P. 370 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

rdgsucuni 36901

If an ordinal number has a predecessor, the value of the recursive definition generator at that number in terms of its predecessor. (Contributed by ML, 17-Oct-2020.)

⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (rec(𝐹, 𝐼)‘𝐵) = (𝐹‘(rec(𝐹, 𝐼)‘∪ 𝐵)))

Theorem

rdgeqoa 36902

If a recursive function with an initial value 𝐴 at step 𝑁 is equal to itself with an initial value 𝐵 at step 𝑀, then every finite number of successor steps will also be equal. (Contributed by ML, 21-Oct-2020.)

⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +_o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +_o 𝑋))))

Theorem

elxp8 36903

Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 8022. (Contributed by ML, 19-Oct-2020.)

⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))

Theorem

cbveud 36904*

Deduction used to change bound variables in an existential uniqueness quantifier, using implicit substitution. (Contributed by ML, 27-Mar-2021.)

⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑦𝜒))

Theorem

cbvreud 36905*

Deduction used to change bound variables in a restricted existential uniqueness quantifier. (Contributed by ML, 27-Mar-2021.)

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

Theorem

difunieq 36906

The difference of unions is a subset of the union of the difference. (Contributed by ML, 29-Mar-2021.)

⊢ (∪ 𝐴 ∖ ∪ 𝐵) ⊆ ∪ (𝐴 ∖ 𝐵)

Theorem

inunissunidif 36907

Theorem about subsets of the difference of unions. (Contributed by ML, 29-Mar-2021.)

⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))

Theorem

rdgellim 36908

Elementhood in a recursive definition at a limit ordinal. (Contributed by ML, 30-Mar-2022.)

⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵)))

Theorem

rdglimss 36909

A recursive definition at a limit ordinal is a superset of itself at any smaller ordinal. (Contributed by ML, 30-Mar-2022.)

⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (rec(𝐹, 𝐴)‘𝐶) ⊆ (rec(𝐹, 𝐴)‘𝐵))

Theorem

rdgssun 36910*

In a recursive definition where each step expands on the previous one using a union, every previous step is a subset of every later step. (Contributed by ML, 1-Apr-2022.)

⊢ 𝐹 = (𝑤 ∈ V ↦ (𝑤 ∪ 𝐵))    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ 𝑋) → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))

Theorem

exrecfnlem 36911*

Lemma for exrecfn 36912. (Contributed by ML, 30-Mar-2022.)

⊢ 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 𝐵 ∈ 𝑊) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥))

Theorem

exrecfn 36912*

Theorem about the existence of infinite recursive sets. 𝑦 should usually be free in 𝐵. (Contributed by ML, 30-Mar-2022.)

⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 𝐵 ∈ 𝑊) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥))

Theorem

exrecfnpw 36913*

For any base set, a set which contains the powerset of all of its own elements exists. (Contributed by ML, 30-Mar-2022.)

⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ∈ 𝑥))

Theorem

finorwe 36914

If the Axiom of Infinity is denied, every total order is a well-order. The notion of a well-order cannot be usefully expressed without the Axiom of Infinity due to the inability to quantify over proper classes. (Contributed by ML, 5-Oct-2023.)

⊢ (¬ ω ∈ V → ( < Or 𝐴 → < We 𝐴))

21.19.2  Cartesian exponentiation

Syntax

cfinxp 36915

Extend the definition of a class to include Cartesian exponentiation.

class (𝑈↑↑𝑁)

Definition

df-finxp 36916*

Define Cartesian exponentiation on a class.

Note that this definition is limited to finite exponents, since it is defined using nested ordered pairs. If tuples of infinite length are needed, or if they might be needed in the future, use df-ixp 8910 or df-map 8840 instead. The main advantage of this definition is that it integrates better with functions and relations. For example if 𝑅 is a subset of (𝐴↑↑2_o), then df-br 5145 can be used on it, and df-fv 6551 can also be used, and so on.

It's also worth keeping in mind that ((𝑈↑↑𝑀) × (𝑈↑↑𝑁)) is generally not equal to (𝑈↑↑(𝑀 +_o 𝑁)).

This definition is technical. Use finxp1o 36924 and finxpsuc 36930 for a more standard recursive experience. (Contributed by ML, 16-Oct-2020.)

⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}

Theorem

dffinxpf 36917*

This theorem is the same as Definition df-finxp 36916, except that the large function is replaced by a class variable for brevity. (Contributed by ML, 24-Oct-2020.)

⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))}

Theorem

finxpeq1 36918

Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)

⊢ (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁))

Theorem

finxpeq2 36919

Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)

⊢ (𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁))

Theorem

csbfinxpg 36920*

Distribute proper substitution through Cartesian exponentiation. (Contributed by ML, 25-Oct-2020.)

⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑈↑↑𝑁) = (⦋𝐴 / 𝑥⦌𝑈↑↑⦋𝐴 / 𝑥⦌𝑁))

Theorem

finxpreclem1 36921*

Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)

⊢ (𝑋 ∈ 𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1_o, 𝑋⟩))

Theorem

finxpreclem2 36922*

Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)

⊢ ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1_o, 𝑋⟩))

Theorem

finxp0 36923

The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.)

⊢ (𝑈↑↑∅) = ∅

Theorem

finxp1o 36924

The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.)

⊢ (𝑈↑↑1_o) = 𝑈

Theorem

finxpreclem3 36925*

Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.)

⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ (((𝑁 ∈ ω ∧ 2_o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1^st ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Theorem

finxpreclem4 36926*

Lemma for ↑↑ recursion theorems. (Contributed by ML, 23-Oct-2020.)

⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ (((𝑁 ∈ ω ∧ 2_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨∪ 𝑁, (1^st ‘𝑦)⟩)‘∪ 𝑁))

Theorem

finxpreclem5 36927*

Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)

⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ ((𝑛 ∈ ω ∧ 1_o ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))

Theorem

finxpreclem6 36928*

Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)

⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ ((𝑁 ∈ ω ∧ 1_o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))

Theorem

finxpsuclem 36929*

Lemma for finxpsuc 36930. (Contributed by ML, 24-Oct-2020.)

⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Theorem

finxpsuc 36930

The value of Cartesian exponentiation at a successor. (Contributed by ML, 24-Oct-2020.)

⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Theorem

finxp2o 36931

The value of Cartesian exponentiation at two. (Contributed by ML, 19-Oct-2020.)

⊢ (𝑈↑↑2_o) = (𝑈 × 𝑈)

Theorem

finxp3o 36932

The value of Cartesian exponentiation at three. (Contributed by ML, 24-Oct-2020.)

⊢ (𝑈↑↑3_o) = ((𝑈 × 𝑈) × 𝑈)

Theorem

finxpnom 36933

Cartesian exponentiation when the exponent is not a natural number defaults to the empty set. (Contributed by ML, 24-Oct-2020.)

⊢ (¬ 𝑁 ∈ ω → (𝑈↑↑𝑁) = ∅)

Theorem

finxp00 36934

Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.)

⊢ (∅↑↑𝑁) = ∅

21.19.3  Topology

Theorem

iunctb2 36935

Using the axiom of countable choice ax-cc 10453, the countable union of countable sets is countable. See iunctb 10592 for a somewhat more general theorem. (Contributed by ML, 10-Dec-2020.)

⊢ (∀𝑥 ∈ ω 𝐵 ≼ ω → ∪ 𝑥 ∈ ω 𝐵 ≼ ω)

Theorem

domalom 36936*

A class which dominates every natural number is not finite. (Contributed by ML, 14-Dec-2020.)

⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ¬ 𝐴 ∈ Fin)

Theorem

isinf2 36937*

The converse of isinf 9278. Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by ML, 14-Dec-2020.)

⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ¬ 𝐴 ∈ Fin)

Theorem

ctbssinf 36938*

Using the axiom of choice, any infinite class has a countable subset. (Contributed by ML, 14-Dec-2020.)

⊢ (¬ 𝐴 ∈ Fin → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))

Theorem

ralssiun 36939*

The index set of an indexed union is a subset of the union when each 𝐵 contains its index. (Contributed by ML, 16-Dec-2020.)

⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Theorem

nlpineqsn 36940*

For every point 𝑝 of a subset 𝐴 of 𝑋 with no limit points, there exists an open set 𝑛 that intersects 𝐴 only at 𝑝. (Contributed by ML, 23-Mar-2021.)

⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))

Theorem

nlpfvineqsn 36941*

Given a subset 𝐴 of 𝑋 with no limit points, there exists a function from each point 𝑝 of 𝐴 to an open set intersecting 𝐴 only at 𝑝. This proof uses the axiom of choice. (Contributed by ML, 23-Mar-2021.)

⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐴 ∈ 𝑉 → ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑓(𝑓:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝑓‘𝑝) ∩ 𝐴) = {𝑝})))

Theorem

fvineqsnf1 36942*

A theorem about functions where the image of every point intersects the domain only at that point. If 𝐽 is a topology and 𝐴 is a set with no limit points, then there exists an 𝐹 such that this antecedent is true. See nlpfvineqsn 36941 for a proof of this fact. (Contributed by ML, 23-Mar-2021.)

⊢ ((𝐹:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴–1-1→𝐽)

Theorem

fvineqsneu 36943*

A theorem about functions where the image of every point intersects the domain only at that point. (Contributed by ML, 27-Mar-2021.)

⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑞 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞 ∈ 𝑥)

Theorem

fvineqsneq 36944*

A theorem about functions where the image of every point intersects the domain only at that point. (Contributed by ML, 28-Mar-2021.)

⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑍 ⊆ ran 𝐹 ∧ 𝐴 ⊆ ∪ 𝑍)) → 𝑍 = ran 𝐹)

21.19.3.1  Pi-base theorems

This section contains a few proofs of theorems found in the pi-base database. The pi-base site can be found at <https://topology.pi-base.org/>.

Definitions of topological properties are theorems labeled pibpN, where N is the property number in pi-base. For example, pibp19 36946 defines countably compact topologies. Proofs of theorems are similarly labeled pibtN, for example pibt2 36949.

Theorem

pibp16 36945*

Property P000016 of pi-base. The class of compact topologies. A space 𝑋 is compact if every open cover of 𝑋 has a finite subcover. This theorem is just a relabeled copy of iscmp 23305. (Contributed by ML, 8-Dec-2020.)

⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Theorem

pibp19 36946*

Property P000019 of pi-base. The class of countably compact topologies. A space 𝑋 is countably compact if every countable open cover of 𝑋 has a finite subcover. (Contributed by ML, 8-Dec-2020.)

⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}    ⇒   ⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽((𝑋 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Theorem

pibp21 36947*

Property P000021 of pi-base. The class of weakly countably compact topologies, or limit point compact topologies. A space 𝑋 is weakly countably compact if every infinite subset of 𝑋 has a limit point. (Contributed by ML, 9-Dec-2020.)

⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑊 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ (𝒫 ∪ 𝑥 ∖ Fin)∃𝑧 ∈ ∪ 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦)}    ⇒   ⊢ (𝐽 ∈ 𝑊 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ (𝒫 𝑋 ∖ Fin)∃𝑧 ∈ 𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))

Theorem

pibt1 36948*

Theorem T000001 of pi-base. A compact topology is also countably compact. See pibp16 36945 and pibp19 36946 for the definitions of the relevant properties. (Contributed by ML, 8-Dec-2020.)

⊢ 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}    ⇒   ⊢ (𝐽 ∈ Comp → 𝐽 ∈ 𝐶)

Theorem

pibt2 36949*

Theorem T000002 of pi-base, a countably compact topology is also weakly countably compact. See pibp19 36946 and pibp21 36947 for the definitions of the relevant properties. This proof uses the axiom of choice. (Contributed by ML, 30-Mar-2021.)

⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}    &   ⊢ 𝑊 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ (𝒫 ∪ 𝑥 ∖ Fin)∃𝑧 ∈ ∪ 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦)}    ⇒   ⊢ (𝐽 ∈ 𝐶 → 𝐽 ∈ 𝑊)

21.20  Mathbox for Wolf Lammen

Theorem

wl-section-prop 36950

Intuitionistic logic is now developed separately, so we need not first focus on intuitionally valid axioms ax-1 6 and ax-2 7 any longer.

Alternatively, I start from Jan Lukasiewicz's axiom system here, i.e., ax-mp 5, ax-luk1 36951, ax-luk2 36952 and ax-luk3 36953. I rather copy this system than use luk-1 1649 to luk-3 1651, since the latter are theorems, while we need axioms here.

(Contributed by Wolf Lammen, 23-Feb-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝜑    ⇒   ⊢ 𝜑

Axiom

ax-luk1 36951

1 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-1 1649 and imim1 83, but introduced as an axiom. It focuses on a basic property of a valid implication, namely that the consequent has to be true whenever the antecedent is. So if 𝜑 and 𝜓 are somehow parametrized expressions, then 𝜑 → 𝜓 states that 𝜑 strengthen 𝜓, in that 𝜑 holds only for a (often proper) subset of those parameters making 𝜓 true. We easily accept, that when 𝜓 is stronger than 𝜒 and, at the same time 𝜑 is stronger than 𝜓, then 𝜑 must be stronger than 𝜒. This transitivity is expressed in this axiom.

A particular result of this strengthening property comes into play if the antecedent holds unconditionally. Then the consequent must hold unconditionally as well. This specialization is the foundational idea behind logical conclusion. Such conclusion is best expressed in so-called immediate versions of this axiom like imim1i 63 or syl 17. Note that these forms are weaker replacements (i.e. just frequent specialization) of the closed form presented here, hence a mere convenience.

We can identify in this axiom up to three antecedents, followed by a consequent. The number of antecedents is not really fixed; the fewer we are willing to "see", the more complex the consequent grows. On the other side, since 𝜒 is a variable capable of assuming an implication itself, we might find even more antecedents after some substitution of 𝜒. This shows that the ideas of antecedent and consequent in expressions like this depends on, and can adapt to, our current interpretation of the whole expression.

In this axiom, up to two antecedents happen to be of complex nature themselves, i.e. are an embedded implication. Logically, this axiom is a compact notion of simpler expressions, which I will later coin implication chains. Herein all antecedents and the consequent appear as simple variables, or their negation. Any propositional expression is equivalent to a set of such chains. This axiom, for example, is dissected into following chains, from which it can be recovered losslessly:

(𝜓 → (𝜒 → (𝜑 → 𝜒))); (¬ 𝜑 → (𝜒 → (𝜑 → 𝜒))); (𝜓 → (¬ 𝜓 → (𝜑 → 𝜒))); (¬ 𝜑 → (¬ 𝜓 → (𝜑 → 𝜒))). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)

⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

Axiom

ax-luk2 36952

2 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-2 1650 or pm2.18 128, but introduced as an axiom. The core idea behind this axiom is, that if something can be implied from both an antecedent, and separately from its negation, then the antecedent is irrelevant to the consequent, and can safely be dropped. This is perhaps better seen from the following slightly extended version (related to pm2.65 192):

((𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑)). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)

⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

Axiom

ax-luk3 36953

3 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-3 1651 and pm2.24 124, but introduced as an axiom. One might think that the similar pm2.21 123 (¬ 𝜑 → (𝜑 → 𝜓)) is a valid replacement for this axiom. But this is not true, ax-3 8 is not derivable from this modification. This can be shown by designing carefully operators ¬ and → on a finite set of primitive statements. In propositional logic such statements are ⊤ and ⊥, but we can assume more and other primitives in our universe of statements. So we denote our primitive statements as phi0 , phi1 and phi2. The actual meaning of the statements are not important in this context, it rather counts how they behave under our operations ¬ and →, and which of them we assume to hold unconditionally (phi1, phi2). For our disproving model, I give that information in tabular form below. The interested reader may check by hand, that all possible interpretations of ax-mp 5, ax-luk1 36951, ax-luk2 36952 and pm2.21 123 result in phi1 or phi2, meaning they always hold. But for wl-luk-ax3 36965 we can find a counter example resulting in phi0, not a statement always true. The verification of a particular set of axioms in a given model is tedious and error prone, so I wrote a computer program, first checking this for me, and second, hunting for a counter example. Here is the result, after 9165 fruitlessly computer generated models:

ax-3 fails for phi2, phi2
number of statements: 3
always true phi1 phi2

Negation is defined as
----------------------------------------------------------------------

-. phi0	-. phi1	-. phi2
phi1	phi0	phi1

Implication is defined as
----------------------------------------------------------------------

p->q	q: phi0	q: phi1	q: phi2
p: phi0	phi1	phi1	phi1
p: phi1	phi0	phi1	phi1
p: phi2	phi0	phi0	phi0

(Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)

⊢ (𝜑 → (¬ 𝜑 → 𝜓))

21.20.1  1. Bootstrapping

Theorem

wl-section-boot 36954

In this section, I provide the first steps needed for convenient proving. The presented theorems follow no common concept other than being useful in themselves, and apt to rederive ax-1 6, ax-2 7 and ax-3 8. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝜑    ⇒   ⊢ 𝜑

Theorem

wl-luk-imim1i 36955

Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. Copy of imim1i 63 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.)

⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜒))

Theorem

wl-luk-syl 36956

An inference version of the transitive laws for implication luk-1 1649. Copy of syl 17 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

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

Theorem

wl-luk-imtrid 36957

A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. Copy of syl5 34 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → (𝜓 → 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜃))

Theorem

wl-luk-pm2.18d 36958

Deduction based on reductio ad absurdum. Copy of pm2.18d 127 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → (¬ 𝜓 → 𝜓))    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

wl-luk-con4i 36959

Inference rule. Copy of con4i 114 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (¬ 𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜓 → 𝜑)

Theorem

wl-luk-pm2.24i 36960

Inference rule. Copy of pm2.24i 150 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝜑    ⇒   ⊢ (¬ 𝜑 → 𝜓)

Theorem

wl-luk-a1i 36961

Inference rule. Copy of a1i 11 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝜑    ⇒   ⊢ (𝜓 → 𝜑)

Theorem

wl-luk-mpi 36962

A nested modus ponens inference. Copy of mpi 20 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

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

Theorem

wl-luk-imim2i 36963

Inference adding common antecedents in an implication. Copy of imim2i 16 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 → 𝜑) → (𝜒 → 𝜓))

Theorem

wl-luk-imtrdi 36964

A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. Copy of syl6 35 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))

Theorem

wl-luk-ax3 36965

ax-3 8 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))

Theorem

wl-luk-ax1 36966

ax-1 6 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜑))

Theorem

wl-luk-pm2.27 36967

This theorem, called "Assertion", can be thought of as closed form of modus ponens ax-mp 5. Theorem *2.27 of [WhiteheadRussell] p. 104. Copy of pm2.27 42 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))

Theorem

wl-luk-com12 36968

Inference that swaps (commutes) antecedents in an implication. Copy of com12 32 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜓 → (𝜑 → 𝜒))

Theorem

wl-luk-pm2.21 36969

From a wff and its negation, anything follows. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. Copy of pm2.21 123 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (¬ 𝜑 → (𝜑 → 𝜓))

Theorem

wl-luk-con1i 36970

A contraposition inference. Copy of con1i 147 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (¬ 𝜑 → 𝜓)    ⇒   ⊢ (¬ 𝜓 → 𝜑)

Theorem

wl-luk-ja 36971

Inference joining the antecedents of two premises. Copy of ja 186 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (¬ 𝜑 → 𝜒)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜒)

Theorem

wl-luk-imim2 36972

A closed form of syllogism (see syl 17). Theorem *2.05 of [WhiteheadRussell] p. 100. Copy of imim2 58 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))

Theorem

wl-luk-a1d 36973

Deduction introducing an embedded antecedent. Copy of imim2 58 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))

