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Type | Label | Description |
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Statement | ||
Theorem | linerflx1 34701 | Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → 𝑃 ∈ (𝑃Line𝑄)) | ||
Theorem | linecom 34702 | Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → (𝑃Line𝑄) = (𝑄Line𝑃)) | ||
Theorem | linerflx2 34703 | Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → 𝑄 ∈ (𝑃Line𝑄)) | ||
Theorem | ellines 34704* | Membership in the set of all lines. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))) | ||
Theorem | linethru 34705 | If 𝐴 is a line containing two distinct points 𝑃 and 𝑄, then 𝐴 is the line through 𝑃 and 𝑄. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ LinesEE ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄)) | ||
Theorem | hilbert1.1 34706* | There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → ∃𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) | ||
Theorem | hilbert1.2 34707* | There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by NM, 17-Jun-2017.) |
⊢ (𝑃 ≠ 𝑄 → ∃*𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) | ||
Theorem | linethrueu 34708* | There is a unique line going through any two distinct points. Theorem 6.19 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → ∃!𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) | ||
Theorem | lineintmo 34709* | Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE ∧ 𝐴 ≠ 𝐵) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | ||
Syntax | cfwddif 34710 | Declare the syntax for the forward difference operator. |
class △ | ||
Definition | df-fwddif 34711* | Define the forward difference operator. This is a discrete analogue of the derivative operator. Definition 2.42 of [GramKnuthPat], p. 47. (Contributed by Scott Fenton, 18-May-2020.) |
⊢ △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥)))) | ||
Syntax | cfwddifn 34712 | Declare the syntax for the nth forward difference operator. |
class
△ | ||
Definition | df-fwddifn 34713* | Define the nth forward difference operator. This works out to be the forward difference operator iterated 𝑛 times. (Contributed by Scott Fenton, 28-May-2020.) |
⊢
△ | ||
Theorem | fwddifval 34714 | Calculate the value of the forward difference operator at a point. (Contributed by Scott Fenton, 18-May-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → (𝑋 + 1) ∈ 𝐴) ⇒ ⊢ (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋))) | ||
Theorem | fwddifnval 34715* | The value of the forward difference operator at a point. (Contributed by Scott Fenton, 28-May-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ 𝐴) ⇒ ⊢ (𝜑 → ((𝑁 △ | ||
Theorem | fwddifn0 34716 | The value of the n-iterated forward difference operator at zero is just the function value. (Contributed by Scott Fenton, 28-May-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → ((0 △ | ||
Theorem | fwddifnp1 34717* | The value of the n-iterated forward difference at a successor. (Contributed by Scott Fenton, 28-May-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑋 + 𝑘) ∈ 𝐴) ⇒ ⊢ (𝜑 → (((𝑁 + 1) △ | ||
Theorem | rankung 34718 | The rank of the union of two sets. Closed form of rankun 9788. (Contributed by Scott Fenton, 15-Jul-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))) | ||
Theorem | ranksng 34719 | The rank of a singleton. Closed form of ranksn 9786. (Contributed by Scott Fenton, 15-Jul-2015.) |
⊢ (𝐴 ∈ 𝑉 → (rank‘{𝐴}) = suc (rank‘𝐴)) | ||
Theorem | rankelg 34720 | The membership relation is inherited by the rank function. Closed form of rankel 9771. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝐵) → (rank‘𝐴) ∈ (rank‘𝐵)) | ||
Theorem | rankpwg 34721 | The rank of a power set. Closed form of rankpw 9775. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ (𝐴 ∈ 𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴)) | ||
Theorem | rank0 34722 | The rank of the empty set is ∅. (Contributed by Scott Fenton, 17-Jul-2015.) |
⊢ (rank‘∅) = ∅ | ||
Theorem | rankeq1o 34723 | The only set with rank 1o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.) |
⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = {∅}) | ||
Syntax | chf 34724 | The constant Hf is a class. |
class Hf | ||
Definition | df-hf 34725 | Define the hereditarily finite sets. These are the finite sets whose elements are finite, and so forth. (Contributed by Scott Fenton, 9-Jul-2015.) |
⊢ Hf = ∪ (𝑅1 “ ω) | ||
Theorem | elhf 34726* | Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.) |
⊢ (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥)) | ||
Theorem | elhf2 34727 | Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω) | ||
Theorem | elhf2g 34728 | Hereditarily finiteness via rank. Closed form of elhf2 34727. (Contributed by Scott Fenton, 15-Jul-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)) | ||
Theorem | 0hf 34729 | The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.) |
⊢ ∅ ∈ Hf | ||
Theorem | hfun 34730 | The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.) |
⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ 𝐵) ∈ Hf ) | ||
Theorem | hfsn 34731 | The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.) |
⊢ (𝐴 ∈ Hf → {𝐴} ∈ Hf ) | ||
Theorem | hfadj 34732 | Adjoining one HF element to an HF set preserves HF status. (Contributed by Scott Fenton, 15-Jul-2015.) |
⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ {𝐵}) ∈ Hf ) | ||
Theorem | hfelhf 34733 | Any member of an HF set is itself an HF set. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ Hf ) → 𝐴 ∈ Hf ) | ||
Theorem | hftr 34734 | The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ Tr Hf | ||
Theorem | hfext 34735* | Extensionality for HF sets depends only on comparison of HF elements. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ Hf (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) | ||
Theorem | hfuni 34736 | The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ Hf ) | ||
Theorem | hfpw 34737 | The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ (𝐴 ∈ Hf → 𝒫 𝐴 ∈ Hf ) | ||
Theorem | hfninf 34738 | ω is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ ¬ ω ∈ Hf | ||
Theorem | a1i14 34739 | Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.) |
⊢ (𝜓 → (𝜒 → 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) | ||
Theorem | a1i24 34740 | Add two antecedents to a wff. Deduction associated with a1i13 27. (Contributed by Jeff Hankins, 5-Aug-2009.) |
⊢ (𝜑 → (𝜒 → 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) | ||
Theorem | exp5d 34741 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → ((𝜃 ∧ 𝜏) → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp5g 34742 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ ((𝜑 ∧ 𝜓) → (((𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp5k 34743 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ (𝜑 → (((𝜓 ∧ (𝜒 ∧ 𝜃)) ∧ 𝜏) → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp56 34744 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp58 34745 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ (((𝜑 ∧ 𝜓) ∧ ((𝜒 ∧ 𝜃) ∧ 𝜏)) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp510 34746 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ ((𝜑 ∧ (((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏)) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp511 34747 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ ((𝜑 ∧ ((𝜓 ∧ (𝜒 ∧ 𝜃)) ∧ 𝜏)) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp512 34748 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏))) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | 3com12d 34749 | Commutation in consequent. Swap 1st and 2nd. (Contributed by Jeff Hankins, 17-Nov-2009.) |
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → (𝜒 ∧ 𝜓 ∧ 𝜃)) | ||
Theorem | imp5p 34750 | A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.) |
⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) ⇒ ⊢ (𝜑 → (𝜓 → ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂))) | ||
Theorem | imp5q 34751 | A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.) |
⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂)) | ||
Theorem | ecase13d 34752 | Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.) |
⊢ (𝜑 → ¬ 𝜒) & ⊢ (𝜑 → ¬ 𝜃) & ⊢ (𝜑 → (𝜒 ∨ 𝜓 ∨ 𝜃)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | subtr 34753 | Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝑌 & ⊢ Ⅎ𝑥𝑍 & ⊢ (𝑥 = 𝐴 → 𝑋 = 𝑌) & ⊢ (𝑥 = 𝐵 → 𝑋 = 𝑍) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → 𝑌 = 𝑍)) | ||
Theorem | subtr2 34754 | Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → (𝜓 ↔ 𝜒))) | ||
Theorem | trer 34755* | A relation intersected with its converse is an equivalence relation if the relation is transitive. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (∀𝑎∀𝑏∀𝑐((𝑎 ≤ 𝑏 ∧ 𝑏 ≤ 𝑐) → 𝑎 ≤ 𝑐) → ( ≤ ∩ ◡ ≤ ) Er dom ( ≤ ∩ ◡ ≤ )) | ||
Theorem | elicc3 34756 | An equivalent membership condition for closed intervals. (Contributed by Jeff Hankins, 14-Jul-2009.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))) | ||
Theorem | finminlem 34757* | A useful lemma about finite sets. If a property holds for a finite set, it holds for a minimal set. (Contributed by Jeff Hankins, 4-Dec-2009.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ Fin 𝜑 → ∃𝑥(𝜑 ∧ ∀𝑦((𝑦 ⊆ 𝑥 ∧ 𝜓) → 𝑥 = 𝑦))) | ||
Theorem | gtinf 34758* | Any number greater than an infimum is greater than some element of the set. (Contributed by Jeff Hankins, 29-Sep-2013.) (Revised by AV, 10-Oct-2021.) |
⊢ (((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ∧ (𝐴 ∈ ℝ ∧ inf(𝑆, ℝ, < ) < 𝐴)) → ∃𝑧 ∈ 𝑆 𝑧 < 𝐴) | ||
Theorem | opnrebl 34759* | A set is open in the standard topology of the reals precisely when every point can be enclosed in an open ball. (Contributed by Jeff Hankins, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.) |
⊢ (𝐴 ∈ (topGen‘ran (,)) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ℝ+ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦)) ⊆ 𝐴)) | ||
Theorem | opnrebl2 34760* | A set is open in the standard topology of the reals precisely when every point can be enclosed in an arbitrarily small ball. (Contributed by Jeff Hankins, 22-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.) |
⊢ (𝐴 ∈ (topGen‘ran (,)) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ (𝑧 ≤ 𝑦 ∧ ((𝑥 − 𝑧)(,)(𝑥 + 𝑧)) ⊆ 𝐴))) | ||
Theorem | nn0prpwlem 34761* | Lemma for nn0prpw 34762. Use strong induction to show that every positive integer has unique prime power divisors. (Contributed by Jeff Hankins, 28-Sep-2013.) |
⊢ (𝐴 ∈ ℕ → ∀𝑘 ∈ ℕ (𝑘 < 𝐴 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐴))) | ||
Theorem | nn0prpw 34762* | Two nonnegative integers are the same if and only if they are divisible by the same prime powers. (Contributed by Jeff Hankins, 29-Sep-2013.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) | ||
Theorem | topbnd 34763 | Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) | ||
Theorem | opnbnd 34764 | A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅)) | ||
Theorem | cldbnd 34765 | A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴)) | ||
Theorem | ntruni 34766* | A union of interiors is a subset of the interior of the union. The reverse inclusion may not hold. (Contributed by Jeff Hankins, 31-Aug-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → ∪ 𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂)) | ||
Theorem | clsun 34767 | A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))) | ||
Theorem | clsint2 34768* | The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ∩ 𝑐 ∈ 𝐶 ((cls‘𝐽)‘𝑐)) | ||
Theorem | opnregcld 34769* | A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜))) | ||
Theorem | cldregopn 34770* | A set if regularly open iff it is the interior of some closed set. (Contributed by Jeff Hankins, 27-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))) | ||
Theorem | neiin 34771 | Two neighborhoods intersect to form a neighborhood of the intersection. (Contributed by Jeff Hankins, 31-Aug-2009.) |
⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑀 ∩ 𝑁) ∈ ((nei‘𝐽)‘(𝐴 ∩ 𝐵))) | ||
Theorem | hmeoclda 34772 | Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.) (Revised by Mario Carneiro, 3-Jun-2014.) |
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽Homeo𝐾)) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑆) ∈ (Clsd‘𝐾)) | ||
Theorem | hmeocldb 34773 | Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.) |
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽Homeo𝐾)) ∧ 𝑆 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑆) ∈ (Clsd‘𝐽)) | ||
Theorem | ivthALT 34774* | An alternate proof of the Intermediate Value Theorem ivth 24802 using topology. (Contributed by Jeff Hankins, 17-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ∃𝑥 ∈ (𝐴(,)𝐵)(𝐹‘𝑥) = 𝑈) | ||
Syntax | cfne 34775 | Extend class definition to include the "finer than" relation. |
class Fne | ||
Definition | df-fne 34776* | Define the fineness relation for covers. (Contributed by Jeff Hankins, 28-Sep-2009.) |
⊢ Fne = {〈𝑥, 𝑦〉 ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))} | ||
Theorem | fnerel 34777 | Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.) |
⊢ Rel Fne | ||
Theorem | isfne 34778* | The predicate "𝐵 is finer than 𝐴". This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))) | ||
Theorem | isfne4 34779 | The predicate "𝐵 is finer than 𝐴 " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) | ||
Theorem | isfne4b 34780 | A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝐵 ∈ 𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)))) | ||
Theorem | isfne2 34781* | The predicate "𝐵 is finer than 𝐴". (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))) | ||
Theorem | isfne3 34782* | The predicate "𝐵 is finer than 𝐴". (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))) | ||
Theorem | fnebas 34783 | A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝐴Fne𝐵 → 𝑋 = 𝑌) | ||
Theorem | fnetg 34784 | A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝐴Fne𝐵 → 𝐴 ⊆ (topGen‘𝐵)) | ||
Theorem | fnessex 34785* | If 𝐵 is finer than 𝐴 and 𝑆 is an element of 𝐴, every point in 𝑆 is an element of a subset of 𝑆 which is in 𝐵. (Contributed by Jeff Hankins, 28-Sep-2009.) |
⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝑆) → ∃𝑥 ∈ 𝐵 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑆)) | ||
Theorem | fneuni 34786* | If 𝐵 is finer than 𝐴, every element of 𝐴 is a union of elements of 𝐵. (Contributed by Jeff Hankins, 11-Oct-2009.) |
⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴) → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥)) | ||
Theorem | fneint 34787* | If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.) |
⊢ (𝐴Fne𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}) | ||
Theorem | fness 34788 | A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Fne𝐵) | ||
Theorem | fneref 34789 | Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴Fne𝐴) | ||
Theorem | fnetr 34790 | Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → 𝐴Fne𝐶) | ||
Theorem | fneval 34791 | Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ ∼ = (Fne ∩ ◡Fne) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∼ 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵))) | ||
Theorem | fneer 34792 | Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ ∼ = (Fne ∩ ◡Fne) ⇒ ⊢ ∼ Er V | ||
Theorem | topfne 34793 | Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐾 ∈ Top ∧ 𝑋 = 𝑌) → (𝐽 ⊆ 𝐾 ↔ 𝐽Fne𝐾)) | ||
Theorem | topfneec 34794 | A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ ∼ = (Fne ∩ ◡Fne) ⇒ ⊢ (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ∼ ↔ (topGen‘𝐴) = 𝐽)) | ||
Theorem | topfneec2 34795 | A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.) |
⊢ ∼ = (Fne ∩ ◡Fne) ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] ∼ = [𝐾] ∼ ↔ 𝐽 = 𝐾)) | ||
Theorem | fnessref 34796* | A cover is finer iff it has a subcover which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝑋 = 𝑌 → (𝐴Fne𝐵 ↔ ∃𝑐(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)))) | ||
Theorem | refssfne 34797* | A cover is a refinement iff it is a subcover of something which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝑋 = 𝑌 → (𝐵Ref𝐴 ↔ ∃𝑐(𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)))) | ||
Theorem | neibastop1 34798* | A collection of neighborhood bases determines a topology. Part of Theorem 4.5 of Stephen Willard's General Topology. (Contributed by Jeff Hankins, 8-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅})) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥))) → ((𝐹‘𝑥) ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅) & ⊢ 𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅} ⇒ ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | ||
Theorem | neibastop2lem 34799* | Lemma for neibastop2 34800. (Contributed by Jeff Hankins, 12-Sep-2009.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅})) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥))) → ((𝐹‘𝑥) ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅) & ⊢ 𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅} & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → 𝑥 ∈ 𝑣) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → ∃𝑡 ∈ (𝐹‘𝑥)∀𝑦 ∈ 𝑡 ((𝐹‘𝑦) ∩ 𝒫 𝑣) ≠ ∅) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ (𝜑 → 𝑁 ⊆ 𝑋) & ⊢ (𝜑 → 𝑈 ∈ (𝐹‘𝑃)) & ⊢ (𝜑 → 𝑈 ⊆ 𝑁) & ⊢ 𝐺 = (rec((𝑎 ∈ V ↦ ∪ 𝑧 ∈ 𝑎 ∪ 𝑥 ∈ 𝑋 ((𝐹‘𝑥) ∩ 𝒫 𝑧)), {𝑈}) ↾ ω) & ⊢ 𝑆 = {𝑦 ∈ 𝑋 ∣ ∃𝑓 ∈ ∪ ran 𝐺((𝐹‘𝑦) ∩ 𝒫 𝑓) ≠ ∅} ⇒ ⊢ (𝜑 → ∃𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑁)) | ||
Theorem | neibastop2 34800* | In the topology generated by a neighborhood base, a set is a neighborhood of a point iff it contains a subset in the base. (Contributed by Jeff Hankins, 9-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅})) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥))) → ((𝐹‘𝑥) ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅) & ⊢ 𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅} & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → 𝑥 ∈ 𝑣) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → ∃𝑡 ∈ (𝐹‘𝑥)∀𝑦 ∈ 𝑡 ((𝐹‘𝑦) ∩ 𝒫 𝑣) ≠ ∅) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁 ⊆ 𝑋 ∧ ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅))) |
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