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Theorem List for Metamath Proof Explorer - 33501-33600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlinedegen 33501 When Line is applied with the same argument, the result is the empty set. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐴Line𝐴) = ∅
 
Theoremfvline 33502* Calculate the value of the Line function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → (𝐴Line𝐵) = {𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩})
 
Theoremliness 33503 A line is a subset of the space its two points lie in. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → (𝐴Line𝐵) ⊆ (𝔼‘𝑁))
 
Theoremfvline2 33504* Alternate definition of a line. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → (𝐴Line𝐵) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩})
 
Theoremlineunray 33505 A line is composed of a point and the two rays emerging from it. Theorem 6.15 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃𝑄𝑃𝑅)) → (𝑃 Btwn ⟨𝑄, 𝑅⟩ → (𝑃Line𝑄) = (((𝑃Ray𝑄) ∪ {𝑃}) ∪ (𝑃Ray𝑅))))
 
Theoremlineelsb2 33506 If 𝑆 lies on 𝑃𝑄, then 𝑃𝑄 = 𝑃𝑆. Theorem 6.16 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃𝑆)) → (𝑆 ∈ (𝑃Line𝑄) → (𝑃Line𝑄) = (𝑃Line𝑆)))
 
Theoremlinerflx1 33507 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) ∧ 𝑃𝑄)) → 𝑃 ∈ (𝑃Line𝑄))
 
Theoremlinecom 33508 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) ∧ 𝑃𝑄)) → (𝑃Line𝑄) = (𝑄Line𝑃))
 
Theoremlinerflx2 33509 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) ∧ 𝑃𝑄)) → 𝑄 ∈ (𝑃Line𝑄))
 
Theoremellines 33510* 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)(𝑝𝑞𝐴 = (𝑝Line𝑞)))
 
Theoremlinethru 33511 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 ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝐴 = (𝑃Line𝑄))
 
Theoremhilbert1.1 33512* 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) ∧ 𝑃𝑄)) → ∃𝑥 ∈ LinesEE (𝑃𝑥𝑄𝑥))
 
Theoremhilbert1.2 33513* 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.)
(𝑃𝑄 → ∃*𝑥 ∈ LinesEE (𝑃𝑥𝑄𝑥))
 
Theoremlinethrueu 33514* 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) ∧ 𝑃𝑄)) → ∃!𝑥 ∈ LinesEE (𝑃𝑥𝑄𝑥))
 
Theoremlineintmo 33515* 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 ∧ 𝐴𝐵) → ∃*𝑥(𝑥𝐴𝑥𝐵))
 
20.9.33  Forward difference
 
Syntaxcfwddif 33516 Declare the syntax for the forward difference operator.
class
 
Definitiondf-fwddif 33517* 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)) − (𝑓𝑥))))
 
Syntaxcfwddifn 33518 Declare the syntax for the nth forward difference operator.
class n
 
Definitiondf-fwddifn 33519* 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.)
n = (𝑛 ∈ ℕ0, 𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))))))
 
Theoremfwddifval 33520 Calculate the value of the forward difference operator at a point. (Contributed by Scott Fenton, 18-May-2020.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝑋𝐴)    &   (𝜑 → (𝑋 + 1) ∈ 𝐴)       (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
 
Theoremfwddifnval 33521* The value of the forward difference operator at a point. (Contributed by Scott Fenton, 28-May-2020.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   ((𝜑𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ 𝐴)       (𝜑 → ((𝑁n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
 
Theoremfwddifn0 33522 The value of the n-iterated forward difference operator at zero is just the function value. (Contributed by Scott Fenton, 28-May-2020.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝑋𝐴)       (𝜑 → ((0 △n 𝐹)‘𝑋) = (𝐹𝑋))
 
Theoremfwddifnp1 33523* The value of the n-iterated forward difference at a successor. (Contributed by Scott Fenton, 28-May-2020.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   ((𝜑𝑘 ∈ (0...(𝑁 + 1))) → (𝑋 + 𝑘) ∈ 𝐴)       (𝜑 → (((𝑁 + 1) △n 𝐹)‘𝑋) = (((𝑁n 𝐹)‘(𝑋 + 1)) − ((𝑁n 𝐹)‘𝑋)))
 
20.9.34  Rank theorems
 
Theoremrankung 33524 The rank of the union of two sets. Closed form of rankun 9273. (Contributed by Scott Fenton, 15-Jul-2015.)
((𝐴𝑉𝐵𝑊) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
 
Theoremranksng 33525 The rank of a singleton. Closed form of ranksn 9271. (Contributed by Scott Fenton, 15-Jul-2015.)
(𝐴𝑉 → (rank‘{𝐴}) = suc (rank‘𝐴))
 
Theoremrankelg 33526 The membership relation is inherited by the rank function. Closed form of rankel 9256. (Contributed by Scott Fenton, 16-Jul-2015.)
((𝐵𝑉𝐴𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
 
Theoremrankpwg 33527 The rank of a power set. Closed form of rankpw 9260. (Contributed by Scott Fenton, 16-Jul-2015.)
(𝐴𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
 
Theoremrank0 33528 The rank of the empty set is . (Contributed by Scott Fenton, 17-Jul-2015.)
(rank‘∅) = ∅
 
Theoremrankeq1o 33529 The only set with rank 1o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)
((rank‘𝐴) = 1o𝐴 = {∅})
 
20.9.35  Hereditarily Finite Sets
 
Syntaxchf 33530 The constant Hf is a class.
class Hf
 
Definitiondf-hf 33531 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)
 
Theoremelhf 33532* Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.)
(𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
 
Theoremelhf2 33533 Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)
𝐴 ∈ V       (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)
 
Theoremelhf2g 33534 Hereditarily finiteness via rank. Closed form of elhf2 33533. (Contributed by Scott Fenton, 15-Jul-2015.)
(𝐴𝑉 → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω))
 
Theorem0hf 33535 The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.)
∅ ∈ Hf
 
Theoremhfun 33536 The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴𝐵) ∈ Hf )
 
Theoremhfsn 33537 The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
(𝐴 ∈ Hf → {𝐴} ∈ Hf )
 
Theoremhfadj 33538 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 )
 
Theoremhfelhf 33539 Any member of an HF set is itself an HF set. (Contributed by Scott Fenton, 16-Jul-2015.)
((𝐴𝐵𝐵 ∈ Hf ) → 𝐴 ∈ Hf )
 
Theoremhftr 33540 The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.)
Tr Hf
 
Theoremhfext 33541* 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 (𝑥𝐴𝑥𝐵)))
 
Theoremhfuni 33542 The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
(𝐴 ∈ Hf → 𝐴 ∈ Hf )
 
Theoremhfpw 33543 The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
(𝐴 ∈ Hf → 𝒫 𝐴 ∈ Hf )
 
Theoremhfninf 33544 ω is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
¬ ω ∈ Hf
 
20.10  Mathbox for Jeff Hankins
 
20.10.1  Miscellany
 
Theorema1i14 33545 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
(𝜓 → (𝜒𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theorema1i24 33546 Add two antecedents to a wff. Deduction associated with a1i13 27. (Contributed by Jeff Hankins, 5-Aug-2009.)
(𝜑 → (𝜒𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp5d 33547 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(((𝜑𝜓) ∧ 𝜒) → ((𝜃𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp5g 33548 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((𝜑𝜓) → (((𝜒𝜃) ∧ 𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp5k 33549 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (((𝜓 ∧ (𝜒𝜃)) ∧ 𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp56 33550 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((((𝜑𝜓) ∧ 𝜒) ∧ (𝜃𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp58 33551 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(((𝜑𝜓) ∧ ((𝜒𝜃) ∧ 𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp510 33552 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((𝜑 ∧ (((𝜓𝜒) ∧ 𝜃) ∧ 𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp511 33553 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((𝜑 ∧ ((𝜓 ∧ (𝜒𝜃)) ∧ 𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp512 33554 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ (𝜃𝜏))) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theorem3com12d 33555 Commutation in consequent. Swap 1st and 2nd. (Contributed by Jeff Hankins, 17-Nov-2009.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑 → (𝜒𝜓𝜃))
 
Theoremimp5p 33556 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → ((𝜒𝜃𝜏) → 𝜂)))
 
Theoremimp5q 33557 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑𝜓) → ((𝜒𝜃𝜏) → 𝜂))
 
Theoremecase13d 33558 Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → ¬ 𝜃)    &   (𝜑 → (𝜒𝜓𝜃))       (𝜑𝜓)
 
Theoremsubtr 33559 Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑌    &   𝑥𝑍    &   (𝑥 = 𝐴𝑋 = 𝑌)    &   (𝑥 = 𝐵𝑋 = 𝑍)       ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵𝑌 = 𝑍))
 
Theoremsubtr2 33560 Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝜓    &   𝑥𝜒    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵 → (𝜓𝜒)))
 
Theoremtrer 33561* 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.)
(∀𝑎𝑏𝑐((𝑎 𝑏𝑏 𝑐) → 𝑎 𝑐) → ( ) Er dom ( ))
 
Theoremelicc3 33562 An equivalent membership condition for closed intervals. (Contributed by Jeff Hankins, 14-Jul-2009.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶𝐶 < 𝐵) ∨ 𝐶 = 𝐵))))
 
Theoremfinminlem 33563* 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 → (𝜑𝜓))       (∃𝑥 ∈ Fin 𝜑 → ∃𝑥(𝜑 ∧ ∀𝑦((𝑦𝑥𝜓) → 𝑥 = 𝑦)))
 
