴 ≤ 𝐻 ∧ 𝐻 ≤ 𝐵))
Theorem	knoppndvlem20 36844	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 18-Aug-2021.)
⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    ⇒   ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+)
Theorem	knoppndvlem21 36845*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 18-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ 𝐺 = (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝐷 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐻 ∈ ℝ)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    &   ⊢ (𝜑 → (((2 · 𝑁)↑-𝐽) / 2) < 𝐷)    &   ⊢ (𝜑 → 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝐽) · 𝐺))    ⇒   ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎 ≤ 𝐻 ∧ 𝐻 ≤ 𝑏) ∧ ((𝑏 − 𝑎) < 𝐷 ∧ 𝑎 ≠ 𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊‘𝑏) − (𝑊‘𝑎))) / (𝑏 − 𝑎))))
Theorem	knoppndvlem22 36846*	Lemma for knoppndv 36847. (Contributed by Asger C. Ipsen, 19-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝐷 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐻 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    ⇒   ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎 ≤ 𝐻 ∧ 𝐻 ≤ 𝑏) ∧ ((𝑏 − 𝑎) < 𝐷 ∧ 𝑎 ≠ 𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊‘𝑏) − (𝑊‘𝑎))) / (𝑏 − 𝑎))))
Theorem	knoppndv 36847*	The continuous nowhere differentiable function 𝑊 ( Knopp, K. (1918). Math. Z. 2, 1-26 ) is, in fact, nowhere differentiable. (Contributed by Asger C. Ipsen, 19-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    ⇒   ⊢ (𝜑 → dom (ℝ D 𝑊) = ∅)
Theorem	knoppf 36848*	Knopp's function is a function. (Contributed by Asger C. Ipsen, 25-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝑊:ℝ⟶ℝ)
Theorem	knoppcn2 36849*	Variant of knoppcn 36817 with different codomain. (Contributed by Asger C. Ipsen, 25-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    ⇒   ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℝ))
Theorem	cnndvlem1 36850*	Lemma for cnndv 36852. (Contributed by Asger C. Ipsen, 25-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    ⇒   ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)
Theorem	cnndvlem2 36851*	Lemma for cnndv 36852. (Contributed by Asger C. Ipsen, 26-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    ⇒   ⊢ ∃𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)
Theorem	cnndv 36852	There exists a continuous nowhere differentiable function. The result follows directly from knoppcn 36817 and knoppndv 36847. (Contributed by Asger C. Ipsen, 26-Aug-2021.)
⊢ ∃𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)
21.19  Mathbox for BJ In this mathbox, we try to respect the ordering of the sections of the main part. There are strengthenings of theorems of the main part, as well as work on reducing axiom dependencies.
21.19.1  Propositional calculus Miscellaneous utility theorems of propositional calculus.
21.19.1.1  Derived rules of inference In this section, we prove a few rules of inference derived from modus ponens ax-mp 5, and which do not depend on any other axioms.
Theorem	bj-mp2c 36853	A double modus ponens inference. Inference associated with mpd 15. (Contributed by BJ, 24-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → 𝜓)    &   ⊢ 𝜑    ⇒   ⊢ 𝜒
Theorem	bj-mp2d 36854	A double modus ponens inference. Inference associated with mpcom 38. (Contributed by BJ, 24-Sep-2019.)
⊢ (𝜓 → (𝜑 → 𝜒))    &   ⊢ (𝜑 → 𝜓)    &   ⊢ 𝜑    ⇒   ⊢ 𝜒
21.19.1.2  A syntactic theorem In this section, we prove a syntactic theorem (bj-0 36855) asserting that some formula is well-formed. Then, we use this syntactic theorem to shorten the proof of a "usual" theorem (bj-1 36856) and explain in the comment of that theorem why this phenomenon is unusual.
