P. 366 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36501-36600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	unblimceq0lem 36501*	Lemma for unblimceq0 36502. (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 36502*	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 36503*	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 36504	Lemma for unbdqndv2 36506. (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 36505*	Lemma for unbdqndv2 36506. (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 36506*	Variant of unbdqndv1 36503 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 36507	Lemma for knoppndv 36529. (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 36508	Lemma for knoppndv 36529. (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 36509	Lemma for knoppndv 36529. (Contributed by Asger C. Ipsen, 15-Jun-2021.)
⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    ⇒   ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
Theorem	knoppndvlem4 36510*	Lemma for knoppndv 36529. (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 36511*	Lemma for knoppndv 36529. (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 36512*	Lemma for knoppndv 36529. (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 36513*	Lemma for knoppndv 36529. (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 36514*	Lemma for knoppndv 36529. (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 36515*	Lemma for knoppndv 36529. (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 36516*	Lemma for knoppndv 36529. (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 36517*	Lemma for knoppndv 36529. (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 36518	Lemma for knoppndv 36529. (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 36519	Lemma for knoppndv 36529. (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 36520*	Lemma for knoppndv 36529. (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 36521*	Lemma for knoppndv 36529. (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 36522	Lemma for knoppndv 36529. (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 36523*	Lemma for knoppndv 36529. (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 36524*	Lemma for knoppndv 36529. (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 36525*	Lemma for knoppndv 36529. (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 36526	Lemma for knoppndv 36529. (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 36527*	Lemma for knoppndv 36529. (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 36528*	Lemma for knoppndv 36529. (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 36529*	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 36530*	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 36531*	Variant of knoppcn 36499 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 36532*	Lemma for cnndv 36534. (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 36533*	Lemma for cnndv 36534. (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 36534	There exists a continuous nowhere differentiable function. The result follows directly from knoppcn 36499 and knoppndv 36529. (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 36535	A double modus ponens inference. Inference associated with mpd 15. (Contributed by BJ, 24-Sep-2019.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ 𝜒
Theorem	bj-mp2d 36536	A double modus ponens inference. Inference associated with mpcom 38. (Contributed by BJ, 24-Sep-2019.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → (𝜑 → 𝜒))    ⇒   ⊢ 𝜒
21.19.1.2  A syntactic theorem In this section, we prove a syntactic theorem (bj-0 36537) asserting that some formula is well-formed. Then, we use this syntactic theorem to shorten the proof of a "usual" theorem (bj-1 36538) and explain in the comment of that theorem why this phenomenon is unusual.
Theorem	bj-0 36537	A syntactic theorem. See the section comment and the comment of bj-1 36538. 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 2110 and weq 1962. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
wff ((𝜑 → 𝜓) → 𝜒)
Theorem	bj-1 36538	In this proof, the use of the syntactic theorem bj-0 36537 allows to reduce the total length by one (non-essential) step. See also the section comment and the comment of bj-0 36537. Since bj-0 36537 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 36537 as a theorem referencing it). The full proof reads $= wph wps wch bj-0 id $. (while, without using bj-0 36537, 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 1962 or wel 2110). 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 36538 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
Theorem	bj-a1k 36539	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 36539 does not require ax-2 7). This shortens the proofs of dfwe2 7753 (937>925), ordunisuc2 7823 (789>777), r111 9735 (558>545), smo11 8336 (1176>1164). (Contributed by BJ, 11-Aug-2020.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜓)))
Theorem	bj-poni 36540	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 36541	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 202 and pm2.27 42 chained using syl 17. (Contributed by BJ, 4-Dec-2023.)
⊢ (((𝜑 → 𝜓) → 𝜑) → ((𝜑 → 𝜓) → 𝜓))
Theorem	bj-nnclavi 36542	Inference associated with bj-nnclav 36541. Its associated inference is an instance of syl 17. Notice the non-intuitionistic proof from bj-peircei 36560 and bj-poni 36540. (Contributed by BJ, 30-Jul-2024.)
⊢ ((𝜑 → 𝜓) → 𝜑)    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜓)
Theorem	bj-nnclavc 36543	Commuted form of bj-nnclav 36541. Notice the non-intuitionistic proof from bj-peircei 36560 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 36544	Inference associated with bj-nnclavc 36543. Its associated inference is an instance of syl 17. Notice the non-intuitionistic proof from peirce 202 and syl 17. (Contributed by BJ, 30-Jul-2024.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜓)
Theorem	bj-jarrii 36545	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 36546	The propositional function (𝜒 → (. → 𝜃)) is decreasing. (Contributed by BJ, 19-Jul-2019.)
⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜓 → 𝜃)) → (𝜒 → (𝜑 → 𝜃))))
Theorem	bj-imim21i 36547	Inference associated with bj-imim21 36546. Its associated inference is syl5 34. (Contributed by BJ, 19-Jul-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 → (𝜓 → 𝜃)) → (𝜒 → (𝜑 → 𝜃)))
Theorem	bj-peircestab 36548	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 36549). (Contributed by BJ, 30-Nov-2023.) Axiom ax-3 8 is only used through Peirce's law peirce 202. (Proof modification is discouraged.)
⊢ (((((𝜑 → 𝜓) → 𝜒) → 𝜑) → 𝜒) → 𝜒)
Theorem	bj-stabpeirce 36549	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 36548). (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.
Theorem	bj-syl66ib 36550	A mixed syllogism inference derived from imbitrdi 251. In addition to bj-dvelimdv1 36847, it can also shorten alexsubALTlem4 23944 (4821>4812), supsrlem 11071 (2868>2863). (Contributed by BJ, 20-Oct-2021.)
⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜃 → 𝜏)    &   ⊢ (𝜏 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	bj-orim2 36551	Proof of orim2 969 from the axiomatic definition of disjunction (olc 868, orc 867, jao 962) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓)))
Theorem	bj-currypeirce 36552	Curry's axiom curryax 893 (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 202 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the elimination axiom jao 962 via its inference form jaoi 857; the introduction axioms olc 868 and orc 867 are not needed). Note that this theorem shows that actually, the standard instance of curryax 893 implies the standard instance of peirce 202, which is not the case for the converse bj-peircecurry 36553. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
Theorem	bj-peircecurry 36553	Peirce's axiom peirce 202 implies Curry's axiom curryax 893 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the introduction axioms olc 868 and orc 867; the elimination axiom jao 962 is not needed). See bj-currypeirce 36552 for the converse. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ∨ (𝜑 → 𝜓))
Theorem	bj-animbi 36554	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 36555	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
Theorem	bj-con2com 36556	A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.)
⊢ (𝜑 → ((𝜓 → ¬ 𝜑) → ¬ 𝜓))
Theorem	bj-con2comi 36557	Inference associated with bj-con2com 36556. Its associated inference is mt2 200. TODO: when in the main part, add to mt2 200 that it is the inference associated with bj-con2comi 36557. (Contributed by BJ, 19-Mar-2020.)
⊢ 𝜑    ⇒   ⊢ ((𝜓 → ¬ 𝜑) → ¬ 𝜓)
Theorem	bj-nimn 36558	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 36559	Inference associated with bj-nimn 36558. (Contributed by BJ, 19-Mar-2020.)
⊢ 𝜑    ⇒   ⊢ ¬ (𝜑 → ¬ 𝜑)
Theorem	bj-peircei 36560	Inference associated with peirce 202. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜑)    ⇒   ⊢ 𝜑
Theorem	bj-looinvi 36561	Inference associated with looinv 203. Its associated inference is bj-looinvii 36562. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜓)    ⇒   ⊢ ((𝜓 → 𝜑) → 𝜑)
Theorem	bj-looinvii 36562	Inference associated with bj-looinvi 36561. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜓)    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	bj-mt2bi 36563	Version of mt2 200 where the major premise is a biconditional. Another proof is also possible via con2bii 357 and mpbi 230. The current mt2bi 363 should be relabeled, maybe to imfal. (Contributed by BJ, 5-Oct-2024.)
⊢ 𝜑    &   ⊢ (𝜓 ↔ ¬ 𝜑)    ⇒   ⊢ ¬ 𝜓
Theorem	bj-ntrufal 36564	The negation of a theorem is equivalent to false. This can shorten dfnul2 4302. (Contributed by BJ, 5-Oct-2024.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜑 ↔ ⊥)
Theorem	bj-fal 36565	Shortening of fal 1554 using bj-mt2bi 36563. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) (Proof modification is discouraged.)
⊢ ¬ ⊥
21.19.1.6  Disjunction A few lemmas about disjunction. The fundamental theorems in this family are the dual statements pm4.71 557 and pm4.72 951. See also biort 935 and biorf 936.
Theorem	bj-jaoi1 36566	Shortens orfa2 38087 (58>53), pm1.2 903 (20>18), pm1.2 903 (20>18), pm2.4 906 (31>25), pm2.41 907 (31>25), pm2.42 944 (38>32), pm3.2ni 880 (43>39), pm4.44 998 (55>51). (Contributed by BJ, 30-Sep-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜓) → 𝜓)
Theorem	bj-jaoi2 36567	Shortens consensus 1052 (110>106), elnn0z 12549 (336>329), pm1.2 903 (20>19), pm3.2ni 880 (43>39), pm4.44 998 (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 848, df-an 396, pm4.64 849, imor 853, pm4.62 856 through pm4.67 398, and, for the De Morgan laws, ianor 983 through pm4.57 992.
Theorem	bj-dfbi4 36568	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
Theorem	bj-dfbi5 36569	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)))
Theorem	bj-dfbi6 36570	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓)))
Theorem	bj-bijust0ALT 36571	Alternate proof of bijust0 204; 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 36572	A self-implication does not imply the negation of a self-implication. Most general theorem of which bijust 205 is an instance (bijust0 204 and bj-bijust0ALT 36571 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 36573	Version of consensus 1052 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1052, 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 36574	Alternate proof of bj-consensus 36573. (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 36575*	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 2709. We reprove the current df-if 4492 from it in bj-dfif 36576. (Contributed by BJ, 20-Sep-2019.) (Proof modification is discouraged.)
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}
Theorem	bj-dfif 36576*	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 36577	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 36578	Uncurried (imported) form of imbi12 346. (Contributed by BJ, 6-May-2019.)
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))
Theorem	bj-falor 36579	Dual of truan 1551 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝜑 ↔ (⊥ ∨ 𝜑))
Theorem	bj-falor2 36580	Dual of truan 1551. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
⊢ ((⊥ ∨ 𝜑) ↔ 𝜑)
Theorem	bj-bibibi 36581	A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝜑 ↔ (𝜓 ↔ (𝜑 ↔ 𝜓)))
Theorem	bj-imn3ani 36582	Duplication of bnj1224 34798. Three-fold version of imnani 400. (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 36583	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 36584	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 36585	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 36586	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 1795 corresponds to the necessitation rule of modal logic, and ax-4 1809 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/ 1809. A basic result in this logic is bj-gl4 36590.
Theorem	bj-axdd2 36587	This implication, proved using only ax-gen 1795 and ax-4 1809 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 36588. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))
Theorem	bj-axd2d 36588	This implication, proved using only ax-gen 1795 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⊤. 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 36587. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤)
Theorem	bj-axtd 36589	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 36587 and bj-axd2d 36588. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑 → 𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))
Theorem	bj-gl4 36590	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 36590 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 36591	Over minimal calculus, the modal axiom (4) (hba1 2293) and the modal axiom (K) (ax-4 1809) together imply axc4 2320. (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 36593 and ax-prv2 36594 and ax-prv3 36595. Note the similarity with ax-gen 1795, ax-4 1809 and hba1 2293 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/ 2293. 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 36598) and Löb's theorem (bj-babylob 36599). See the comments of these theorems for details.
Syntax	cprvb 36592	Syntax for the provability predicate.
wff Prv 𝜑
Axiom	ax-prv1 36593	First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ 𝜑    ⇒   ⊢ Prv 𝜑
Axiom	ax-prv2 36594	Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓))
Axiom	ax-prv3 36595	Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (Prv 𝜑 → Prv Prv 𝜑)
Theorem	prvlem1 36596	An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (Prv 𝜑 → Prv 𝜓)
Theorem	prvlem2 36597	An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒))
Theorem	bj-babygodel 36598	See the section header comments for the context. The first hypothesis reads "𝜑 is true if and only if it is not provable in T" (and having this first hypothesis means that we can prove this fact in T). The wff 𝜑 is a formal version of the sentence "This sentence is not provable". The hard part of the proof of Gödel's theorem is to construct such a 𝜑, called a "Gödel–Rosser sentence", for a first-order theory T which is effectively axiomatizable and contains Robinson arithmetic, through Gödel diagonalization (this can be done in primitive recursive arithmetic). The second hypothesis means that ⊥ is not provable in T, that is, that the theory T is consistent (and having this second hypothesis means that we can prove in T that the theory T is consistent). The conclusion is the falsity, so having the conclusion means that T can prove the falsity, that is, T is inconsistent. Therefore, taking the contrapositive, this theorem expresses that if a first-order theory is consistent (and one can prove in it that some formula is true if and only if it is not provable in it), then this theory does not prove its own consistency. This proof is due to George Boolos, Gödel's Second Incompleteness Theorem Explained in Words of One Syllable, Mind, New Series, Vol. 103, No. 409 (January 1994), pp. 1--3. (Contributed by BJ, 3-Apr-2019.)
⊢ (𝜑 ↔ ¬ Prv 𝜑)    &   ⊢ ¬ Prv ⊥    ⇒   ⊢ ⊥
Theorem	bj-babylob 36599	See the section header comments for the context, as well as the comments for bj-babygodel 36598. Löb's theorem when the Löb sentence is given as a hypothesis (the hard part of the proof of Löb's theorem is to construct this Löb sentence; this can be done, using Gödel diagonalization, for any first-order effectively axiomatizable theory containing Robinson arithmetic). More precisely, the present theorem states that if a first-order theory proves that the provability of a given sentence entails its truth (and if one can construct in this theory a provability predicate and a Löb sentence, given here as the first hypothesis), then the theory actually proves that sentence. See for instance, Eliezer Yudkowsky, The Cartoon Guide to Löb's Theorem (available at http://yudkowsky.net/rational/lobs-theorem/ 36598). (Contributed by BJ, 20-Apr-2019.)
⊢ (𝜓 ↔ (Prv 𝜓 → 𝜑))    &   ⊢ (Prv 𝜑 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	bj-godellob 36600	Proof of Gödel's theorem from Löb's theorem (see comments at bj-babygodel 36598 and bj-babylob 36599 for details). (Contributed by BJ, 20-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ↔ ¬ Prv 𝜑)    &   ⊢ ¬ Prv ⊥    ⇒   ⊢ ⊥