Theorem

wl-luk-ax2 36974

ax-2 7 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

Theorem

wl-luk-id 36975

Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. Copy of id 22 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → 𝜑)

Theorem

wl-luk-notnotr 36976

Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102. In classical logic (our logic) this is always true. In intuitionistic logic this is not always true; in intuitionistic logic, when this is true for some 𝜑, then 𝜑 is stable. Copy of notnotr 130 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (¬ ¬ 𝜑 → 𝜑)

Theorem

wl-luk-pm2.04 36977

Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. This was the third axiom in Frege's logic system, specifically Proposition 8 of [Frege1879] p. 35. Copy of pm2.04 90 with a different proof. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))

21.20.2  Implication chains

Theorem

wl-section-impchain 36978

An implication like (𝜓 → 𝜑) with one antecedent can easily be extended by prepending more and more antecedents, as in (𝜒 → (𝜓 → 𝜑)) or (𝜃 → (𝜒 → (𝜓 → 𝜑))). I call these expressions implication chains, and the number of antecedents (number of nodes minus one) denotes their length. A given length often marks just a required minimum value, since the consequent 𝜑 itself may represent an implication, or even an implication chain, such hiding part of the whole chain. As an extension, it is useful to consider a single variable 𝜑 as a degenerate implication chain of length zero.

Implication chains play a particular role in logic, as all propositional expressions turn out to be convertible to one or more implication chains, their nodes as simple as a variable, or its negation.

So there is good reason to focus on implication chains as a sort of normalized expressions, and build some general theorems around them, with proofs using recursive patterns. This allows for theorems referring to longer and longer implication chains in an automated way.

The theorem names in this section contain the text fragment 'impchain' to point out their relevance to implication chains, followed by a number indicating the (minimal) length of the longest chain involved. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝜑    ⇒   ⊢ 𝜑

Theorem

wl-impchain-mp-x 36979

This series of theorems provide a means of exchanging the consequent of an implication chain via a simple implication. In the main part, Theorems ax-mp 5, syl 17, syl6 35, syl8 76 form the beginning of this series. These theorems are replicated here, but with proofs that aim at a recursive scheme, allowing to base a proof on that of the previous one in the series. (Contributed by Wolf Lammen, 17-Nov-2019.)

⊢ ⊤

Theorem

wl-impchain-mp-0 36980

This theorem is the start of a proof recursion scheme where we replace the consequent of an implication chain. The number '0' in the theorem name indicates that the modified chain has no antecedents.

This theorem is in fact a copy of ax-mp 5, and is repeated here to emphasize the recursion using similar theorem names. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝜓    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ 𝜑

Theorem

wl-impchain-mp-1 36981

This theorem is in fact a copy of wl-luk-syl 36956, and repeated here to demonstrate a recursive proof scheme. The number '1' in the theorem name indicates that a chain of length 1 is modified. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

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

Theorem

wl-impchain-mp-2 36982

This theorem is in fact a copy of wl-luk-imtrdi 36964, and repeated here to demonstrate a recursive proof scheme. The number '2' in the theorem name indicates that a chain of length 2 is modified. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜃 → (𝜒 → 𝜓))    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ (𝜃 → (𝜒 → 𝜑))

Theorem

wl-impchain-com-1.x 36983

It is often convenient to have the antecedent under focus in first position, so we can apply immediate theorem forms (as opposed to deduction, tautology form). This series of theorems swaps the first with the last antecedent in an implication chain. This kind of swapping is self-inverse, whence we prefer it over, say, rotating theorems. A consequent can hide a tail of a longer chain, so theorems of this series appear as swapping a pair of antecedents with fixed offsets. This form of swapping antecedents is flexible enough to allow for any permutation of antecedents in an implication chain.

The first elements of this series correspond to com12 32, com13 88, com14 96 and com15 101 in the main part.

The proofs of this series aim at automated proving using a simple recursive scheme. It employs the previous theorem in the series along with a sample from the wl-impchain-mp-x 36979 series developed before. (Contributed by Wolf Lammen, 17-Nov-2019.)

⊢ ⊤

Theorem

wl-impchain-com-1.1 36984

A degenerate form of antecedent swapping. The number '1' in the theorem name indicates that it handles a chain of length 1.

Since there is just one antecedent in the chain, there is nothing to swap. Nondegenerated forms begin with wl-impchain-com-1.2 36985, for more see there. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜓 → 𝜑)    ⇒   ⊢ (𝜓 → 𝜑)

Theorem

wl-impchain-com-1.2 36985

This theorem is in fact a copy of wl-luk-com12 36968, and repeated here to demonstrate a simple proof scheme. The number '2' in the theorem name indicates that a chain of length 2 is modified.

See wl-impchain-com-1.x 36983 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜒 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → (𝜒 → 𝜑))

Theorem

wl-impchain-com-1.3 36986

This theorem is in fact a copy of com13 88, and repeated here to demonstrate a simple proof scheme. The number '3' in the theorem name indicates that a chain of length 3 is modified.

See wl-impchain-com-1.x 36983 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜃 → (𝜒 → (𝜓 → 𝜑)))    ⇒   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜑)))

Theorem

wl-impchain-com-1.4 36987

This theorem is in fact a copy of com14 96, and repeated here to demonstrate a simple proof scheme. The number '4' in the theorem name indicates that a chain of length 4 is modified.

See wl-impchain-com-1.x 36983 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜂 → (𝜃 → (𝜒 → (𝜓 → 𝜑))))    ⇒   ⊢ (𝜓 → (𝜃 → (𝜒 → (𝜂 → 𝜑))))

Theorem

wl-impchain-com-n.m 36988

This series of theorems allow swapping any two antecedents in an implication chain. The theorem names follow a pattern wl-impchain-com-n.m with integral numbers n < m, that swaps the m-th antecedent with n-th one in an implication chain. It is sufficient to restrict the length of the chain to m, too, since the consequent can be assumed to be the tail right of the m-th antecedent of any arbitrary sized implication chain. We further assume n > 1, since the wl-impchain-com-1.x 36983 series already covers the special case n = 1.

Being able to swap any two antecedents in an implication chain lays the foundation of permuting its antecedents arbitrarily.

The proofs of this series aim at automated proofing using a simple scheme. Any instance of this series is a triple step of swapping the first and n-th antecedent, then the first and the m-th, then the first and the n-th antecedent again. Each of these steps is an instance of the wl-impchain-com-1.x 36983 series. (Contributed by Wolf Lammen, 17-Nov-2019.)

⊢ ⊤

Theorem

wl-impchain-com-2.3 36989

This theorem is in fact a copy of com23 86. It starts a series of theorems named after wl-impchain-com-n.m 36988. For more information see there. (Contributed by Wolf Lammen, 12-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜃 → (𝜒 → (𝜓 → 𝜑)))    ⇒   ⊢ (𝜃 → (𝜓 → (𝜒 → 𝜑)))

Theorem

wl-impchain-com-2.4 36990

This theorem is in fact a copy of com24 95. It is another instantiation of theorems named after wl-impchain-com-n.m 36988. For more information see there. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜂 → (𝜃 → (𝜒 → (𝜓 → 𝜑))))    ⇒   ⊢ (𝜂 → (𝜓 → (𝜒 → (𝜃 → 𝜑))))

Theorem

wl-impchain-com-3.2.1 36991

This theorem is in fact a copy of com3r 87. The proof is an example of how to arrive at arbitrary permutations of antecedents, using only swapping theorems. The recursion principle is to first swap the correct antecedent to the position just before the consequent, and then employ a theorem handling an implication chain of length one less to reorder the others. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜃 → (𝜒 → (𝜓 → 𝜑)))    ⇒   ⊢ (𝜓 → (𝜃 → (𝜒 → 𝜑)))

Theorem

wl-impchain-a1-x 36992

If an implication chain is assumed (hypothesis) or proven (theorem) to hold, then we may add any extra antecedent to it, without changing its truth. This is expressed in its simplest form in wl-luk-a1i 36961, that allows prepending an arbitrary antecedent to an implication chain. Using our antecedent swapping theorems described in wl-impchain-com-n.m 36988, we may then move such a prepended antecedent to any desired location within all antecedents. The first series of theorems of this kind adds a single antecedent somewhere to an implication chain. The appended number in the theorem name indicates its position within all antecedents, 1 denoting the head position. A second theorem series extends this idea to multiple additions (TODO).

Adding antecedents to an implication chain usually weakens their universality. The consequent afterwards dependends on more conditions than before, which renders the implication chain less versatile. So you find this proof technique mostly when you adjust a chain to a hypothesis of a rule. A common case are syllogisms merging two implication chains into one.

The first elements of the first series correspond to a1i 11, a1d 25 and a1dd 50 in the main part.

The proofs of this series aim at automated proving using a simple recursive scheme. It employs the previous theorem in the series along with a sample from the wl-impchain-com-1.x 36983 series developed before. (Contributed by Wolf Lammen, 20-Jun-2020.)

⊢ ⊤

Theorem

wl-impchain-a1-1 36993

Inference rule, a copy of a1i 11. Head start of a recursive proof pattern. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝜑    ⇒   ⊢ (𝜓 → 𝜑)

Theorem

wl-impchain-a1-2 36994

Inference rule, a copy of a1d 25. First recursive proof based on the previous instance. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))

Theorem

wl-impchain-a1-3 36995

Inference rule, a copy of a1dd 50. A recursive proof depending on previous instances, and demonstrating the proof pattern. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

21.20.3  Theorems around the conditional operator

Theorem

wl-ifp-ncond1 36996

If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.)

⊢ (¬ 𝜓 → (if-(𝜑, 𝜓, 𝜒) ↔ (¬ 𝜑 ∧ 𝜒)))

Theorem

wl-ifp-ncond2 36997

If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.)

⊢ (¬ 𝜒 → (if-(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓)))

Theorem

wl-ifpimpr 36998

If one case of an if- condition is a consequence of the other, the expression in df-ifp 1061 can be shortened. (Contributed by Wolf Lammen, 12-Jun-2024.)

⊢ ((𝜒 → 𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ 𝜒)))

Theorem

wl-ifp4impr 36999

If one case of an if- condition is a consequence of the other, the expression in dfifp4 1064 can be shortened. (Contributed by Wolf Lammen, 18-Jun-2024.)

⊢ ((𝜒 → 𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ 𝜓)))

21.20.4  Alternative development of hadd, cadd

Theorem

wl-df-3xor 37000

Alternative definition of whad 1586 based on hadifp 1598. See df-had 1587 to learn how it is currently introduced. The only use case so far is being a binary addition primitive for df-sad 16420. If inputs are viewed as binary digits (true is 1, false is 0), the result is what a binary single-bit addition with carry-in yields in the low bit of their sum.

The core meaning is to check whether an odd number of three inputs are true. The ⊻ operation tests this for two inputs. So, if the first input is true, the two remaining inputs need to amount to an even (or: not an odd) number, else to an odd number.

The idea of an odd number of inputs being true carries over to other than 3 inputs by recursion: In an informal notation we depend the case with n+1 inputs, 𝜑 being the additional one, recursively on that of n inputs: "(n+1)-xor" ↔ if-(𝜑, ¬ "n-xor" , "n-xor" ). The base case is "0-xor" being ⊥, because zero inputs never contain an odd number among them. Then we find, after simplifying, in our informal notation:

"2-xor" (𝜑, 𝜓) ↔ (𝜑 ⊻ 𝜓) (see wl-2xor 37015).

Our definition here follows exactly the above pattern.

In microprocessor technology an addition limited to a range (a one-bit range in our case) is called a "wrap-around operation". The name "had", as in df-had 1587, by contrast, is somehow suggestive of a "half adder" instead. Such a circuit, for one, takes two inputs only, no carry-in, and then yields two outputs - both sum and carry. That's why we use "3xor" instead of "had" here. (Contributed by Wolf Lammen, 24-Apr-2024.)

⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒)))