Theorem	bj-0 36855	A syntactic theorem. See the section comment and the comment of bj-1 36856. The full proof (that is, with the syntactic, non-essential steps) does not appear on this webpage. It has five steps and reads $= wph wps wi wch wi $. The only other syntactic theorems in the main part of set.mm are wel 2120 and weq 1969. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
wff ((𝜑 → 𝜓) → 𝜒)
Theorem	bj-1 36856	In this proof, the use of the syntactic theorem bj-0 36855 allows to reduce the total length by one (non-essential) step. See also the section comment and the comment of bj-0 36855. Since bj-0 36855 is used in a non-essential step, this use does not appear on this webpage (but the present theorem appears on the webpage for bj-0 36855 as a theorem referencing it). The full proof reads $= wph wps wch bj-0 id $. (while, without using bj-0 36855, it would read $= wph wps wi wch wi id $.). Now we explain why syntactic theorems are not useful in set.mm. Suppose that the syntactic theorem thm-0 proves that PHI is a well-formed formula, and that thm-0 is used to shorten the proof of thm-1. Assume that PHI does have proper non-atomic subformulas (which is not the case of the formula proved by weq 1969 or wel 2120). Then, the proof of thm-1 does not construct all the proper non-atomic subformulas of PHI (if it did, then using thm-0 would not shorten it). Therefore, thm-1 is a special instance of a more general theorem with essentially the same proof. In the present case, bj-1 36856 is a special instance of id 22. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜑 → 𝜓) → 𝜒))
21.19.1.3  Minimal implicational calculus Minimal implicational calculus, or intuitionistic implicational calculus, is the logical calculus with axioms ax-mp 5, ax-1 6, ax-2 7.
Theorem	bj-a1k 36857	Weakening of ax-1 6. As a consequence, its associated inference is an instance (where we allow extra hypotheses) of ax-1 6. Its commuted form is 2a1 28 (but bj-a1k 36857 does not require ax-2 7). Shortens dfwe2 7724 (937>925), ordunisuc2 7791 (789>777), r111 9697 (558>545), smo11 8301 (1176>1164). (Contributed by BJ, 11-Aug-2020.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜓)))
Theorem	bj-poni 36858	Inference associated with "pon", pm2.27 42. Its associated inference is ax-mp 5. (Contributed by BJ, 30-Jul-2024.)
⊢ 𝜑    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜓)
Theorem	bj-nnclav 36859	When ⊥ is substituted for 𝜓, this formula is the Clavius law with a doubly negated consequent, which is therefore a minimalistic tautology. Notice the non-intuitionistic proof from peirce 203 and pm2.27 42 chained using syl 17. (Contributed by BJ, 4-Dec-2023.)
⊢ (((𝜑 → 𝜓) → 𝜑) → ((𝜑 → 𝜓) → 𝜓))
Theorem	bj-nnclavi 36860	Inference associated with bj-nnclav 36859. Its associated inference is an instance of syl 17. Notice the non-intuitionistic proof from bj-peircei 36882 and bj-poni 36858. (Contributed by BJ, 30-Jul-2024.)
⊢ ((𝜑 → 𝜓) → 𝜑)    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜓)
Theorem	bj-nnclavc 36861	Commuted form of bj-nnclav 36859. Notice the non-intuitionistic proof from bj-peircei 36882 and imim1i 63. (Contributed by BJ, 30-Jul-2024.) A proof which is shorter when compressed uses embantd 59. (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜑) → 𝜓))
Theorem	bj-nnclavci 36862	Inference associated with bj-nnclavc 36861. Its associated inference is an instance of syl 17. Notice the non-intuitionistic proof from peirce 203 and syl 17. (Contributed by BJ, 30-Jul-2024.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜓)
Theorem	bj-jarrii 36863	Inference associated with jarri 107. Contrary to it, it does not require ax-2 7, but only ax-mp 5 and ax-1 6. (Contributed by BJ, 29-Mar-2020.) (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → 𝜒)    &   ⊢ 𝜓    ⇒   ⊢ 𝜒
Theorem	bj-imim21 36864	The propositional function (𝜑 → (. → 𝜓)) is decreasing. (Contributed by BJ, 19-Jul-2019.)
⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜓 → 𝜃)) → (𝜒 → (𝜑 → 𝜃))))
Theorem	bj-imim21i 36865	The propositional function (𝜑 → (. → 𝜓)) is decreasing. Its associated inference is syl5 34. (Contributed by BJ, 19-Jul-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 → (𝜓 → 𝜃)) → (𝜒 → (𝜑 → 𝜃)))
Theorem	bj-imim11 36866	The propositional function ((. → 𝜑) → 𝜓) is increasing. (Contributed by BJ, 3-Apr-2026.)
⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → 𝜃) → ((𝜓 → 𝜒) → 𝜃)))
Theorem	bj-imim11i 36867	The propositional function ((. → 𝜑) → 𝜓) is increasing. Its associated inference is wl-syls2 37887. (Contributed by BJ, 3-Apr-2026.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (((𝜑 → 𝜒) → 𝜃) → ((𝜓 → 𝜒) → 𝜃))
Theorem	bj-peircestab 36868	Over minimal implicational calculus, Peirce's law implies the double negation of the stability of any formula (that is the interpretation when ⊥ is substituted for 𝜓 and for 𝜒). Therefore, the double negation of the stability of any formula is provable in classical refutability calculus. It is also provable in intuitionistic calculus (see iset.mm/bj-nnst) but it is not provable in minimal calculus (see bj-stabpeirce 36869). (Contributed by BJ, 30-Nov-2023.) Axiom ax-3 8 is only used through Peirce's law peirce 203. (Proof modification is discouraged.)
⊢ (((((𝜑 → 𝜓) → 𝜒) → 𝜑) → 𝜒) → 𝜒)
Theorem	bj-stabpeirce 36869	This minimal implicational calculus tautology is used in the following argument: When 𝜑, 𝜓, 𝜒, 𝜃, 𝜏 are replaced respectively by (𝜑 → ⊥), ⊥, 𝜑, ⊥, ⊥, the antecedent becomes ¬ ¬ (¬ ¬ 𝜑 → 𝜑), that is, the double negation of the stability of 𝜑. If that statement were provable in minimal calculus, then, since ⊥ plays no particular role in minimal calculus, also the statement with 𝜓 in place of ⊥ would be provable. The corresponding consequent is (((𝜓 → 𝜑) → 𝜓) → 𝜓), that is, the non-intuitionistic Peirce law. Therefore, the double negation of the stability of any formula is not provable in minimal calculus. However, it is provable both in intuitionistic calculus (see iset.mm/bj-nnst) and in classical refutability calculus (see bj-peircestab 36868). (Contributed by BJ, 30-Nov-2023.) (Revised by BJ, 30-Jul-2024.) (Proof modification is discouraged.)
⊢ (((((𝜑 → 𝜓) → 𝜒) → 𝜃) → 𝜏) → (((𝜓 → 𝜒) → 𝜃) → 𝜏))
21.19.1.4  Positive calculus Positive calculus is understood to be intuitionistic. Its primitive connectives are implication, equivalence, conjunction, and disjunction. Due to the current axiomatization of set.mm, axiom ax-3 8, which does not belong to positive calculus, may appear as a dependency of some theorems in this section.
Theorem	bj-bisimpl 36870	Implication from equivalence with a conjunct. Its associated inference is simplbi 497. (Contributed by BJ, 20-Mar-2026.)
⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜑 → 𝜓))
Theorem	bj-bisimpr 36871	Implication from equivalence with a conjunct. Its associated inference is simprbi 498. (Contributed by BJ, 20-Mar-2026.)
⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜑 → 𝜒))
Theorem	bj-syl66ib 36872	A mixed syllogism inference derived from imbitrdi 252. Shortens bj-dvelimdv1 37212, alexsubALTlem4 24040 (4821>4812), supsrlem 11032 (2868>2863). (Contributed by BJ, 20-Oct-2021.)
⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜃 → 𝜏)    &   ⊢ (𝜏 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	bj-orim2 36873	Proof of orim2 975 from the axiomatic definition of disjunction (olc 874, orc 873, jao 968) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓)))
Theorem	bj-currypeirce 36874	Curry's axiom curryax 899 (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 203 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the elimination axiom jao 968 via its inference form jaoi 863; the introduction axioms olc 874 and orc 873 are not needed). Note that this theorem shows that actually, the standard instance of curryax 899 implies the standard instance of peirce 203, which is not the case for the converse bj-peircecurry 36875. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
Theorem	bj-peircecurry 36875	Peirce's axiom peirce 203 implies Curry's axiom curryax 899 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the introduction axioms olc 874 and orc 873; the elimination axiom jao 968 is not needed). See bj-currypeirce 36874 for the converse. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ∨ (𝜑 → 𝜓))
Theorem	bj-animbi 36876	Conjunction in terms of implication and biconditional. Note that the proof is intuitionistic (use of ax-3 8 comes from the unusual definition of the biconditional in set.mm). (Contributed by BJ, 23-Sep-2023.)
⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ↔ (𝜑 → 𝜓)))
Theorem	bj-currypara 36877	Curry's paradox. Note that the proof is intuitionistic (use of ax-3 8 comes from the unusual definition of the biconditional in set.mm). The paradox comes from the case where 𝜑 is the self-referential sentence "If this sentence is true, then 𝜓", so that one can prove everything. Therefore, a consistent system cannot allow the formation of such self-referential sentences. This has lead to the study of logics rejecting contraction pm2.43 56, such as affine logic and linear logic. (Contributed by BJ, 23-Sep-2023.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → 𝜓)
21.19.1.5  Implication and negation Some theorems of propositional calculus in the language of implication and negation.
Theorem	bj-con2com 36878	A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.)
⊢ (𝜑 → ((𝜓 → ¬ 𝜑) → ¬ 𝜓))
Theorem	bj-con2comi 36879	Inference associated with bj-con2com 36878. Its associated inference is mt2 201. TODO: when in the main part, add to mt2 201 that it is the inference associated with bj-con2comi 36879. (Contributed by BJ, 19-Mar-2020.)
⊢ 𝜑    ⇒   ⊢ ((𝜓 → ¬ 𝜑) → ¬ 𝜓)
Theorem	bj-nimn 36880	If a formula is true, then it does not imply its negation. (Contributed by BJ, 19-Mar-2020.) A shorter proof is possible using id 22 and jc 161, however, the present proof uses theorems that are more basic than jc 161. (Proof modification is discouraged.)
⊢ (𝜑 → ¬ (𝜑 → ¬ 𝜑))
Theorem	bj-nimni 36881	Inference associated with bj-nimn 36880. (Contributed by BJ, 19-Mar-2020.)
⊢ 𝜑    ⇒   ⊢ ¬ (𝜑 → ¬ 𝜑)
Theorem	bj-peircei 36882	Inference associated with peirce 203. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜑)    ⇒   ⊢ 𝜑
Theorem	bj-looinvi 36883	Inference associated with looinv 204. Its associated inference is bj-looinvii 36884. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜓)    ⇒   ⊢ ((𝜓 → 𝜑) → 𝜑)
Theorem	bj-looinvii 36884	Inference associated with bj-looinvi 36883. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜓)    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	bj-mt2bi 36885	Version of mt2 201 where the major premise is a biconditional. Shortens fal 1561 (see bj-fal 36886) . Another proof is also possible via con2bii 358 and mpbi 231. The current mt2bi 364 should be relabeled, maybe to imfal. (Contributed by BJ, 5-Oct-2024.)
