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Theorem List for Metamath Proof Explorer - 31501-31600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremknoppcnlem11 31501* Lemma for knoppcn 31502. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹𝑧)‘𝑚)))):ℕ0⟶(ℝ–cn→ℂ))
 
Theoremknoppcn 31502* 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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (abs‘𝐶) < 1)       (𝜑𝑊 ∈ (ℝ–cn→ℂ))
 
Theoremknoppcld 31503* 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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (abs‘𝐶) < 1)       (𝜑 → (𝑊𝐴) ∈ ℂ)
 
Theoremaddgtge0d 31504 Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → 0 < (𝐴 + 𝐵))
 
Theoremunblimceq0lem 31505* Lemma for unblimceq0 31506. (Contributed by Asger C. Ipsen, 12-May-2021.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐹:𝑆⟶ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑 → ∀𝑏 ∈ ℝ+𝑑 ∈ ℝ+𝑥𝑆 ((abs‘(𝑥𝐴)) < 𝑑𝑏 ≤ (abs‘(𝐹𝑥))))       (𝜑 → ∀𝑐 ∈ ℝ+𝑑 ∈ ℝ+𝑦𝑆 (𝑦𝐴 ∧ (abs‘(𝑦𝐴)) < 𝑑𝑐 ≤ (abs‘(𝐹𝑦))))
 
Theoremunblimceq0 31506* If 𝐹 is unbounded near 𝐴 it has no limit at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐹:𝑆⟶ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑 → ∀𝑏 ∈ ℝ+𝑑 ∈ ℝ+𝑥𝑆 ((abs‘(𝑥𝐴)) < 𝑑𝑏 ≤ (abs‘(𝐹𝑥))))       (𝜑 → (𝐹 lim 𝐴) = ∅)
 
Theoremunbdqndv1 31507* If the difference quotient (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) is unbounded near 𝐴 then 𝐹 is not differentiable at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.)
𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)))    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑 → ∀𝑏 ∈ ℝ+𝑑 ∈ ℝ+𝑥 ∈ (𝑋 ∖ {𝐴})((abs‘(𝑥𝐴)) < 𝑑𝑏 ≤ (abs‘(𝐺𝑥))))       (𝜑 → ¬ 𝐴 ∈ dom (𝑆 D 𝐹))
 
Theoremunbdqndv2lem1 31508 Lemma for unbdqndv2 31510. (Contributed by Asger C. Ipsen, 12-May-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐷 ≠ 0)    &   (𝜑 → (2 · 𝐸) ≤ (abs‘((𝐴𝐵) / 𝐷)))       (𝜑 → ((𝐸 · (abs‘𝐷)) ≤ (abs‘(𝐴𝐶)) ∨ (𝐸 · (abs‘𝐷)) ≤ (abs‘(𝐵𝐶))))
 
Theoremunbdqndv2lem2 31509* Lemma for unbdqndv2 31510. (Contributed by Asger C. Ipsen, 12-May-2021.)
𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)))    &   𝑊 = if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝑈𝑋)    &   (𝜑𝑉𝑋)    &   (𝜑𝑈𝑉)    &   (𝜑𝑈𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑 → (𝑉𝑈) < 𝐷)    &   (𝜑 → (2 · 𝐵) ≤ ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))       (𝜑 → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
 
Theoremunbdqndv2 31510* Variant of unbdqndv1 31507 with the hypothesis that (((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥)) is unbounded where 𝑥𝐴 and 𝐴𝑦. (Contributed by Asger C. Ipsen, 12-May-2021.)
(𝜑𝑋 ⊆ ℝ)    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑 → ∀𝑏 ∈ ℝ+𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐴𝐴𝑦) ∧ ((𝑦𝑥) < 𝑑𝑥𝑦) ∧ 𝑏 ≤ ((abs‘((𝐹𝑦) − (𝐹𝑥))) / (𝑦𝑥))))       (𝜑 → ¬ 𝐴 ∈ dom (ℝ D 𝐹))
 
Theoremknoppndvlem1 31511 Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐽 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ)
 
Theoremknoppndvlem2 31512 Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ ℤ)    &   (𝜑𝐽 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐽 < 𝐼)       (𝜑 → (((2 · 𝑁)↑𝐼) · ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)) ∈ ℤ)
 
Theoremknoppndvlem3 31513 Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 15-Jun-2021.)
(𝜑𝐶 ∈ (-1(,)1))       (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
 
Theoremknoppndvlem4 31514* Lemma for knoppndv 31533. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → seq0( + , (𝐹𝐴)) ⇝ (𝑊𝐴))
 
Theoremknoppndvlem5 31515* Lemma for knoppndv 31533. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹𝐴)‘𝑖) ∈ ℝ)
 
Theoremknoppndvlem6 31516* Lemma for knoppndv 31533. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝑊𝐴) = Σ𝑖 ∈ (0...𝐽)((𝐹𝐴)‘𝑖))
 
Theoremknoppndvlem7 31517* Lemma for knoppndv 31533. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ((𝐹𝐴)‘𝐽) = ((𝐶𝐽) · (𝑇‘(𝑀 / 2))))
 
Theoremknoppndvlem8 31518* Lemma for knoppndv 31533. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 2 ∥ 𝑀)       (𝜑 → ((𝐹𝐴)‘𝐽) = 0)
 
Theoremknoppndvlem9 31519* Lemma for knoppndv 31533. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝑀)       (𝜑 → ((𝐹𝐴)‘𝐽) = ((𝐶𝐽) / 2))
 
Theoremknoppndvlem10 31520* Lemma for knoppndv 31533. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (abs‘(((𝐹𝐵)‘𝐽) − ((𝐹𝐴)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))
 
Theoremknoppndvlem11 31521* Lemma for knoppndv 31533. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹𝐴)‘𝑖))) ≤ ((abs‘(𝐵𝐴)) · Σ𝑖 ∈ (0...(𝐽 − 1))(((2 · 𝑁) · (abs‘𝐶))↑𝑖)))
 
Theoremknoppndvlem12 31522 Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 29-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)))
 
Theoremknoppndvlem13 31523 Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑𝐶 ≠ 0)
 
Theoremknoppndvlem14 31524* Lemma for knoppndv 31533. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹𝐴)‘𝑖))) ≤ ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))
 
Theoremknoppndvlem15 31525* Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 6-Jul-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ (abs‘((𝑊𝐵) − (𝑊𝐴))))
 
Theoremknoppndvlem16 31526 Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 19-Jul-2021.)
𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐵𝐴) = (((2 · 𝑁)↑-𝐽) / 2))
 
Theoremknoppndvlem17 31527* Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 12-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ((((2 · 𝑁) · (abs‘𝐶))↑𝐽) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ ((abs‘((𝑊𝐵) − (𝑊𝐴))) / (𝐵𝐴)))
 
Theoremknoppndvlem18 31528* Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 14-Aug-2021.)
(𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐺 ∈ ℝ+)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ∃𝑗 ∈ ℕ0 ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · 𝐺)))
 
Theoremknoppndvlem19 31529* Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 17-Aug-2021.)
𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑚)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑚 + 1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝐻 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ∃𝑚 ∈ ℤ (𝐴𝐻𝐻𝐵))
 
Theoremknoppndvlem20 31530 Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 18-Aug-2021.)
(𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+)
 
Theoremknoppndvlem21 31531* Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 18-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   𝐺 = (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐻 ∈ ℝ)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    &   (𝜑 → (((2 · 𝑁)↑-𝐽) / 2) < 𝐷)    &   (𝜑𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝐽) · 𝐺))       (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎𝐻𝐻𝑏) ∧ ((𝑏𝑎) < 𝐷𝑎𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊𝑏) − (𝑊𝑎))) / (𝑏𝑎))))
 
Theoremknoppndvlem22 31532* Lemma for knoppndv 31533. (Contributed by Asger C. Ipsen, 19-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐻 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎𝐻𝐻𝑏) ∧ ((𝑏𝑎) < 𝐷𝑎𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊𝑏) − (𝑊𝑎))) / (𝑏𝑎))))
 
Theoremknoppndv 31533* 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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → dom (ℝ D 𝑊) = ∅)
 
Theoremknoppf 31534* 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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)       (𝜑𝑊:ℝ⟶ℝ)
 
Theoremknoppcn2 31535* Variant of knoppcn 31502 with different codomain. (Contributed by Asger C. Ipsen, 25-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ (-1(,)1))       (𝜑𝑊 ∈ (ℝ–cn→ℝ))
 
Theoremcnndvlem1 31536* Lemma for cnndv 31538. (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 ((𝐹𝑤)‘𝑖))       (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)
 
Theoremcnndvlem2 31537* Lemma for cnndv 31538. (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 ((𝐹𝑤)‘𝑖))       𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)
 
Theoremcnndv 31538 There exists a continuous nowhere differentiable function. The result follows directly from knoppcn 31502 and knoppndv 31533. (Contributed by Asger C. Ipsen, 26-Aug-2021.)
𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)
 
20.14  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.

 
20.14.1  Propositional calculus

Miscellaneous utility theorems of propositional calculus.

 
20.14.1.1  Derived rules of inference

In this section, we prove a few rules of inference derived from modus ponens, and which do not depend on any axioms.

 
Theorembj-mp2c 31539 A double modus ponens inference. (Contributed by BJ, 24-Sep-2019.)
𝜑    &   (𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       𝜒
 
Theorembj-mp2d 31540 A double modus ponens inference. (Contributed by BJ, 24-Sep-2019.)
𝜑    &   (𝜑𝜓)    &   (𝜓 → (𝜑𝜒))       𝜒
 
20.14.1.2  A syntactic theorem

In this section, we prove a syntactic theorem (bj-0 31541) asserting that some formula is well-formed. Then, we use this syntactic theorem to shorten the proof of a "usual" theorem (bj-1 31542) and explain in the comment of that theorem why this phenomenon is unusual.

 
Theorembj-0 31541 A syntactic theorem. See the section comment and the comment of bj-1 31542. 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 1939 and weq 1824. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
wff ((𝜑𝜓) → 𝜒)
 
Theorembj-1 31542 In this proof, the use of the syntactic theorem bj-0 31541 allows to reduce the total length by one (non-essential) step. See also the section comment and the comment of bj-0 31541. Since bj-0 31541 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 31541 as a theorem referencing it). The full proof reads $= wph wps wch bj-0 id $. (while, without using bj-0 31541, 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 1824 or wel 1939). 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 31542 is a special instance of id 22. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

(((𝜑𝜓) → 𝜒) → ((𝜑𝜓) → 𝜒))
 
20.14.1.3  Minimal implicational calculus
 
Theorembj-a1k 31543 Weakening of ax-1 6. This shortens the proofs of dfwe2 6754, ordunisuc2 6817, r111 8401, smo11 7228. (Contributed by BJ, 11-Aug-2020.)
(𝜑 → (𝜓 → (𝜒𝜓)))
 
Theorembj-jarri 31544 Inference associated with jarr 103. Its associated inference is bj-jarrii 31545. (Contributed by BJ, 29-Mar-2020.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)
 
Theorembj-jarrii 31545 Inference associated with bj-jarri 31544. (Contributed by BJ, 29-Mar-2020.)
((𝜑𝜓) → 𝜒)    &   𝜓       𝜒
 
Theorembj-imim2ALT 31546 More direct proof of imim2 55. Note that imim2i 16 and imim2d 54 can be proved as usual from this closed form (i.e., using ax-mp 5 and syl 17 respectively). (Contributed by BJ, 19-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))
 
Theorembj-imim21 31547 The propositional function (𝜒 → (. → 𝜃)) is decreasing. (Contributed by BJ, 19-Jul-2019.)
((𝜑𝜓) → ((𝜒 → (𝜓𝜃)) → (𝜒 → (𝜑𝜃))))
 
Theorembj-imim21i 31548 Inference associated with bj-imim21 31547. Its associated inference is syl5 33. (Contributed by BJ, 19-Jul-2019.)
(𝜑𝜓)       ((𝜒 → (𝜓𝜃)) → (𝜒 → (𝜑𝜃)))
 
20.14.1.4  Positive calculus
 
Theorembj-orim2 31549 Proof of orim2 881 from the axiomatic definition of disjunction (olc 397, orc 398, jao 532) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))
 
Theorembj-curry 31550 A non-intuitionistic positive statement, sometimes called a paradox of material implication. Sometimes called Curry's axiom. (Contributed by BJ, 4-Apr-2021.)
(𝜑 ∨ (𝜑𝜓))
 
Theorembj-peirce 31551 Proof of peirce 191 from minimal implicational calculus, the axiomatic definition of disjunction (olc 397, orc 398, jao 532), and Curry's axiom bj-curry 31550. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)
 
Theorembj-currypeirce 31552 Curry's axiom (a non-intuitionistic statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 191 over minimal implicational calculus and the axiomatic definition of disjunction (olc 397, orc 398, jao 532). A shorter proof from bj-orim2 31549, pm1.2 533, syl6com 36 is possible if we accept to use pm1.2 533, itself a direct consequence of jao 532. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∨ (𝜑𝜓)) → (((𝜑𝜓) → 𝜑) → 𝜑))
 
Theorembj-peircecurry 31553 Peirce's axiom peirce 191 implies Curry's axiom over minimal implicational calculus and the axiomatic definition of disjunction (olc 397, orc 398, jao 532). See comment of bj-currypeirce 31552. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ∨ (𝜑𝜓))
 
20.14.1.5  Implication and negation
 
Theorempm4.81ALT 31554 Alternate proof of pm4.81 379. (Contributed by BJ, 30-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) ↔ 𝜑)
 
Theorembj-con4iALT 31555 Alternate proof of con4i 111. Probably the original proof. (Contributed by BJ, 29-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑 → ¬ 𝜓)       (𝜓𝜑)
 
Theorembj-con2com 31556 A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.)
(𝜑 → ((𝜓 → ¬ 𝜑) → ¬ 𝜓))
 
Theorembj-con2comi 31557 Inference associated with bj-con2com 31556. Its associated inference is mt2 189. TODO: when in the main part, add to mt2 189 that it is the inference associated with bj-con2comi 31557. (Contributed by BJ, 19-Mar-2020.)
𝜑       ((𝜓 → ¬ 𝜑) → ¬ 𝜓)
 
Theorembj-pm2.01i 31558 Inference associated with pm2.01 178. (Contributed by BJ, 30-Mar-2020.)
(𝜑 → ¬ 𝜑)        ¬ 𝜑
 
Theorembj-nimn 31559 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 157, however, the present proof uses theorems that are more basic than jc 157. (Proof modification is discouraged.)
(𝜑 → ¬ (𝜑 → ¬ 𝜑))
 
Theorembj-nimni 31560 Inference associated with bj-nimn 31559. (Contributed by BJ, 19-Mar-2020.)
𝜑        ¬ (𝜑 → ¬ 𝜑)
 
Theorembj-peircei 31561 Inference associated with peirce 191. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜑)       𝜑
 
Theorembj-looinvi 31562 Inference associated with looinv 192. Its associated inference is bj-looinvii 31563. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜓)       ((𝜓𝜑) → 𝜑)
 
Theorembj-looinvii 31563 Inference associated with bj-looinvi 31562. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜓)    &   (𝜓𝜑)       𝜑
 
20.14.1.6  Disjunction

A few lemmas about disjunction. The fundamental theorems in this family are the dual statements pm4.71 659 and pm4.72 915. See also biort 935 and biorf 418.

 
Theorembj-jaoi1 31564 Shortens 11 proofs by a total of around 60 bytes. (Contributed by BJ, 30-Sep-2019.)
(𝜑𝜓)       ((𝜑𝜓) → 𝜓)
 
Theorembj-jaoi2 31565 Shortens 9 proofs by a total of around 50 bytes. (Contributed by BJ, 30-Sep-2019.)
(𝜑𝜓)       ((𝜓𝜑) → 𝜓)
 
20.14.1.7  Logical equivalence

A few other characterizations of the bicondional. The inter-definability of logical connectives offers many ways to express a given statement. Some useful theorems in this regard are df-or 383, df-an 384, pm4.64 385, imor 426, pm4.62 433 through pm4.67 442, and, for the De Morgan laws, ianor 507 through pm4.57 516.

 
Theorembj-dfbi4 31566 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
 
Theorembj-dfbi5 31567 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) → (𝜑𝜓)))
 
Theorembj-dfbi6 31568 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))
 
Theorembj-bijust0 31569 The general statement that bijust 193 proves (with a shorter proof). (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Revised by BJ, 19-Mar-2020.)
¬ ((𝜑𝜑) → ¬ (𝜑𝜑))
 
20.14.1.8  The conditional operator for propositions
 
Theorembj-consensus 31570 Version of consensus 989 expressed using the conditional operator. (Remark: it may be better to express it as consensus 989, 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) ∨ (𝜓𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))
 
Theorembj-consensusALT 31571 Alternate proof of bj-consensus 31570. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))
 
Theorembj-dfifc2 31572* This should be the alternate definition of "ifc" if "if-" enters the main part. (Contributed by BJ, 20-Sep-2019.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))}
 
Theorembj-df-ifc 31573* The definition of "ifc" if "if-" enters the main part. This is in line with the definition of a class as the extension of a predicate in df-clab 2501. (Contributed by BJ, 20-Sep-2019.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
 
Theorembj-ififc 31574* A theorem linking if- and if. (Contributed by BJ, 24-Sep-2019.)
(𝑥 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑥𝐴, 𝑥𝐵))
 
20.14.1.9  Propositional calculus: miscellaneous

Miscellaneous theorems of propositional calculus.

 
Theoremsylancl2 31575 Shortens 5 proofs. (Contributed by BJ, 25-Apr-2019.)
(𝜑𝜓)    &   𝜒    &   ((𝜓𝜒) ↔ 𝜃)       (𝜑𝜃)
 
Theoremsylancl3 31576 Shortens 11 proofs by a total of around 150 bytes. (Contributed by BJ, 25-Apr-2019.)
(𝜑𝜓)    &   𝜒    &   (𝜃 ↔ (𝜓𝜒))       (𝜑𝜃)
 
Theorembj-imbi12 31577 Imported form (uncurried form) of imbi12 334. (Contributed by BJ, 6-May-2019.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) ↔ (𝜓𝜃)))
 
Theorembj-trut 31578 A proposition is equivalent to it being implied by . Closed form of trud 1483 (which it can shorten); dual of dfnot 1492. It is to tbtru 1484 what a1bi 350 is to tbt 357, and this appears in their respective proofs. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
(𝜑 ↔ (⊤ → 𝜑))
 
Theorembj-biorfi 31579 This should be labeled "biorfi" while the current biorfi 420 should be labeled "biorfri". The dual of biorf 418 is not biantr 967 but iba 522 (and ibar 523). So there should also be a "biorfr". (Note that these four statements can actually be strengthened to biconditionals.) (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
¬ 𝜑       (𝜓 ↔ (𝜑𝜓))
 
Theorembj-falor 31580 Dual of truan 1491 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
(𝜑 ↔ (⊥ ∨ 𝜑))
 
Theorembj-falor2 31581 Dual of truan 1491. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
((⊥ ∨ 𝜑) ↔ 𝜑)
 
Theorembj-bibibi 31582 A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
(𝜑 ↔ (𝜓 ↔ (𝜑𝜓)))
 
Theorembj-imn3ani 31583 Duplication of bnj1224 29975. Three-fold version of imnani 437. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 22-Oct-2019.) (Proof modification is discouraged.)
¬ (𝜑𝜓𝜒)       ((𝜑𝜓) → ¬ 𝜒)
 
Theorembj-andnotim 31584 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) ↔ ((𝜑𝜓) ∨ 𝜒))
 
Theorembj-bi3ant 31585 This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)
(𝜑 → (𝜓𝜒))       (((𝜃𝜏) → 𝜑) → (((𝜏𝜃) → 𝜓) → ((𝜃𝜏) → 𝜒)))
 
Theorembj-bisym 31586 This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)
(((𝜑𝜓) → (𝜒𝜃)) → (((𝜓𝜑) → (𝜃𝜒)) → ((𝜑𝜓) → (𝜒𝜃))))
 
20.14.2  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 dv conditions between 𝑥 and any other metavariables appearing in the statements.)

For instance, ax-gen 1700 corresponds to the necessitation rule of modal logic, and ax-4 1713 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/. A basic result in this logic is bj-gl4 31591.

 
Theorembj-axdd2 31587 This implication, proved using only ax-gen 1700 and ax-4 1713 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 31588. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))
 
Theorembj-axd2d 31588 This implication, proved using only ax-gen 1700 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 31587. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤)
 
Theorembj-axtd 31589 This implication, proved from propositional calculus only (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme (∀𝑥𝜑𝜑) (modal T) implies the axiom scheme (∀𝑥𝜑 → ∃𝑥𝜑) (modal D). See also bj-axdd2 31587 and bj-axd2d 31588. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))
 
Theorembj-gl4lem 31590 Lemma for bj-gl4 31591. Note that this proof holds in the modal logic (K). (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)))
 
Theorembj-gl4 31591 In a normal modal logic, the modal axiom GL implies the modal axiom (4). Note that the antecedent of bj-gl4 31591 is an instance of the axiom GL, with 𝜑 replaced by (∀𝑥𝜑𝜑), sometimes called the "strong necessity" of 𝜑. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)) → ∀𝑥(∀𝑥𝜑𝜑)) → (∀𝑥𝜑 → ∀𝑥𝑥𝜑))
 
Theorembj-axc4 31592 Over minimal calculus, the modal axiom (4) (hba1 2026) and the modal axiom (K) (ax-4 1713) together imply axc4 1991. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥𝜑 → ∀𝑥𝑥𝜑) → ((∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝑥𝜑 → ∀𝑥𝜓)) → (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))))
 
20.14.3  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 31594 and ax-prv2 31595 and ax-prv3 31596. Note the similarity with ax-gen 1700, ax-4 1713 and hba1 2026 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/.

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 31599) and Löb's theorem (bj-babylob 31600). See the comments of these theorems for details.

 
Syntaxcprvb 31593 Syntax for the provability predicate.
wff Prv 𝜑
 
Axiomax-prv1 31594 First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
𝜑       Prv 𝜑
 
Axiomax-prv2 31595 Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(Prv (𝜑𝜓) → (Prv 𝜑 → Prv 𝜓))
 
Axiomax-prv3 31596 Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(Prv 𝜑 → Prv Prv 𝜑)
 
Theoremprvlem1 31597 An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(𝜑𝜓)       (Prv 𝜑 → Prv 𝜓)
 
Theoremprvlem2 31598 An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(𝜑 → (𝜓𝜒))       (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒))
 
Theorembj-babygodel 31599 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 ⊥       
 
Theorembj-babylob 31600 See the section header comments for the context, as well as the comments for bj-babygodel 31599.

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/).

(Contributed by BJ, 20-Apr-2019.)

(𝜓 ↔ (Prv 𝜓𝜑))    &   (Prv 𝜑𝜑)       𝜑
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