Theoremgtinf 33564* 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 ∀𝑦𝑆 𝑥𝑦) ∧ (𝐴 ∈ ℝ ∧ inf(𝑆, ℝ, < ) < 𝐴)) → ∃𝑧𝑆 𝑧 < 𝐴)
 
Theoremopnrebl 33565* 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 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+ ((𝑥𝑦)(,)(𝑥 + 𝑦)) ⊆ 𝐴))
 
Theoremopnrebl2 33566* 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 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝑧𝑦 ∧ ((𝑥𝑧)(,)(𝑥 + 𝑧)) ⊆ 𝐴)))
 
Theoremnn0prpwlem 33567* Lemma for nn0prpw 33568. 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 ¬ ((𝑝𝑛) ∥ 𝑘 ↔ (𝑝𝑛) ∥ 𝐴)))
 
Theoremnn0prpw 33568* 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 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝𝑛) ∥ 𝐴 ↔ (𝑝𝑛) ∥ 𝐵)))
 
20.10.2  Basic topological facts
 
Theoremtopbnd 33569 Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
 
Theoremopnbnd 33570 A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))
 
Theoremcldbnd 33571 A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴))
 
Theoremntruni 33572* 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) → 𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
 
Theoremclsun 33573 A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
 
Theoremclsint2 33574* 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) ⊆ 𝑐𝐶 ((cls‘𝐽)‘𝑐))
 
Theoremopnregcld 33575* 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 ∧ 𝐴𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
 
Theoremcldregopn 33576* 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 ∧ 𝐴𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))
 
Theoremneiin 33577 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)) → (𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)))
 
Theoremhmeoclda 33578 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)) → (𝐹𝑆) ∈ (Clsd‘𝐾))
 
Theoremhmeocldb 33579 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.)
(((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽Homeo𝐾)) ∧ 𝑆 ∈ (Clsd‘𝐾)) → (𝐹𝑆) ∈ (Clsd‘𝐽))
 
20.10.3  Topology of the real numbers
 
TheoremivthALT 33580* An alternate proof of the Intermediate Value Theorem ivth 23982 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) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ∃𝑥 ∈ (𝐴(,)𝐵)(𝐹𝑥) = 𝑈)
 
20.10.4  Refinements
 
Syntaxcfne 33581 Extend class definition to include the "finer than" relation.
class Fne
 
Definitiondf-fne 33582* Define the fineness relation for covers. (Contributed by Jeff Hankins, 28-Sep-2009.)
Fne = {⟨𝑥, 𝑦⟩ ∣ ( 𝑥 = 𝑦 ∧ ∀𝑧𝑥 𝑧 (𝑦 ∩ 𝒫 𝑧))}
 
Theoremfnerel 33583 Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)
Rel Fne
 
Theoremisfne 33584* 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       (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
 
Theoremisfne4 33585 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 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
 
Theoremisfne4b 33586 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       (𝐵𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
 
Theoremisfne2 33587* 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       (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
 
Theoremisfne3 33588* 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       (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦))))
 
Theoremfnebas 33589 A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝐴Fne𝐵𝑋 = 𝑌)
 
Theoremfnetg 33590 A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝐴Fne𝐵𝐴 ⊆ (topGen‘𝐵))
 
Theoremfnessex 33591* 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.)
((𝐴Fne𝐵𝑆𝐴𝑃𝑆) → ∃𝑥𝐵 (𝑃𝑥𝑥𝑆))
 
Theoremfneuni 33592* If 𝐵 is finer than 𝐴, every element of 𝐴 is a union of elements of 𝐵. (Contributed by Jeff Hankins, 11-Oct-2009.)
((𝐴Fne𝐵𝑆𝐴) → ∃𝑥(𝑥𝐵𝑆 = 𝑥))
 
Theoremfneint 33593* If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)
(𝐴Fne𝐵 {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥})
 
Theoremfness 33594 A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)
𝑋 = 𝐴    &   𝑌 = 𝐵       ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Fne𝐵)
 
Theoremfneref 33595 Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
(𝐴𝑉𝐴Fne𝐴)
 
Theoremfnetr 33596 Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
((𝐴Fne𝐵𝐵Fne𝐶) → 𝐴Fne𝐶)
 
Theoremfneval 33597 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)       ((𝐴𝑉𝐵𝑊) → (𝐴 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵)))
 
Theoremfneer 33598 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
 
Theoremtopfne 33599 Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.)
𝑋 = 𝐽    &   𝑌 = 𝐾       ((𝐾 ∈ Top ∧ 𝑋 = 𝑌) → (𝐽𝐾𝐽Fne𝐾))
 
Theoremtopfneec 33600 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))
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