⊢ (𝜓 ↔ ¬ 𝜑)    &   ⊢ 𝜑    ⇒   ⊢ ¬ 𝜓
Theorem	bj-fal 36886	Shortening of fal 1561 using bj-mt2bi 36885. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) (Proof modification is discouraged.)
⊢ ¬ ⊥
Theorem	bj-ntrufal 36887	The negation of a theorem is equivalent to false. Shortens dfnul2 4271 (see bj-dfnul2 36888). (Contributed by BJ, 5-Oct-2024.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜑 ↔ ⊥)
Theorem	bj-dfnul2 36888	Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2152, ax-11 2168, and ax-12 2189. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.) (Proof modification is discouraged.)
⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
21.19.1.6  Disjunction A few lemmas about disjunction. The fundamental theorems in this family are the dual statements pm4.71 562 and pm4.72 957. See also biort 941 and biorf 942.
Theorem	bj-jaoi1 36889	Shortens orfa2 38460 (58>53), pm1.2 909 (20>18), pm1.2 909 (20>18), pm2.4 912 (31>25), pm2.41 913 (31>25), pm2.42 950 (38>32), pm3.2ni 886 (43>39), pm4.44 1004 (55>51). (Contributed by BJ, 30-Sep-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜓) → 𝜓)
Theorem	bj-jaoi2 36890	Shortens consensus 1058 (110>106), elnn0z 12535 (336>329), pm1.2 909 (20>19), pm3.2ni 886 (43>39), pm4.44 1004 (55>51). (Contributed by BJ, 30-Sep-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜓 ∨ 𝜑) → 𝜓)
21.19.1.7  Logical equivalence A few other characterizations of the biconditional. The inter-definability of logical connectives offers many ways to express a given statement. Some useful theorems in this regard are df-or 854, df-an 397, pm4.64 855, imor 859, pm4.62 862 through pm4.67 399, and, for the De Morgan laws, ianor 989 through pm4.57 998.
Theorem	bj-dfbi4 36891	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
Theorem	bj-dfbi5 36892	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)))
Theorem	bj-dfbi6 36893	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓)))
Theorem	bj-bijust0ALT 36894	Alternate proof of bijust0 205; shorter but using additional intermediate results. (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Revised by BJ, 19-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ ((𝜑 → 𝜑) → ¬ (𝜑 → 𝜑))
Theorem	bj-bijust00 36895	A self-implication does not imply the negation of a self-implication. Most general theorem of which bijust 206 is an instance (bijust0 205 and bj-bijust0ALT 36894 are therefore also instances of it). (Contributed by BJ, 7-Sep-2022.)
⊢ ¬ ((𝜑 → 𝜑) → ¬ (𝜓 → 𝜓))
21.19.1.8  The conditional operator for propositions
Theorem	bj-consensus 36896	Version of consensus 1058 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1058, using only binary connectives, and hinting at the fact that it is a Boolean algebra identity, like the absorption identities.) (Contributed by BJ, 30-Sep-2019.)
⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))
Theorem	bj-consensusALT 36897	Alternate proof of bj-consensus 36896. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))
Theorem	bj-df-ifc 36898*	Candidate definition for the conditional operator for classes. This is in line with the definition of a class as the extension of a predicate in df-clab 2719. We reprove the current df-if 4462 from it in bj-dfif 36899. (Contributed by BJ, 20-Sep-2019.) (Proof modification is discouraged.)
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}
Theorem	bj-dfif 36899*	Alternate definition of the conditional operator for classes, which used to be the main definition. (Contributed by BJ, 26-Dec-2023.) (Proof modification is discouraged.)
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))}
Theorem	bj-ififc 36900	A biconditional connecting the conditional operator for propositions and the conditional operator for classes. Note that there is no sethood hypothesis on 𝑋: it is implied by either side. (Contributed by BJ, 24-Sep-2019.) Generalize statement from setvar 𝑥 to class 𝑋. (Revised by BJ, 26-Dec-2023.)
⊢ (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵))