P. 126 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

negfcncf 12501*

The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)

⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ -(𝐹‘𝑥))    ⇒   ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐺 ∈ (𝐴–cn→ℂ))

Theorem

mulcncflem 12502*

Lemma for mulcncf 12503. (Contributed by Jim Kingdon, 29-May-2023.)

⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → 𝑉 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)    &   ⊢ (𝜑 → 𝐹 ∈ ℝ⁺)    &   ⊢ (𝜑 → 𝐺 ∈ ℝ⁺)    &   ⊢ (𝜑 → 𝑆 ∈ ℝ⁺)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)    &   ⊢ (𝜑 → ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑆 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) < 𝐹))    &   ⊢ (𝜑 → ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑇 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) < 𝐺))    &   ⊢ (𝜑 → ∀𝑢 ∈ 𝑋 (((abs‘(⦋𝑢 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑢 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) → (abs‘((⦋𝑢 / 𝑥⦌𝐴 · ⦋𝑢 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸))    ⇒   ⊢ (𝜑 → ∃𝑑 ∈ ℝ⁺ ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑑 → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑢) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸))

Theorem

mulcncf 12503*

The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋–cn→ℂ))

Theorem

expcncf 12504*

The power function on complex numbers, for fixed exponent N, is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

⊢ (𝑁 ∈ ℕ₀ → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ))

PART 7  BASIC REAL AND COMPLEX ANALYSIS

7.1  Derivatives

7.1.1  Real and complex differentiation

7.1.1.1  Derivatives of functions of one complex or real variable

Syntax

climc 12505

The limit operator.

class lim_ℂ

Syntax

cdv 12506

The derivative operator.

class D

Definition

df-limced 12507*

Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)

⊢ lim_ℂ = (𝑓 ∈ (ℂ ↑_pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦 ∈ ℂ ∣ ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) → (abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒)))})

Definition

df-dvap 12508*

Define the derivative operator. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set 𝑠 here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of ℂ and is well-behaved when 𝑠 contains no isolated points, we will restrict our attention to the cases 𝑠 = ℝ or 𝑠 = ℂ for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)

⊢ D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑_pm 𝑠) ↦ ∪ 𝑥 ∈ ((int‘((MetOpen‘(abs ∘ − )) ↾_t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ {𝑤 ∈ dom 𝑓 ∣ 𝑤 # 𝑥} ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)))

Theorem

limcrcl 12509

Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.)

⊢ (𝐶 ∈ (𝐹 lim_ℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))

Theorem

limccl 12510

Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)

⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ

Theorem

ellimc3ap 12511*

Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) Use apartness. (Revised by Jim Kingdon, 3-Jun-2023.)

⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))

Theorem

limcdifap 12512*

It suffices to consider functions which are not defined at 𝐵 to define the limit of a function. In particular, the value of the original function 𝐹 at 𝐵 does not affect the limit of 𝐹. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)

⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    ⇒   ⊢ (𝜑 → (𝐹 lim_ℂ 𝐵) = ((𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵}) lim_ℂ 𝐵))

Theorem

limcimolemlt 12513*

Lemma for limcimo 12514. (Contributed by Jim Kingdon, 3-Jul-2023.)

⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐾 ↾_t 𝑆))    &   ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → {𝑞 ∈ 𝐶 ∣ 𝑞 # 𝐵} ⊆ 𝐴)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐹 lim_ℂ 𝐵))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐹 lim_ℂ 𝐵))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝐷) → (abs‘((𝐹‘𝑧) − 𝑋)) < ((abs‘(𝑋 − 𝑌)) / 2)))    &   ⊢ (𝜑 → 𝐺 ∈ ℝ⁺)    &   ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 ((𝑤 # 𝐵 ∧ (abs‘(𝑤 − 𝐵)) < 𝐺) → (abs‘((𝐹‘𝑤) − 𝑌)) < ((abs‘(𝑋 − 𝑌)) / 2)))    ⇒   ⊢ (𝜑 → (abs‘(𝑋 − 𝑌)) < (abs‘(𝑋 − 𝑌)))

Theorem

limcimo 12514*

Conditions which ensure there is at most one limit value of 𝐹 at 𝐵. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 8-Jul-2023.)

⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐾 ↾_t 𝑆))    &   ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → {𝑞 ∈ 𝐶 ∣ 𝑞 # 𝐵} ⊆ 𝐴)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐹 lim_ℂ 𝐵))

Theorem

limcresi 12515

Any limit of 𝐹 is also a limit of the restriction of 𝐹. (Contributed by Mario Carneiro, 28-Dec-2016.)

⊢ (𝐹 lim_ℂ 𝐵) ⊆ ((𝐹 ↾ 𝐶) lim_ℂ 𝐵)

Theorem

cnplimcim 12516

If a function is continuous at 𝐵, its limit at 𝐵 equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Jun-2023.)

⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝐽 = (𝐾 ↾_t 𝐴)    ⇒   ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → (𝐹:𝐴⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 lim_ℂ 𝐵))))

Theorem

cnlimcim 12517*

If 𝐹 is a continuous function, the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 16-Jun-2023.)

⊢ (𝐴 ⊆ ℂ → (𝐹 ∈ (𝐴–cn→ℂ) → (𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐹 lim_ℂ 𝑥))))

Theorem

cnlimci 12518

If 𝐹 is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)

⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐷))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 lim_ℂ 𝐵))

Theorem

cnmptlimc 12519*

If 𝐹 is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)

⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑋) ∈ (𝐴–cn→𝐷))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝑥 = 𝐵 → 𝑋 = 𝑌)    ⇒   ⊢ (𝜑 → 𝑌 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑋) lim_ℂ 𝐵))

Theorem

limccnpcntop 12520

If the limit of 𝐹 at 𝐵 is 𝐶 and 𝐺 is continuous at 𝐶, then the limit of 𝐺 ∘ 𝐹 at 𝐵 is 𝐺(𝐶). (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 18-Jun-2023.)

⊢ (𝜑 → 𝐹:𝐴⟶𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℂ)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝐽 = (𝐾 ↾_t 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐹 lim_ℂ 𝐵))    &   ⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶))    ⇒   ⊢ (𝜑 → (𝐺‘𝐶) ∈ ((𝐺 ∘ 𝐹) lim_ℂ 𝐵))

Theorem

reldvg 12521

The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)

⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → Rel (𝑆 D 𝐹))

Theorem

dvlemap 12522*

Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)

⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    &   ⊢ (𝜑 → 𝐷 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑤 ∈ 𝐷 ∣ 𝑤 # 𝐵}) → (((𝐹‘𝐴) − (𝐹‘𝐵)) / (𝐴 − 𝐵)) ∈ ℂ)

Theorem

dvfvalap 12523*

Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)

⊢ 𝑇 = (𝐾 ↾_t 𝑆)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → ((𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑥} ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)) ∧ (𝑆 D 𝐹) ⊆ (((int‘𝑇)‘𝐴) × ℂ)))

Theorem

eldvap 12524*

The differentiable predicate. A function 𝐹 is differentiable at 𝐵 with derivative 𝐶 iff 𝐹 is defined in a neighborhood of 𝐵 and the difference quotient has limit 𝐶 at 𝐵. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)

⊢ 𝑇 = (𝐾 ↾_t 𝑆)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝐺 = (𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝐵} ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → (𝐵(𝑆 D 𝐹)𝐶 ↔ (𝐵 ∈ ((int‘𝑇)‘𝐴) ∧ 𝐶 ∈ (𝐺 lim_ℂ 𝐵))))

Theorem

dvcl 12525

The derivative function takes values in the complex numbers. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    ⇒   ⊢ ((𝜑 ∧ 𝐵(𝑆 D 𝐹)𝐶) → 𝐶 ∈ ℂ)

Theorem

dvbssntrcntop 12526

The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)

⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    &   ⊢ 𝐽 = (𝐾 ↾_t 𝑆)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ ((int‘𝐽)‘𝐴))

Theorem

dvbss 12527

The set of differentiable points is a subset of the domain of the function. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝐴)

Theorem

dvbsssg 12528

The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)

⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → dom (𝑆 D 𝐹) ⊆ 𝑆)

Theorem

recnprss 12529

Both ℝ and ℂ are subsets of ℂ. (Contributed by Mario Carneiro, 10-Feb-2015.)

⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)

Theorem

dvfgg 12530

Explicitly write out the functionality condition on derivative for 𝑆 = ℝ and ℂ. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)

⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)

Theorem

dvfpm 12531

The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 28-Jul-2023.)

⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ)

Theorem

dvfcnpm 12532

The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jul-2023.)

⊢ (𝐹 ∈ (ℂ ↑_pm ℂ) → (ℂ D 𝐹):dom (ℂ D 𝐹)⟶ℂ)

Theorem

dvidlemap 12533*

Lemma for dvid 12535 and dvconst 12534. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)

⊢ (𝜑 → 𝐹:ℂ⟶ℂ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 # 𝑥)) → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) = 𝐵)    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝜑 → (ℂ D 𝐹) = (ℂ × {𝐵}))

Theorem

dvconst 12534

Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)

⊢ (𝐴 ∈ ℂ → (ℂ D (ℂ × {𝐴})) = (ℂ × {0}))

Theorem

dvid 12535

Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)

⊢ (ℂ D ( I ↾ ℂ)) = (ℂ × {1})

PART 8  GUIDES AND MISCELLANEA

8.1  Guides (conventions, explanations, and examples)

8.1.1  Conventions

This section describes the conventions we use. However, these conventions often refer to existing mathematical practices, which are discussed in more detail in other references. The following sources lay out how mathematics is developed without the law of the excluded middle. Of course, there are a greater number of sources which assume excluded middle and most of what is in them applies here too (especially in a treatment such as ours which is built on first order logic and set theory, rather than, say, type theory). Studying how a topic is treated in the Metamath Proof Explorer and the references therein is often a good place to start (and is easy to compare with the Intuitionistic Logic Explorer). The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

• Axioms of propositional calculus - Stanford Encyclopedia of Philosophy or [Heyting].
• Axioms of predicate calculus - our axioms are adapted from the ones in the Metamath Proof Explorer.
• Theorems of propositional calculus - [Heyting].
• Theorems of pure predicate calculus - Metamath Proof Explorer.
• Theorems of equality and substitution - Metamath Proof Explorer.
• Axioms of set theory - [Crosilla].
• Development of set theory - Chapter 10 of [HoTT].
• Construction of real and complex numbers - Chapter 11 of [HoTT]; [BauerTaylor].
• Theorems about real numbers - [Geuvers].

Theorem

conventions 12536

Unless there is a reason to diverge, we follow the conventions of the Metamath Proof Explorer (MPE, set.mm). This list of conventions is intended to be read in conjunction with the corresponding conventions in the Metamath Proof Explorer, and only the differences are described below.

• Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, our choice of IZF (Intuitionistic Zermelo-Fraenkel) over CZF (Constructive Zermelo-Fraenkel, a weaker system) was just an expedient choice because IZF is easier to formalize in Metamath. You can find some development using CZF in BJ's mathbox starting at wbd 12591 (and the section header just above it). As for the axiom of choice, the full axiom of choice implies excluded middle as seen at acexmid 5705, although some authors will use countable choice or dependent choice. For example, countable choice or excluded middle is needed to show that the Cauchy reals coincide with the Dedekind reals - Corollary 11.4.3 of [HoTT], p. (varies).
• Junk/undefined results. Much of the discussion of this topic in the Metamath Proof Explorer applies except that certain techniques are not available to us. For example, the Metamath Proof Explorer will often say "if a function is evaluated within its domain, a certain result follows; if the function is evaluated outside its domain, the same result follows. Since the function must be evaluated within its domain or outside it, the result follows unconditionally" (the use of excluded middle in this argument is perhaps obvious when stated this way). Often, the easiest fix will be to prove we are evaluating functions within their domains, other times it will be possible to use a theorem like relelfvdm 5385 which says that if a function value produces an inhabited set, then the function is being evaluated within its domain.
• Bibliography references. The bibliography for the Intuitionistic Logic Explorer is separate from the one for the Metamath Proof Explorer but feel free to copy-paste a citation in either direction in order to cite it.

Label naming conventions

Here are a few of the label naming conventions:

• Suffixes. We follow the conventions of the Metamath Proof Explorer with a few additions. A biconditional in set.mm which is an implication in iset.mm should have a "r" (for the reverse direction), or "i"/"im" (for the forward direction) appended. A theorem in set.mm which has a decidability condition added should add "dc" to the theorem name. A theorem in set.mm where "nonempty class" is changed to "inhabited class" should add "m" (for member) to the theorem name.
• iset.mm versus set.mm names

  Theorems which are the same as in set.mm should be named the same (that is, where the statement of the theorem is the same; the proof can differ without a new name being called for). Theorems which are different should be named differently (we do have a small number of intentional exceptions to this rule but on the whole it serves us well).

  As for how to choose names so they are different between iset.mm and set.mm, when possible choose a name which reflect the difference in the theorems. For example, if a theorem in set.mm is an equality and the iset.mm analogue is a subset, add "ss" to the iset.mm name. If need be, add "i" to the iset.mm name (usually as a prefix to some portion of the name).

  As with set.mm, we welcome suggestions for better names (such as names which are more consistent with naming conventions).

  We do try to keep set.mm and iset.mm similar where we can. For example, if a theorem exists in both places but the name in set.mm isn't great, we tend to keep that name for iset.mm, or change it in both files together. This is mainly to make it easier to copy theorems, but also to generally help people browse proofs, find theorems, write proofs, etc.

The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.

For the "g" abbreviation, this is related to the set.mm usage, in which "is a set" conditions are converted from hypotheses to antecedents, but is also used where "is a set" conditions are added relative to similar set.mm theorems.

Abbreviation	Mnenomic/Meaning	Source	Expression	Syntax?	Example(s)
ap	apart	df-ap 8210		Yes	apadd1 8236, apne 8251
g	with "is a set" condition			No	1stvalg 5971, brtposg 6081, setsmsbasg 12407
seq3, sum3	recursive sequence	df-seqfrec 10060		Yes	seq3-1 10074, fsum3 10995

Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/forum/#!forum/metamath.

(Contributed by Jim Kingdon, 24-Feb-2020.)

⊢ 𝜑    ⇒   ⊢ 𝜑

8.1.2  Definitional examples

Theorem

ex-or 12537

Example for ax-io 671. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

⊢ (2 = 3 ∨ 4 = 4)

Theorem

ex-an 12538

Example for ax-ia1 105. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

⊢ (2 = 2 ∧ 3 = 3)

Theorem

1kp2ke3k 12539

Example for df-dec 9035, 1000 + 2000 = 3000.

This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with (2 + 1) = 3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 9035 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

⊢ (;;;1000 + ;;;2000) = ;;;3000

Theorem

ex-fl 12540

Example for df-fl 9884. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2)

Theorem

ex-ceil 12541

Example for df-ceil 9885. (Contributed by AV, 4-Sep-2021.)

⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)

Theorem

ex-exp 12542

Example for df-exp 10134. (Contributed by AV, 4-Sep-2021.)

⊢ ((5↑2) = ;25 ∧ (-3↑-2) = (1 / 9))

Theorem

ex-fac 12543

Example for df-fac 10313. (Contributed by AV, 4-Sep-2021.)

⊢ (!‘5) = ;;120

Theorem

ex-bc 12544

Example for df-bc 10335. (Contributed by AV, 4-Sep-2021.)

⊢ (5C3) = ;10

Theorem

ex-dvds 12545

Example for df-dvds 11289: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

⊢ 3 ∥ 6

Theorem

ex-gcd 12546

Example for df-gcd 11431. (Contributed by AV, 5-Sep-2021.)

⊢ (-6 gcd 9) = 3

PART 9  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)

9.1  Mathboxes for user contributions

9.1.1  Mathbox guidelines

Theorem

mathbox 12547

(This theorem is a dummy placeholder for these guidelines. The name of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of iset.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of iset.mm.

Guidelines:

Mathboxes in iset.mm follow the same practices as in set.mm, so refer to the mathbox guidelines there for more details.

(Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (New usage is discouraged.)

⊢ 𝜑    ⇒   ⊢ 𝜑

9.2  Mathbox for BJ

9.2.1  Propositional calculus

Theorem

nnexmid 12548

Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but, of course, does not prove excluded middle) for any formula. (Contributed by BJ, 9-Oct-2019.)

⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑)

Theorem

nndc 12549

Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but, of course, does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.)

⊢ ¬ ¬ DECID 𝜑

Theorem

dcdc 12550

Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.)

⊢ (DECID DECID 𝜑 ↔ DECID 𝜑)

9.2.2  Predicate calculus

Theorem

bj-ex 12551*

Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1545 and 19.9ht 1588 or 19.23ht 1441). (Proof modification is discouraged.)

⊢ (∃𝑥𝜑 → 𝜑)

Theorem

bj-hbalt 12552

Closed form of hbal 1421 (copied from set.mm). (Contributed by BJ, 2-May-2019.)

⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Theorem

bj-nfalt 12553

Closed form of nfal 1523 (copied from set.mm). (Contributed by BJ, 2-May-2019.)

⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)

Theorem

spimd 12554

Deduction form of spim 1684. (Contributed by BJ, 17-Oct-2019.)

⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Theorem

2spim 12555*

Double substitution, as in spim 1684. (Contributed by BJ, 17-Oct-2019.)

⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑧𝜒    &   ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑧∀𝑥𝜓 → 𝜒)

Theorem

ch2var 12556*

Implicit substitution of 𝑦 for 𝑥 and 𝑡 for 𝑧 into a theorem. (Contributed by BJ, 17-Oct-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑧𝜓    &   ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓

Theorem

ch2varv 12557*

Version of ch2var 12556 with non-freeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)

⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓

Theorem

bj-exlimmp 12558

Lemma for bj-vtoclgf 12564. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜒 → 𝜓))

Theorem

bj-exlimmpi 12559

Lemma for bj-vtoclgf 12564. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → 𝜑)    &   ⊢ (𝜒 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)

Theorem

bj-sbimedh 12560

A strengthening of sbiedh 1728 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜒))

Theorem

bj-sbimeh 12561

A strengthening of sbieh 1731 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

Theorem

bj-sbime 12562

A strengthening of sbie 1732 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

9.2.3  Extensionality

Various utility theorems using FOL and extensionality.

Theorem

bj-vtoclgft 12563

Weakening two hypotheses of vtoclgf 2699. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → 𝜑)    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓))

Theorem

bj-vtoclgf 12564

Weakening two hypotheses of vtoclgf 2699. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → 𝜑)    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Theorem

elabgf0 12565

Lemma for elabgf 2780. (Contributed by BJ, 21-Nov-2019.)

⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))

Theorem

elabgft1 12566

One implication of elabgf 2780, in closed form. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))

Theorem

elabgf1 12567

One implication of elabgf 2780. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Theorem

elabgf2 12568

One implication of elabgf 2780. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))

Theorem

elabf1 12569*

One implication of elabf 2781. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Theorem

elabf2 12570*

One implication of elabf 2781. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Theorem

elab1 12571*

One implication of elab 2782. (Contributed by BJ, 21-Nov-2019.)

⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Theorem

elab2a 12572*

One implication of elab 2782. (Contributed by BJ, 21-Nov-2019.)

⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Theorem

elabg2 12573*

One implication of elabg 2783. (Contributed by BJ, 21-Nov-2019.)

⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))

Theorem

bj-rspgt 12574

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2741 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)))

Theorem

bj-rspg 12575

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2741 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))

Theorem

cbvrald 12576*

Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)

⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))

Theorem

bj-intabssel 12577

Version of intss1 3733 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))

Theorem

bj-intabssel1 12578

Version of intss1 3733 using a class abstraction and implicit substitution. Closed form of intmin3 3745. (Contributed by BJ, 29-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))

Theorem

bj-elssuniab 12579

Version of elssuni 3711 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → 𝐴 ⊆ ∪ {𝑥 ∣ 𝜑}))

Theorem

bj-sseq 12580

If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.)

⊢ (𝜑 → (𝜓 ↔ 𝐴 ⊆ 𝐵))    &   ⊢ (𝜑 → (𝜒 ↔ 𝐵 ⊆ 𝐴))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝐴 = 𝐵))

9.2.4  Decidability of classes

The question of decidability is essential in intuitionistic logic. In intuitionistic set theories, it is natural to define decidability of a set (or class) as decidability of membership in it. One can parameterize this notion with another set (or class) since it is often important to assess decidability of membership in one class among elements of another class. Namely, one will say that "𝐴 is decidable in 𝐵 " if ∀𝑥 ∈ 𝐵DECID 𝑥 ∈ 𝐴 (see df-dcin 12582).

Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 12620).

Syntax

wdcin 12581

Syntax for decidability of a class in another.

wff 𝐴 DECID_in 𝐵

Definition

df-dcin 12582*

Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.)

⊢ (𝐴 DECID_in 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)

Theorem

decidi 12583

Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)

⊢ (𝐴 DECID_in 𝐵 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))

Theorem

decidr 12584*

Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)

⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))    ⇒   ⊢ (𝜑 → 𝐴 DECID_in 𝐵)

Theorem

decidin 12585

If A is a decidable subclass of B (meaning: it is a subclass of B and it is decidable in B), and B is decidable in C, then A is decidable in C. (Contributed by BJ, 19-Feb-2022.)

⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐴 DECID_in 𝐵)    &   ⊢ (𝜑 → 𝐵 DECID_in 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 DECID_in 𝐶)

Theorem

uzdcinzz 12586

An upperset of integers is decidable in the integers. Reformulation of eluzdc 9254. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)

⊢ (𝑀 ∈ ℤ → (ℤ_≥‘𝑀) DECID_in ℤ)

Theorem

sumdc2 12587*

Alternate proof of sumdc 10966, without disjoint variable condition on 𝑁, 𝑥 (longer because the statement is taylored to the proof sumdc 10966). (Contributed by BJ, 19-Feb-2022.)

⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))    &   ⊢ (𝜑 → ∀𝑥 ∈ (ℤ_≥‘𝑀)DECID 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → DECID 𝑁 ∈ 𝐴)

9.2.5  Disjoint union

Theorem

djucllem 12588*

Lemma for djulcl 6851 and djurcl 6852. (Contributed by BJ, 4-Jul-2022.)

⊢ 𝑋 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)    ⇒   ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))

Theorem

djulclALT 12589

Shortening of djulcl 6851 using djucllem 12588. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵))

Theorem

djurclALT 12590

Shortening of djurcl 6852 using djucllem 12588. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵))

9.2.6  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes

This section develops constructive Zermelo--Fraenkel set theory (CZF) on top of intuitionistic logic. It is a constructive theory in the sense that its logic is intuitionistic and it is predicative. "Predicative" means that new sets can be constructed only from already constructed sets. In particular, the axiom of separation ax-sep 3986 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 12663. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 3983 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 12765 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 12724. Similarly, the axiom of powerset ax-pow 4038 is not predicative (checking whether a set is included in another requires to universally quantifier over that "not yet constructed" set) and is replaced in CZF by the axiom of fullness or the axiom of subset collection ax-sscoll 12770.

In an intuitionistic context, the axiom of regularity is stated in IZF as well as in CZF as the axiom of set induction ax-setind 4390. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 12751.

For more details on CZF, a useful set of notes is

Peter Aczel and Michael Rathjen, CST Book draft. (available at http://www1.maths.leeds.ac.uk/~rathjen/book.pdf)

and an interesting article is

Michael Shulman, Comparing material and structural set theories, Annals of Pure and Applied Logic, Volume 170, Issue 4 (Apr. 2019), 465--504. (available at https://arxiv.org/abs/1808.05204)

I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results.

9.2.6.1  Bounded formulas

The present definition of bounded formulas emerged from a discussion on GitHub between Jim Kingdon, Mario Carneiro and I, started 23-Sept-2019 (see https://github.com/metamath/set.mm/issues/1173 and links therein).

In order to state certain axiom schemes of Constructive Zermelo–Fraenkel (CZF) set theory, like the axiom scheme of bounded (or restricted, or Δ₀) separation, it is necessary to distinguish certain formulas, called bounded (or restricted, or Δ₀) formulas. The necessity of considering bounded formulas also arises in several theories of bounded arithmetic, both classical or intuitonistic, for instance to state the axiom scheme of Δ₀-induction.

To formalize this in Metamath, there are several choices to make.

A first choice is to either create a new type for bounded formulas, or to create a predicate on formulas that indicates whether they are bounded. In the first case, one creates a new type "wff0" with a new set of metavariables (ph₀ ...) and an axiom "$a wff ph₀ " ensuring that bounded formulas are formulas, so that one can reuse existing theorems, and then axioms take the form "$a wff0 ( ph₀ -> ps₀ )", etc. In the second case, one introduces a predicate "BOUNDED " with the intended meaning that "BOUNDED 𝜑 " is a formula meaning that 𝜑 is a bounded formula. We choose the second option, since the first would complicate the grammar, risking to make it ambiguous. (TODO: elaborate.)

A second choice is to view "bounded" either as a syntactic or a semantic property. For instance, ∀𝑥⊤ is not syntactically bounded since it has an unbounded universal quantifier, but it is semantically bounded since it is equivalent to ⊤ which is bounded. We choose the second option, so that formulas using defined symbols can be proved bounded.

A third choice is in the form of the axioms, either in closed form or in inference form. One cannot state all the axioms in closed form, especially ax-bd0 12592. Indeed, if we posited it in closed form, then we could prove for instance ⊢ (𝜑 → BOUNDED 𝜑) and ⊢ (¬ 𝜑 → BOUNDED 𝜑) which is problematic (with the law of excluded middle, this would entail that all formulas are bounded, but even without it, too many formulas could be proved bounded...). (TODO: elaborate.)

Having ax-bd0 12592 in inference form ensures that a formula can be proved bounded only if it is equivalent *for all values of the free variables* to a syntactically bounded one. The other axioms (ax-bdim 12593 through ax-bdsb 12601) can be written either in closed or inference form. The fact that ax-bd0 12592 is an inference is enough to ensure that the closed forms cannot be "exploited" to prove that some unbounded formulas are bounded. (TODO: check.) However, we state all the axioms in inference form to make it clear that we do not exploit any over-permissiveness.

Finally, note that our logic has no terms, only variables. Therefore, we cannot prove for instance that 𝑥 ∈ ω is a bounded formula. However, since ω can be defined as "the 𝑦 such that PHI" a proof using the fact that 𝑥 ∈ ω is bounded can be converted to a proof in iset.mm by replacing ω with 𝑦 everywhere and prepending the antecedent PHI, since 𝑥 ∈ 𝑦 is bounded by ax-bdel 12600. For a similar method, see bj-omtrans 12739.

Note that one cannot add an axiom ⊢ BOUNDED 𝑥 ∈ 𝐴 since by bdph 12629 it would imply that every formula is bounded.

Syntax

wbd 12591

Syntax for the predicate BOUNDED.

wff BOUNDED 𝜑

Axiom

ax-bd0 12592

If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.)

⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)

Axiom

ax-bdim 12593

An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)

⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 → 𝜓)

Axiom

ax-bdan 12594

The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)

⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ∧ 𝜓)

Axiom

ax-bdor 12595

The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)

⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ∨ 𝜓)

Axiom

ax-bdn 12596

The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)

⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ¬ 𝜑

Axiom

ax-bdal 12597*

A bounded universal quantification of a bounded formula is bounded. Note the disjoint variable condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.)

⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝜑

Axiom

ax-bdex 12598*

A bounded existential quantification of a bounded formula is bounded. Note the disjoint variable condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.)

⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑

Axiom

ax-bdeq 12599

An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED 𝑥 = 𝑦

Axiom

ax-bdel 12600

An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)

⊢ BOUNDED 𝑥 ∈ 𝑦