Home | Metamath
Proof Explorer Theorem List (p. 323 of 425) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-26947) |
Hilbert Space Explorer
(26948-28472) |
Users' Mathboxes
(28473-42426) |
Type | Label | Description | ||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Statement | ||||||||||||||||||||||||
Theorem | topdifinfindis 32201* | Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is the trivial topology when 𝐴 is finite. (Contributed by ML, 14-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} ⇒ ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴}) | ||||||||||||||||||||||||
Theorem | topdifinffinlem 32202* | This is the core of the proof of topdifinffin 32203, but to avoid the distinct variables on the definition, we need to split this proof into two. (Contributed by ML, 17-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} ⇒ ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin) | ||||||||||||||||||||||||
Theorem | topdifinffin 32203* | Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is a topology only if 𝐴 is finite. (Contributed by ML, 17-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} ⇒ ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin) | ||||||||||||||||||||||||
Theorem | topdifinf 32204* | Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is a topology if and only if 𝐴 is finite, in which case it is the trivial topology. (Contributed by ML, 17-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} ⇒ ⊢ ((𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin) ∧ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴})) | ||||||||||||||||||||||||
Theorem | topdifinfeq 32205* | Two different ways of defining the collection from Exercise 3 of [Munkres] p. 83. (Contributed by ML, 18-Jul-2020.) | ||||||||||||||||||||||
⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴))} = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} | ||||||||||||||||||||||||
Theorem | icorempt2 32206* | Closed-below, open-above intervals of reals. (Contributed by ML, 26-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = ([,) ↾ (ℝ × ℝ)) ⇒ ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | ||||||||||||||||||||||||
Theorem | icoreresf 32207 | Closed-below, open-above intervals of reals map to subsets of reals. (Contributed by ML, 25-Jul-2020.) | ||||||||||||||||||||||
⊢ ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ | ||||||||||||||||||||||||
Theorem | icoreval 32208* | Value of the closed-below, open-above interval function on reals. (Contributed by ML, 26-Jul-2020.) | ||||||||||||||||||||||
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) = {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)}) | ||||||||||||||||||||||||
Theorem | icoreelrnab 32209* | Elementhood in the set of closed-below, open-above intervals of reals. (Contributed by ML, 27-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) | ||||||||||||||||||||||||
Theorem | isbasisrelowllem1 32210* | Lemma for isbasisrelowl 32213. (Contributed by ML, 27-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑)) → (𝑥 ∩ 𝑦) ∈ 𝐼) | ||||||||||||||||||||||||
Theorem | isbasisrelowllem2 32211* | Lemma for isbasisrelowl 32213. (Contributed by ML, 27-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼) | ||||||||||||||||||||||||
Theorem | icoreclin 32212* | The set of closed-below, open-above intervals of reals is closed under finite intersection. (Contributed by ML, 27-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼) | ||||||||||||||||||||||||
Theorem | isbasisrelowl 32213 | The set of all closed-below, open-above intervals of reals form a basis. (Contributed by ML, 27-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ 𝐼 ∈ TopBases | ||||||||||||||||||||||||
Theorem | icoreunrn 32214 | The union of all closed-below, open-above intervals of reals is the set of reals. (Contributed by ML, 27-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ℝ = ∪ 𝐼 | ||||||||||||||||||||||||
Theorem | istoprelowl 32215 | The set of all closed-below, open-above intervals of reals generate a topology on the reals. (Contributed by ML, 27-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ (topGen‘𝐼) ∈ (TopOn‘ℝ) | ||||||||||||||||||||||||
Theorem | icoreelrn 32216* | A class abstraction which is an element of the set of closed-below, open-above intervals of reals. (Contributed by ML, 1-Aug-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)} ∈ 𝐼) | ||||||||||||||||||||||||
Theorem | iooelexlt 32217* | An element of an open interval is not its smallest element. (Contributed by ML, 2-Aug-2020.) | ||||||||||||||||||||||
⊢ (𝑋 ∈ (𝐴(,)𝐵) → ∃𝑦 ∈ (𝐴(,)𝐵)𝑦 < 𝑋) | ||||||||||||||||||||||||
Theorem | relowlssretop 32218 | The lower limit topology on the reals is finer than the standard topology. (Contributed by ML, 1-Aug-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ (topGen‘ran (,)) ⊆ (topGen‘𝐼) | ||||||||||||||||||||||||
Theorem | relowlpssretop 32219 | The lower limit topology on the reals is strictly finer than the standard topology. (Contributed by ML, 2-Aug-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ (topGen‘ran (,)) ⊊ (topGen‘𝐼) | ||||||||||||||||||||||||
Theorem | sucneqond 32220 | Inequality of an ordinal set with its successor. Does not use the axiom of regularity. (Contributed by ML, 18-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝑋 = suc 𝑌) & ⊢ (𝜑 → 𝑌 ∈ On) ⇒ ⊢ (𝜑 → 𝑋 ≠ 𝑌) | ||||||||||||||||||||||||
Theorem | sucneqoni 32221 | Inequality of an ordinal set with its successor. Does not use the axiom of regularity. (Contributed by ML, 18-Oct-2020.) | ||||||||||||||||||||||
⊢ 𝑋 = suc 𝑌 & ⊢ 𝑌 ∈ On ⇒ ⊢ 𝑋 ≠ 𝑌 | ||||||||||||||||||||||||
Theorem | onsucuni3 32222 | If an ordinal number has a predecessor, then it is successor of that predecessor. (Contributed by ML, 17-Oct-2020.) | ||||||||||||||||||||||
⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 = suc ∪ 𝐵) | ||||||||||||||||||||||||
Theorem | 1oequni2o 32223 | The ordinal number 1_{𝑜} is the predecessor of the ordinal number 2_{𝑜}. (Contributed by ML, 19-Oct-2020.) | ||||||||||||||||||||||
⊢ 1_{𝑜} = ∪ 2_{𝑜} | ||||||||||||||||||||||||
Theorem | rdgsucuni 32224 | 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 32225 | 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 32226 | Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 6967. (Contributed by ML, 19-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1^{st} ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) | ||||||||||||||||||||||||
Syntax | cfinxp 32227 | Extend the definition of a class to include Cartesian exponentiation. | ||||||||||||||||||||||
class (𝑈↑↑𝑁) | ||||||||||||||||||||||||
Definition | df-finxp 32228* |
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 7671 or df-map 7622 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 4482 can be used on it, and df-fv 5697 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 32236 and finxpsuc 32242 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 32229* | This theorem is the same as the definition df-finxp 32228, 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 32230 | Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁)) | ||||||||||||||||||||||||
Theorem | finxpeq2 32231 | Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁)) | ||||||||||||||||||||||||
Theorem | csbfinxpg 32232* | 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 32233* | 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 32234* | 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 32235 | The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑∅) = ∅ | ||||||||||||||||||||||||
Theorem | finxp1o 32236 | The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑1_{𝑜}) = 𝑈 | ||||||||||||||||||||||||
Theorem | finxpreclem3 32237* | 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 32238* | 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 32239* | 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 32240* | 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 32241* | Lemma for finxpsuc 32242. (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 32242 | 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 32243 | The value of Cartesian exponentiation at two. (Contributed by ML, 19-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑2_{𝑜}) = (𝑈 × 𝑈) | ||||||||||||||||||||||||
Theorem | finxp3o 32244 | The value of Cartesian exponentiation at three. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑3_{𝑜}) = ((𝑈 × 𝑈) × 𝑈) | ||||||||||||||||||||||||
Theorem | finxpnom 32245 | 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 32246 | Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ (∅↑↑𝑁) = ∅ | ||||||||||||||||||||||||
Theorem | wl-section-prop 32247 |
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 32248, ax-luk2 32249 and ax-luk3 32250. I rather copy this system than use luk-1 1570 to luk-3 1572, 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 32248 |
1 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-1 1570 and imim1 80, 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 60 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 32249 |
2 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-2 1571 or pm2.18 120, 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 pm2.65 182):
((𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑)). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ ((¬ 𝜑 → 𝜑) → 𝜑) | ||||||||||||||||||||||||
Axiom | ax-luk3 32250 |
3 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-3 1572 and pm2.24 119, but introduced as an axiom.
One might think that the similar pm2.21 118 (¬ 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 per
hand, that all possible interpretations of ax-mp 5, ax-luk1 32248, ax-luk2 32249
and pm2.21 118 result in phi1 or phi2, meaning they always hold. But for
wl-ax3 32262 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 ----------------------------------------------------------------------
Implication is defined as ----------------------------------------------------------------------
(Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (¬ 𝜑 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-section-boot 32251 | 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-imim1i 32252 | Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. Copy of imim1i 60 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-syl 32253 | An inference version of the transitive laws for implication luk-1 1570. 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-syl5 32254 | A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. Copy of syl5 33 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-pm2.18d 32255 | Deduction based on reductio ad absurdum. Copy of pm2.18d 122 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-con4i 32256 | Inference rule. Copy of con4i 111 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-pm2.24i 32257 | Inference rule. Copy of pm2.24i 144 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ 𝜑 ⇒ ⊢ (¬ 𝜑 → 𝜓) | ||||||||||||||||||||||||
Theorem | wl-a1i 32258 | 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-mpi 32259 | 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-imim2i 32260 | 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-syl6 32261 | A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. Copy of syl6 34 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-ax3 32262 | 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-ax1 32263 | 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-pm2.27 32264 | 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 40 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-com12 32265 | 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-pm2.21 32266 | From a wff and its negation, anything follows. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. Copy of pm2.21 118 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-con1i 32267 | A contraposition inference. Copy of con1i 142 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-ja 32268 | Inference joining the antecedents of two premises. Copy of ja 171 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-imim2 32269 | A closed form of syllogism (see syl 17). Theorem *2.05 of [WhiteheadRussell] p. 100. Copy of imim2 55 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-a1d 32270 | Deduction introducing an embedded antecedent. Copy of imim2 55 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-ax2 32271 | 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-id 32272 | 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-notnotr 32273 | 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 123 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (¬ ¬ 𝜑 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-pm2.04 32274 | 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 87 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))) | ||||||||||||||||||||||||
Theorem | wl-section-impchain 32275 |
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 32276 | This series of theorems provide a means of exchanging the consequent of an implication chain via a simple implication. In the main part, the theorems ax-mp 5, syl 17, syl6 34, syl8 73 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 32277 |
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 32278 | This theorem is in fact a copy of wl-syl 32253, 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 32279 | This theorem is in fact a copy of wl-syl6 32261, 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 32280 |
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 85, com14 93 and com15 98 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 32276 series developed before. (Contributed by Wolf Lammen, 17-Nov-2019.) | ||||||||||||||||||||||
⊢ ⊤ | ||||||||||||||||||||||||
Theorem | wl-impchain-com-1.1 32281 |
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. Non-degenerated forms begin with wl-impchain-com-1.2 32282, 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 32282 |
This theorem is in fact a copy of wl-com12 32265, 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 32280 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 32283 |
This theorem is in fact a copy of com13 85, 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 32280 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 32284 |
This theorem is in fact a copy of com14 93, 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 32280 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 32285 |
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 32280 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 32280 series. (Contributed by Wolf Lammen, 17-Nov-2019.) | ||||||||||||||||||||||
⊢ ⊤ | ||||||||||||||||||||||||
Theorem | wl-impchain-com-2.3 32286 | This theorem is in fact a copy of com23 83. It starts a series of theorems named after wl-impchain-com-n.m 32285. 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 32287 | This theorem is in fact a copy of com24 92. It is another instantiation of theorems named after wl-impchain-com-n.m 32285. 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 32288 | This theorem is in fact a copy of com3r 84. 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 32289 |
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-a1i 32258, that allows
us prepending an arbitrary antecedent to an implication chain. Using our
antecedent swapping theorems described in wl-impchain-com-n.m 32285, 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 47 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 32280 series developed before. (Contributed by Wolf Lammen, 20-Jun-2020.) | ||||||||||||||||||||||
⊢ ⊤ | ||||||||||||||||||||||||
Theorem | wl-impchain-a1-1 32290 | 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 32291 | 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 32292 | Inference rule, a copy of a1dd 47. 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))) | ||||||||||||||||||||||||
Theorem | wl-section-nf 32293 |
The current definition of 'not free', df-nf 1699, has its downsides. In
particular, it often drags axioms ax-10 1966 and ax-12 1983 into proofs, that
are not needed otherwise. This is because of the particular structure
of the term ∀𝑥(𝜑 → ∀𝑥𝜑). It does not allow an easy
transition 𝜑--> ¬ 𝜑 (see nfn 2031).
The mix of both
quantified and simple 𝜑 requires explicit use of sp 1990 (or
ax-12 1983)
in many instances. And, finally, the nesting of the quantifier ∀𝑥
sometimes requires an invocation of theorems like nfa1 2027.
All of this
mandates the use of ax-10 1966 and/or ax-12 1983.
On the other hand, the current definition is structurally better aligned with both the hb* series of theorems and ax-5 1793. In total, I personally prefer nf2 2090 ∃𝑥𝜑 → ∀𝑥𝜑 (or nf4 2092 ∀𝑥𝜑 ∨ ∀𝑥(¬ 𝜑)) over df-nf 1699. Apart from df-nf 1699, nfi 1705 (but virtually no theorem using them!) I know of no theorem depending on more axioms after switching to nf2 2090 style. A note on ax-10 1966: The obvious content of this axiom is, that the ¬ operator does not change the not-free state of a set varable. This is in fact only one aspect of this axiom. The other one, more hidden, states that 𝑥 is not free in ∀𝑥𝜑. A simple transformation renders this axiom as (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑), from which the second aspect is better deduced. Both aspects are not needed in simple applications of the nf2 2090 style definition, while use of the current df-nf 1699 incurs these. A note on ax-12 1983: This axiom enters proofs using Ⅎ via sp 1990 or 19.8a 1988 or their variants. In a context where both mixed quantified and simple variables 𝜑 appear (like 19.21 2036), this axiom is almost always required, no matter how 'not free' is defined. But in a context, where a variable 𝜑 appears only quantified, chances are, this axiom can be evaded when using nf2 2090, but not when using df-nf 1699. A note on the technique used in this section: I restate nf2 2090 as an axiom instead of defining Ⅎ new. This method allows us to easily determine what axioms a proof depends on (after ignoring the auxiliary axiom) without overloading definitions. Almost all theorems proved in this section correspond to one in the Main part based on the definition df-nf 1699. I will link to it, so a quick comparison on axiom usage is ready at hand. When checking the definition list in proofs in this section, df-nf 1699 must not occur, as this would create a circular dependency. (Contributed by Wolf Lammen, 11-Sep-2021.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ 𝜑 ⇒ ⊢ 𝜑 | ||||||||||||||||||||||||
Axiom | ax-wl-dfnf 32294 |
This axiom actually provides a definition of Ⅎ
different from that
in the Main part. Using an axiom instead of a definition, which would
be more appropriate in other contexts, avoids overloading the symbol
Ⅎ. It keeps the notations consistent
while the disturbation of
the axiom lists is easily corrigible by humans. Simply ignore this
axiom where it appears, and add a 'df-nf' to the definitions list.
We weaken the definition df-nf 1699 here. Instead of finding a particular element for which 𝜑 holds, we are already satisfied, if we know one exists. In the presence of axioms listed in nf2 2090, this semantic change does not matter. In logic models, particularly those, where ax-10 1966 or sp 1990 do not hold, we might see differences where the symbol Ⅎ is used. (Contributed by Wolf Lammen, 11-Sep-2021.) | ||||||||||||||||||||||
⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | ||||||||||||||||||||||||
Theorem | wl-nf2 32295 |
Revision of nf2 2090 using (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) as a
definition for 'not free'. [ -ax-mp -ax-1 -ax-2 -ax-3 -ax-gen -ax-4 -ax-5
-ax-6 -ax-7 -ax-10 -ax-12 ]
A definition of 'not free', which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.) Definition change. (Revised by Wolf Lammen, 11-Sep-2021.) | ||||||||||||||||||||||
⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | ||||||||||||||||||||||||
Theorem | wl-nf3 32296 | Alternate definition of non-freeness. [see bj-nf2 31600] [The numbers in "wl-nf." may change once they go to Main; see also nf3 2091, nf4 2092] (Contributed by BJ, 16-Sep-2021.) | ||||||||||||||||||||||
⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑)) | ||||||||||||||||||||||||
Theorem | wl-nf4 32297 | Alternate definition of non-freeness. [see bj-nf3 31601, nf4 2092] [ -ax-gen -ax-4 -ax-5 -ax-6 -ax-7 -ax-10 -ax-12 ] (Contributed by BJ, 16-Sep-2021.) | ||||||||||||||||||||||
⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-nf5 32298 | Alternate definition of non-freeness. [see bj-nf4 31602] This definition uses only primitive symbols. (Contributed by BJ, 16-Sep-2021.) | ||||||||||||||||||||||
⊢ (Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-nf2i 32299 |
Counterpast of nfi 1705 using nf2 2090 as a definition for 'not free'.
Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Wolf Lammen, 15-Sep-2021.) | ||||||||||||||||||||||
⊢ (∃𝑥𝜑 → ∀𝑥𝜑) ⇒ ⊢ Ⅎ𝑥𝜑 | ||||||||||||||||||||||||
Theorem | wl-nf2ri 32300 |
Counterpart of to nfri 2005 using nf2 2090 as a definition for 'not free'.
Consequence of the definition of not-free. (Contributed by Wolf Lammen, 16-Sep-2021.) | ||||||||||||||||||||||
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥𝜑 → ∀𝑥𝜑) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |