P. 373 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37201-37300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	knoppcn 37201*	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 37202*	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 37203*	Lemma for unblimceq0 37204. (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 37204*	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 37205*	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 37206	Lemma for unbdqndv2 37208. (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 37207*	Lemma for unbdqndv2 37208. (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 37208*	Variant of unbdqndv1 37205 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 37209	Lemma for knoppndv 37231. (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 37210	Lemma for knoppndv 37231. (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 37211	Lemma for knoppndv 37231. (Contributed by Asger C. Ipsen, 15-Jun-2021.)
⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    ⇒   ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
Theorem	knoppndvlem4 37212*	Lemma for knoppndv 37231. (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 37213*	Lemma for knoppndv 37231. (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 37214*	Lemma for knoppndv 37231. (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 37215*	Lemma for knoppndv 37231. (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 37216*	Lemma for knoppndv 37231. (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 37217*	Lemma for knoppndv 37231. (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 37218*	Lemma for knoppndv 37231. (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 37219*	Lemma for knoppndv 37231. (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 37220	Lemma for knoppndv 37231. (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 37221	Lemma for knoppndv 37231. (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 37222*	Lemma for knoppndv 37231. (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 37223*	Lemma for knoppndv 37231. (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 37224	Lemma for knoppndv 37231. (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 37225*	Lemma for knoppndv 37231. (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 37226*	Lemma for knoppndv 37231. (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 37227*	Lemma for knoppndv 37231. (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 37228	Lemma for knoppndv 37231. (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 37229*	Lemma for knoppndv 37231. (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 37230*	Lemma for knoppndv 37231. (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 37231*	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 37232*	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 37233*	Variant of knoppcn 37201 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 37234*	Lemma for cnndv 37236. (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 37235*	Lemma for cnndv 37236. (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 37236	There exists a continuous nowhere differentiable function. The result follows directly from knoppcn 37201 and knoppndv 37231. (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 37237	A double modus ponens inference. Inference associated with mpd 16. (Contributed by BJ, 24-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → 𝜓)    &   ⊢ 𝜑    ⇒   ⊢ 𝜒
Theorem	bj-mp2d 37238	A double modus ponens inference. Inference associated with mpcom 39. (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 37239) asserting that some formula is well-formed. Then, we use this syntactic theorem to shorten the proof of a "usual" theorem (bj-1 37240) and explain in the comment of that theorem why this phenomenon is unusual.
Theorem	bj-0 37239	A syntactic theorem. See the section comment and the comment of bj-1 37240. 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 2146 and weq 1995. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
wff ((𝜑 → 𝜓) → 𝜒)
Theorem	bj-1 37240	In this proof, the use of the syntactic theorem bj-0 37239 allows to reduce the total length by one (non-essential) step. See also the section comment and the comment of bj-0 37239. Since bj-0 37239 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 37239 as a theorem referencing it). The full proof reads $= wph wps wch bj-0 id $. (while, without using bj-0 37239, 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 1995 or wel 2146). 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 37240 is a special instance of id 23. (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-poni 37241	Inference associated with "pon", pm2.27 43. Its associated inference is ax-mp 5. (Contributed by BJ, 30-Jul-2024.)
⊢ 𝜑    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜓)
Theorem	bj-nnclav 37242	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 205 and pm2.27 43 chained using syl 18. (Contributed by BJ, 4-Dec-2023.)
⊢ (((𝜑 → 𝜓) → 𝜑) → ((𝜑 → 𝜓) → 𝜓))
Theorem	bj-nnclavi 37243	Inference associated with bj-nnclav 37242. Its associated inference is an instance of syl 18. Notice the non-intuitionistic proof from bj-peircei 37265 and bj-poni 37241. (Contributed by BJ, 30-Jul-2024.)
⊢ ((𝜑 → 𝜓) → 𝜑)    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜓)
Theorem	bj-nnclavc 37244	Commuted form of bj-nnclav 37242. Notice the non-intuitionistic proof from bj-peircei 37265 and imim1i 64. (Contributed by BJ, 30-Jul-2024.) A proof which is shorter when compressed uses embantd 60. (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜑) → 𝜓))
Theorem	bj-nnclavci 37245	Inference associated with bj-nnclavc 37244. Its associated inference is an instance of syl 18. Notice the non-intuitionistic proof from peirce 205 and syl 18. (Contributed by BJ, 30-Jul-2024.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜓)
Theorem	bj-jarrii 37246	Inference associated with jarri 108. 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 37247	The propositional function (𝜑 → (. → 𝜓)) is decreasing. (Contributed by BJ, 19-Jul-2019.)
⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜓 → 𝜃)) → (𝜒 → (𝜑 → 𝜃))))
Theorem	bj-imim21i 37248	The propositional function (𝜑 → (. → 𝜓)) is decreasing. Its associated inference is syl5 35. (Contributed by BJ, 19-Jul-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 → (𝜓 → 𝜃)) → (𝜒 → (𝜑 → 𝜃)))
Theorem	bj-imim11 37249	The propositional function ((. → 𝜑) → 𝜓) is increasing. (Contributed by BJ, 3-Apr-2026.)
⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → 𝜃) → ((𝜓 → 𝜒) → 𝜃)))
Theorem	bj-imim11i 37250	The propositional function ((. → 𝜑) → 𝜓) is increasing. Its associated inference is wl-syls2 38272. (Contributed by BJ, 3-Apr-2026.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (((𝜑 → 𝜒) → 𝜃) → ((𝜓 → 𝜒) → 𝜃))
Theorem	bj-peircestab 37251	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 37252). (Contributed by BJ, 30-Nov-2023.) Axiom ax-3 8 is only used through Peirce's law peirce 205. (Proof modification is discouraged.)
⊢ (((((𝜑 → 𝜓) → 𝜒) → 𝜑) → 𝜒) → 𝜒)
Theorem	bj-stabpeirce 37252	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 37251). (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 37253	Implication from equivalence with a conjunct. Its associated inference is simplbi 502. (Contributed by BJ, 20-Mar-2026.)
⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜑 → 𝜓))
Theorem	bj-bisimpr 37254	Implication from equivalence with a conjunct. Its associated inference is simprbi 503. (Contributed by BJ, 20-Mar-2026.)
⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜑 → 𝜒))
Theorem	bj-syl66ib 37255	A mixed syllogism inference derived from imbitrdi 254. Shortens bj-dvelimdv1 37595, alexsubALTlem4 24276 (4821>4812), supsrlem 11120 (2868>2863). (Contributed by BJ, 20-Oct-2021.)
⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜃 → 𝜏)    &   ⊢ (𝜏 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	bj-orim2 37256	Proof of orim2 983 from the axiomatic definition of disjunction (olc 882, orc 881, jao 975) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓)))
Theorem	bj-currypeirce 37257	Curry's axiom curryax 907 (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 205 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the elimination axiom jao 975 via its inference form jaoi 871; the introduction axioms olc 882 and orc 881 are not needed). Note that this theorem shows that actually, the standard instance of curryax 907 implies the standard instance of peirce 205, which is not the case for the converse bj-peircecurry 37258. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
Theorem	bj-peircecurry 37258	Peirce's axiom peirce 205 implies Curry's axiom curryax 907 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the introduction axioms olc 882 and orc 881; the elimination axiom jao 975 is not needed). See bj-currypeirce 37257 for the converse. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ∨ (𝜑 → 𝜓))
Theorem	bj-animbi 37259	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 37260	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 57, 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 37261	A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.)
⊢ (𝜑 → ((𝜓 → ¬ 𝜑) → ¬ 𝜓))
Theorem	bj-con2comi 37262	Inference associated with bj-con2com 37261. Its associated inference is mt2 203. TODO: when in the main part, add to mt2 203 that it is the inference associated with bj-con2comi 37262. (Contributed by BJ, 19-Mar-2020.)
⊢ 𝜑    ⇒   ⊢ ((𝜓 → ¬ 𝜑) → ¬ 𝜓)
Theorem	bj-nimn 37263	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 23 and jc 162, however, the present proof uses theorems that are more basic than jc 162. (Proof modification is discouraged.)
⊢ (𝜑 → ¬ (𝜑 → ¬ 𝜑))
Theorem	bj-nimni 37264	Inference associated with bj-nimn 37263. (Contributed by BJ, 19-Mar-2020.)
⊢ 𝜑    ⇒   ⊢ ¬ (𝜑 → ¬ 𝜑)
Theorem	bj-peircei 37265	Inference associated with peirce 205. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜑)    ⇒   ⊢ 𝜑
Theorem	bj-looinvi 37266	Inference associated with looinv 206. Its associated inference is bj-looinvii 37267. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜓)    ⇒   ⊢ ((𝜓 → 𝜑) → 𝜑)
Theorem	bj-looinvii 37267	Inference associated with bj-looinvi 37266. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜓)    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	bj-mt2bi 37268	Version of mt2 203 where the major premise is a biconditional. Shortens fal 1584 (see bj-fal 37269) . Another proof is also possible via con2bii 360 and mpbi 233. The current mt2bi 366 should be relabeled, maybe to imfal. (Contributed by BJ, 5-Oct-2024.)
⊢ (𝜓 ↔ ¬ 𝜑)    &   ⊢ 𝜑    ⇒   ⊢ ¬ 𝜓
Theorem	bj-fal 37269	Shortening of fal 1584 using bj-mt2bi 37268. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) (Proof modification is discouraged.)
⊢ ¬ ⊥
Theorem	bj-ntrufal 37270	The negation of a theorem is equivalent to false. Shortens dfnul2 4282 (see bj-dfnul2 37271). (Contributed by BJ, 5-Oct-2024.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜑 ↔ ⊥)
Theorem	bj-dfnul2 37271	Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2178, ax-11 2194, and ax-12 2213. (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 pm4.71 567 and pm4.72 964. See also biort 949 and biorf 950.
Theorem	bj-jaoi1 37272	Shortens orfa2 38836 (58>53), pm1.2 917 (20>18), pm1.2 917 (20>18), pm2.4 920 (31>25), pm2.41 921 (31>25), pm2.42 957 (38>32), pm3.2ni 894 (43>39), pm4.44 1012 (55>51). (Contributed by BJ, 30-Sep-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜓) → 𝜓)
Theorem	bj-jaoi2 37273	Shortens consensus 1068 (110>106), elnn0z 12628 (336>329), pm1.2 917 (20>19), pm3.2ni 894 (43>39), pm4.44 1012 (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 862, df-an 402, pm4.64 863, imor 867, pm4.62 870 through pm4.67 404, and, for the De Morgan laws, ianor 997 through pm4.57 1006.
Theorem	bj-dfbi4 37274	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
Theorem	bj-dfbi5 37275	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)))
Theorem	bj-dfbi6 37276	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓)))
Theorem	bj-bijust0ALT 37277	Alternate proof of bijust0 207; 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 37278	A self-implication does not imply the negation of a self-implication. Most general theorem of which bijust 208 is an instance (bijust0 207 and bj-bijust0ALT 37277 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 37279	Version of consensus 1068 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1068, 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 37280	Alternate proof of bj-consensus 37279. (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 37281*	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 2739. We reprove the current df-if 4483 from it in bj-dfif 37282. (Contributed by BJ, 20-Sep-2019.) (Proof modification is discouraged.)
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}
Theorem	bj-dfif 37282*	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 37283	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))
21.19.1.9  Propositional calculus: miscellaneous Miscellaneous theorems of propositional calculus.
Theorem	bj-imbi12 37284	Uncurried (imported) form of imbi12 349. (Contributed by BJ, 6-May-2019.)
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))
Theorem	bj-falor 37285	Dual of truan 1581 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝜑 ↔ (⊥ ∨ 𝜑))
Theorem	bj-falor2 37286	Dual of truan 1581. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
⊢ ((⊥ ∨ 𝜑) ↔ 𝜑)
Theorem	bj-bibibi 37287	A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝜑 ↔ (𝜓 ↔ (𝜑 ↔ 𝜓)))
Theorem	bj-imn3ani 37288	Duplication of bnj1224 35310. Three-fold version of imnani 406. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 22-Oct-2019.) (Proof modification is discouraged.)
⊢ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒)
Theorem	bj-andnotim 37289	Two ways of expressing a certain ternary connective. Note the respective positions of the three formulas on each side of the biconditional. (Contributed by BJ, 6-Oct-2018.)
⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) ∨ 𝜒))
Theorem	bj-bi3ant 37290	This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (((𝜃 → 𝜏) → 𝜑) → (((𝜏 → 𝜃) → 𝜓) → ((𝜃 ↔ 𝜏) → 𝜒)))
Theorem	bj-bisym 37291	This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)
⊢ (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → (((𝜓 → 𝜑) → (𝜃 → 𝜒)) → ((𝜑 ↔ 𝜓) → (𝜒 ↔ 𝜃))))
Theorem	bj-bixor 37292	Equivalence of two ternary operations. Note the identical order and parenthesizing of the three arguments in both expressions. (Contributed by BJ, 31-Dec-2023.)
⊢ ((𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ⊻ (𝜓 ↔ 𝜒)))
21.19.2  Modal logic In this section, we prove some theorems related to modal logic. For modal logic, we refer to https://en.wikipedia.org/wiki/Kripke_semantics, https://en.wikipedia.org/wiki/Modal_logic and https://plato.stanford.edu/entries/logic-modal/. Monadic first-order logic (i.e., with quantification over only one variable) is bi-interpretable with modal logic, by mapping ∀𝑥 to "necessity" (generally denoted by a box) and ∃𝑥 to "possibility" (generally denoted by a diamond). Therefore, we use these quantifiers so as not to introduce new symbols. (To be strictly within modal logic, we should add disjoint variable conditions between 𝑥 and any other metavariables appearing in the statements.) For instance, ax-gen 1828 corresponds to the necessitation rule of modal logic, and ax-4 1842 corresponds to the distributivity axiom (K) of modal logic, also called the Kripke scheme. Modal logics satisfying these rule and axiom are called "normal modal logics", of which the most important modal logics are. The minimal normal modal logic is also denoted by (K). Here are a few normal modal logics with their axiomatizations (on top of (K)): (K) axiomatized by no supplementary axioms; (T) axiomatized by the axiom T; (K4) axiomatized by the axiom 4; (S4) axiomatized by the axioms T,4; (S5) axiomatized by the axioms T,5 or D,B,4; (GL) axiomatized by the axiom GL. The last one, called Gödel–Löb logic or provability logic, is important because it describes exactly the properties of provability in Peano arithmetic, as proved by Robert Solovay. See for instance https://plato.stanford.edu/entries/logic-provability/ 1842. A basic result in this logic is bj-gl4 37296.
Theorem	bj-axdd2 37293	This implication, proved using only ax-gen 1828 and ax-4 1842 on top of propositional calculus (hence holding, up to the standard interpretation, in any normal modal logic), shows that the axiom scheme ⊢ ∃𝑥⊤ implies the axiom scheme ⊢ (∀𝑥𝜑 → ∃𝑥𝜑). These correspond to the modal axiom (D), and in predicate calculus, they assert that the universe of discourse is nonempty. For the converse, see bj-axd2d 37294. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))
Theorem	bj-axd2d 37294	This implication, proved using only ax-gen 1828 on top of propositional calculus (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme ⊢ (∀𝑥𝜑 → ∃𝑥𝜑) implies the axiom scheme ⊢ ∃𝑥⊤ (substitute ⊤ for 𝜑). These correspond to the modal axiom (D), and in predicate calculus, they assert that the universe of discourse is nonempty. For the converse, see bj-axdd2 37293. (Contributed by BJ, 16-May-2019.) Generalize from its instance with ⊤ substituted for 𝜑. (Revised by BJ, 20-Mar-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥⊤ → ∃𝑥𝜑) → ∃𝑥𝜑)
Theorem	bj-axtd 37295	This implication, proved from propositional calculus only (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme ⊢ (∀𝑥𝜑 → 𝜑) (modal T) implies the axiom scheme ⊢ (∀𝑥𝜑 → ∃𝑥𝜑) (modal D). See also bj-axdd2 37293 and bj-axd2d 37294. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑 → 𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))
Theorem	bj-gl4 37296	In a normal modal logic, the modal axiom GL implies the modal axiom (4). Translated to first-order logic, Axiom GL reads ⊢ (∀𝑥(∀𝑥𝜑 → 𝜑) → ∀𝑥𝜑). Note that the antecedent of bj-gl4 37296 is an instance of the axiom GL, with 𝜑 replaced by (∀𝑥𝜑 ∧ 𝜑), which is a modality sometimes called the "strong necessity" of 𝜑. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)) → ∀𝑥(∀𝑥𝜑 ∧ 𝜑)) → (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑))
Theorem	bj-axc4 37297	Over minimal calculus, the modal axiom (4) (hba1 2326) and the modal axiom (K) (ax-4 1842) together imply axc4 2351. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) → ((∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥∀𝑥𝜑 → ∀𝑥𝜓)) → (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))))
21.19.3  Provability logic In this section, we assume that, on top of propositional calculus, there is given a provability predicate Prv satisfying the three axioms ax-prv1 37299 and ax-prv2 37300 and ax-prv3 37301. Note the similarity with ax-gen 1828, ax-4 1842 and hba1 2326 respectively. These three properties of Prv are often called the Hilbert–Bernays–Löb derivability conditions, or the Hilbert–Bernays provability conditions. This corresponds to the modal logic (K4) (see previous section for modal logic). The interpretation of provability logic is the following: we are given a background first-order theory T, the wff Prv 𝜑 means "𝜑 is provable in T", and the turnstile ⊢ indicates provability in T. Beware that "provability logic" often means (K) augmented with the Gödel–Löb axiom GL, which we do not assume here (at least for the moment). See for instance https://plato.stanford.edu/entries/logic-provability/ 2326. Provability logic is worth studying because whenever T is a first-order theory containing Robinson arithmetic (a fragment of Peano arithmetic), one can prove (using Gödel numbering, and in the much weaker primitive recursive arithmetic) that there exists in T a provability predicate Prv satisfying the above three axioms. (We do not construct this predicate in this section; this is still a project.) The main theorems of this section are the "easy parts" of the proofs of Gödel's second incompleteness theorem (bj-babygodel 37304) and Löb's theorem (bj-babylob 37305). See the comments of these theorems for details.
Syntax	cprvb 37298	Syntax for the provability predicate.
wff Prv 𝜑
Axiom	ax-prv1 37299	First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ 𝜑    ⇒   ⊢ Prv 𝜑
Axiom	ax-prv2 37300	Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓))