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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | filnetlem4 32501* | Lemma for filnet 32502. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.) |
⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) & ⊢ 𝐷 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} ⇒ ⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)))) | ||
Theorem | filnet 32502* | A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.) |
⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)))) | ||
Theorem | tb-ax1 32503 | The first of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | ||
Theorem | tb-ax2 32504 | The second of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜑)) | ||
Theorem | tb-ax3 32505 |
The third of three axioms in the Tarski-Bernays axiom system.
This axiom, along with ax-mp 5, tb-ax1 32503, and tb-ax2 32504, can be used to derive any theorem or rule that uses only →. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑) | ||
Theorem | tbsyl 32506 | The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | re1ax2lem 32507 | Lemma for re1ax2 32508. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | re1ax2 32508 | ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 32503 is replaced with this theorem to make a "standard" system. This is because this theorem is easier to work with, despite it being longer. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | ||
Theorem | naim1 32509 | Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 ⊼ 𝜒) → (𝜑 ⊼ 𝜒))) | ||
Theorem | naim2 32510 | Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑))) | ||
Theorem | naim1i 32511 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 ⊼ 𝜒) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Theorem | naim2i 32512 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 ⊼ 𝜓) ⇒ ⊢ (𝜒 ⊼ 𝜑) | ||
Theorem | naim12i 32513 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜃) & ⊢ (𝜓 ⊼ 𝜃) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Theorem | nabi1 32514 | Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒))) | ||
Theorem | nabi2 32515 | Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓))) | ||
Theorem | nabi1i 32516 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜓 ⊼ 𝜒) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Theorem | nabi2i 32517 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ⊼ 𝜓) ⇒ ⊢ (𝜒 ⊼ 𝜑) | ||
Theorem | nabi12i 32518 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) & ⊢ (𝜓 ⊼ 𝜃) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Syntax | w3nand 32519 | The double nand. |
wff (𝜑 ⊼ 𝜓 ⊼ 𝜒) | ||
Definition | df-3nand 32520 | The double nand. This definition allows us to express the input of three variables only being false if all three are true. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒))) | ||
Theorem | df3nandALT1 32521 | The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 ⊼ ((𝜓 ⊼ 𝜒) ⊼ (𝜓 ⊼ 𝜒)))) | ||
Theorem | df3nandALT2 32522 | The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)) | ||
Theorem | andnand1 32523 | Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ⊼ (𝜑 ⊼ 𝜓 ⊼ 𝜒))) | ||
Theorem | imnand2 32524 | An → nand relation. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ ((¬ 𝜑 → 𝜓) ↔ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))) | ||
Theorem | allt 32525 | For all sets, ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ∀𝑥⊤ | ||
Theorem | alnof 32526 | For all sets, ⊥ is not true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ∀𝑥 ¬ ⊥ | ||
Theorem | nalf 32527 | Not all sets hold ⊥ as true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∀𝑥⊥ | ||
Theorem | extt 32528 | There exists a set that holds ⊤ as true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ∃𝑥⊤ | ||
Theorem | nextnt 32529 | There does not exist a set, such that ⊤ is not true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃𝑥 ¬ ⊤ | ||
Theorem | nextf 32530 | There does not exist a set, such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃𝑥⊥ | ||
Theorem | unnf 32531 | There does not exist exactly one set, such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃!𝑥⊥ | ||
Theorem | unnt 32532 | There does not exist exactly one set, such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃!𝑥⊤ | ||
Theorem | mont 32533 | There does not exist at most one set, such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃*𝑥⊤ | ||
Theorem | mof 32534 | There exist at most one set, such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ∃*𝑥⊥ | ||
Theorem | meran1 32535 | A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜃 ∨ 𝜑) ∨ (𝜒 ∨ (𝜏 ∨ 𝜑)))) | ||
Theorem | meran2 32536 | A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜏 ∨ 𝜃) ∨ (𝜒 ∨ (𝜑 ∨ 𝜃)))) | ||
Theorem | meran3 32537 | A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜒 ∨ 𝜑) ∨ (𝜏 ∨ (𝜃 ∨ 𝜑)))) | ||
Theorem | waj-ax 32538 | A single axiom for propositional calculus offered by Wajsberg. (Contributed by Anthony Hart, 13-Aug-2011.) |
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ (((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))) ⊼ (𝜑 ⊼ (𝜑 ⊼ 𝜓)))) | ||
Theorem | lukshef-ax2 32539 | A single axiom for propositional calculus offered by Lukasiewicz. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜑)) ⊼ ((𝜃 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) | ||
Theorem | arg-ax 32540 | ? (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜒 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) | ||
Theorem | negsym1 32541 |
In the paper "On Variable Functors of Propositional Arguments",
Lukasiewicz introduced a system that can handle variable connectives.
This was done by introducing a variable, marked with a lowercase delta,
which takes a wff as input. In the system, "delta 𝜑 "
means that
"something is true of 𝜑." "delta 𝜑 "
can be substituted with
¬ 𝜑, 𝜓 ∧ 𝜑, ∀𝑥𝜑, etc.
Later on, Meredith discovered a single axiom, in the form of ( delta delta ⊥ → delta 𝜑 ). This represents the shortest theorem in the extended propositional calculus that cannot be derived as an instance of a theorem in propositional calculus. A symmetry with ¬. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ (¬ ¬ ⊥ → ¬ 𝜑) | ||
Theorem | imsym1 32542 |
A symmetry with →.
See negsym1 32541 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 → (𝜓 → ⊥)) → (𝜓 → 𝜑)) | ||
Theorem | bisym1 32543 |
A symmetry with ↔.
See negsym1 32541 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓 ↔ 𝜑)) | ||
Theorem | consym1 32544 |
A symmetry with ∧.
See negsym1 32541 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑)) | ||
Theorem | dissym1 32545 |
A symmetry with ∨.
See negsym1 32541 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 ∨ (𝜓 ∨ ⊥)) → (𝜓 ∨ 𝜑)) | ||
Theorem | nandsym1 32546 |
A symmetry with ⊼.
See negsym1 32541 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓 ⊼ 𝜑)) | ||
Theorem | unisym1 32547 |
A symmetry with ∀.
See negsym1 32541 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
⊢ (∀𝑥∀𝑥⊥ → ∀𝑥𝜑) | ||
Theorem | exisym1 32548 |
A symmetry with ∃.
See negsym1 32541 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ (∃𝑥∃𝑥⊥ → ∃𝑥𝜑) | ||
Theorem | unqsym1 32549 |
A symmetry with ∃!.
See negsym1 32541 for more information. (Contributed by Anthony Hart, 6-Sep-2011.) |
⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑) | ||
Theorem | amosym1 32550 |
A symmetry with ∃*.
See negsym1 32541 for more information. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑) | ||
Theorem | subsym1 32551 |
A symmetry with [𝑥 / 𝑦].
See negsym1 32541 for more information. (Contributed by Anthony Hart, 11-Sep-2011.) |
⊢ ([𝑥 / 𝑦][𝑥 / 𝑦]⊥ → [𝑥 / 𝑦]𝜑) | ||
Theorem | ontopbas 32552 | An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.) |
⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases) | ||
Theorem | onsstopbas 32553 | The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.) |
⊢ On ⊆ TopBases | ||
Theorem | onpsstopbas 32554 | The class of ordinal numbers is a proper subclass of the class of topological bases. (Contributed by Chen-Pang He, 9-Oct-2015.) |
⊢ On ⊊ TopBases | ||
Theorem | ontgval 32555 | The topology generated from an ordinal number 𝐵 is suc ∪ 𝐵. (Contributed by Chen-Pang He, 10-Oct-2015.) |
⊢ (𝐵 ∈ On → (topGen‘𝐵) = suc ∪ 𝐵) | ||
Theorem | ontgsucval 32556 | The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.) |
⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴) | ||
Theorem | onsuctop 32557 | A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top) | ||
Theorem | onsuctopon 32558 | One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴)) | ||
Theorem | ordtoplem 32559 | Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.) |
⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆) ⇒ ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆)) | ||
Theorem | ordtop 32560 | An ordinal is a topology iff it is not its supremum (union), proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 1-Nov-2015.) |
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) | ||
Theorem | onsucconni 32561 | A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.) |
⊢ 𝐴 ∈ On ⇒ ⊢ suc 𝐴 ∈ Conn | ||
Theorem | onsucconn 32562 | A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Conn) | ||
Theorem | ordtopconn 32563 | An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.) |
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn)) | ||
Theorem | onintopssconn 32564 | An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.) |
⊢ (On ∩ Top) ⊆ Conn | ||
Theorem | onsuct0 32565 | A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Kol2) | ||
Theorem | ordtopt0 32566 | An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.) |
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2)) | ||
Theorem | onsucsuccmpi 32567 | The successor of a successor ordinal number is a compact topology, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 18-Oct-2015.) |
⊢ 𝐴 ∈ On ⇒ ⊢ suc suc 𝐴 ∈ Comp | ||
Theorem | onsucsuccmp 32568 | The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015.) |
⊢ (𝐴 ∈ On → suc suc 𝐴 ∈ Comp) | ||
Theorem | limsucncmpi 32569 | The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.) |
⊢ Lim 𝐴 ⇒ ⊢ ¬ suc 𝐴 ∈ Comp | ||
Theorem | limsucncmp 32570 | The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.) |
⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp) | ||
Theorem | ordcmp 32571 | An ordinal topology is compact iff the underlying set is its supremum (union) only when the ordinal is 1𝑜. (Contributed by Chen-Pang He, 1-Nov-2015.) |
⊢ (Ord 𝐴 → (𝐴 ∈ Comp ↔ (∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1𝑜))) | ||
Theorem | ssoninhaus 32572 | The ordinal topologies 1𝑜 and 2𝑜 are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.) |
⊢ {1𝑜, 2𝑜} ⊆ (On ∩ Haus) | ||
Theorem | onint1 32573 | The ordinal T1 spaces are 1𝑜 and 2𝑜, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015.) |
⊢ (On ∩ Fre) = {1𝑜, 2𝑜} | ||
Theorem | oninhaus 32574 | The ordinal Hausdorff spaces are 1𝑜 and 2𝑜. (Contributed by Chen-Pang He, 10-Nov-2015.) |
⊢ (On ∩ Haus) = {1𝑜, 2𝑜} | ||
Theorem | fveleq 32575 | Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.) |
⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃))) | ||
Theorem | findfvcl 32576* | Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.) |
⊢ (𝜑 → (𝐹‘∅) ∈ 𝑃) & ⊢ (𝑦 ∈ ω → (𝜑 → ((𝐹‘𝑦) ∈ 𝑃 → (𝐹‘suc 𝑦) ∈ 𝑃))) ⇒ ⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 𝑃)) | ||
Theorem | findreccl 32577* | Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.) |
⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃) ⇒ ⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃)) | ||
Theorem | findabrcl 32578* | Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃) ⇒ ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ 𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) ∈ 𝑃) | ||
Theorem | nnssi2 32579 | Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
⊢ ℕ ⊆ 𝐷 & ⊢ (𝐵 ∈ ℕ → 𝜑) & ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑) → 𝜓) ⇒ ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓) | ||
Theorem | nnssi3 32580 | Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
⊢ ℕ ⊆ 𝐷 & ⊢ (𝐶 ∈ ℕ → 𝜑) & ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ∧ 𝜑) → 𝜓) ⇒ ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓) | ||
Theorem | nndivsub 32581 | Please add description here. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴 / 𝐶) ∈ ℕ ∧ 𝐴 < 𝐵)) → ((𝐵 / 𝐶) ∈ ℕ ↔ ((𝐵 − 𝐴) / 𝐶) ∈ ℕ)) | ||
Theorem | nndivlub 32582 | A factor of a positive integer cannot exceed it. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) ∈ ℕ → 𝐵 ≤ 𝐴)) | ||
Syntax | cgcdOLD 32583 | Extend class notation to include the gdc function. (New usage is discouraged.) |
class gcdOLD (𝐴, 𝐵) | ||
Definition | df-gcdOLD 32584* | gcdOLD (𝐴, 𝐵) is the largest positive integer that evenly divides both 𝐴 and 𝐵. (Contributed by Jeff Hoffman, 17-Jun-2008.) (New usage is discouraged.) |
⊢ gcdOLD (𝐴, 𝐵) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, < ) | ||
Theorem | ee7.2aOLD 32585 | Lemma for Euclid's Elements, Book 7, proposition 2. The original mentions the smaller measure being 'continually subtracted' from the larger. Many authors interpret this phrase as 𝐴 mod 𝐵. Here, just one subtraction step is proved to preserve the gcdOLD. The rec function will be used in other proofs for iterated subtraction. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 → gcdOLD (𝐴, 𝐵) = gcdOLD (𝐴, (𝐵 − 𝐴)))) | ||
Theorem | dnival 32586* | Value of the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) ⇒ ⊢ (𝐴 ∈ ℝ → (𝑇‘𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) | ||
Theorem | dnicld1 32587 | Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ ℝ) | ||
Theorem | dnicld2 32588* | Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑇‘𝐴) ∈ ℝ) | ||
Theorem | dnif 32589 | The "distance to nearest integer" function is a function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) ⇒ ⊢ 𝑇:ℝ⟶ℝ | ||
Theorem | dnizeq0 32590* | The distance to nearest integer is zero for integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑇‘𝐴) = 0) | ||
Theorem | dnizphlfeqhlf 32591* | The distance to nearest integer is a half for half-integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑇‘(𝐴 + (1 / 2))) = (1 / 2)) | ||
Theorem | rddif2 32592 | Variant of rddif 14124. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝐴 ∈ ℝ → 0 ≤ ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) | ||
Theorem | dnibndlem1 32593* | Lemma for dnibnd 32606. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ 𝑆 ↔ (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ 𝑆)) | ||
Theorem | dnibndlem2 32594* | Lemma for dnibnd 32606. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) = (⌊‘(𝐴 + (1 / 2)))) ⇒ ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | ||
Theorem | dnibndlem3 32595 | Lemma for dnibnd 32606. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) = ((⌊‘(𝐴 + (1 / 2))) + 1)) ⇒ ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) = (abs‘((𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))) + (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)))) | ||
Theorem | dnibndlem4 32596 | Lemma for dnibnd 32606. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝐵 ∈ ℝ → 0 ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2)))) | ||
Theorem | dnibndlem5 32597 | Lemma for dnibnd 32606. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝐴 ∈ ℝ → 0 < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) | ||
Theorem | dnibndlem6 32598 | Lemma for dnibnd 32606. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) + ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))) | ||
Theorem | dnibndlem7 32599 | Lemma for dnibnd 32606. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2)))) | ||
Theorem | dnibndlem8 32600 | Lemma for dnibnd 32606. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ≤ (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) |
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