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Type | Label | Description |
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Statement | ||
Definition | df-gru 9801* | A Grothendieck universe is a set that is closed with respect to all the operations that are common in set theory: pairs, powersets, unions, intersections, Cartesian products etc. Grothendieck and alii, Séminaire de Géométrie Algébrique 4, Exposé I, p. 185. It was designed to give a precise meaning to the concepts of categories of sets, groups... (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑢))} | ||
Theorem | elgrug 9802* | Properties of a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈)))) | ||
Theorem | grutr 9803 | A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝑈 ∈ Univ → Tr 𝑈) | ||
Theorem | gruelss 9804 | A Grothendieck universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈) | ||
Theorem | grupw 9805 | A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈) | ||
Theorem | gruss 9806 | Any subset of an element of a Grothendieck universe is also an element. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑈) | ||
Theorem | grupr 9807 | A Grothendieck universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈) | ||
Theorem | gruurn 9808 | A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 9809 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈) | ||
Theorem | gruiun 9809* | If 𝐵(𝑥) is a family of elements of 𝑈 and the index set 𝐴 is an element of 𝑈, then the indexed union ∪ 𝑥 ∈ 𝐴𝐵 is also an element of 𝑈, where 𝑈 is a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) | ||
Theorem | gruuni 9810 | A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝐴 ∈ 𝑈) | ||
Theorem | grurn 9811 | A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 9809 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈) | ||
Theorem | gruima 9812 | A Grothendieck universe contains image sets drawn from its members. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) → (𝐴 ∈ 𝑈 → (𝐹 “ 𝐴) ∈ 𝑈)) | ||
Theorem | gruel 9813 | Any element of an element of a Grothendieck universe is also an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝑈) | ||
Theorem | grusn 9814 | A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈) | ||
Theorem | gruop 9815 | A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 〈𝐴, 𝐵〉 ∈ 𝑈) | ||
Theorem | gruun 9816 | A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) ∈ 𝑈) | ||
Theorem | gruxp 9817 | A Grothendieck universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈) | ||
Theorem | grumap 9818 | A Grothendieck universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ↑𝑚 𝐵) ∈ 𝑈) | ||
Theorem | gruixp 9819* | A Grothendieck universe contains indexed cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → X𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) | ||
Theorem | gruiin 9820* | A Grothendieck universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) | ||
Theorem | gruf 9821 | A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝐹 ∈ 𝑈) | ||
Theorem | gruen 9822 | A Grothendieck universe contains all subsets of itself that are equipotent to an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ⊆ 𝑈 ∧ (𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴)) → 𝐴 ∈ 𝑈) | ||
Theorem | gruwun 9823 | A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni) | ||
Theorem | intgru 9824 | The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Univ) | ||
Theorem | ingru 9825* | The intersection of a universe with a class that acts like a universe is another universe. (Contributed by Mario Carneiro, 10-Jun-2013.) |
⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))) → (𝑈 ∈ Univ → (𝑈 ∩ 𝐴) ∈ Univ)) | ||
Theorem | wfgru 9826 | The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.) |
⊢ (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ) | ||
Theorem | grudomon 9827 | Each ordinal that is comparable with an element of the universe is in the universe. (Contributed by Mario Carneiro, 10-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝐵 ∈ 𝑈 ∧ 𝐴 ≼ 𝐵)) → 𝐴 ∈ 𝑈) | ||
Theorem | gruina 9828 | If a Grothendieck universe 𝑈 is nonempty, then the height of the ordinals in 𝑈 is a strongly inaccessible cardinal. (Contributed by Mario Carneiro, 17-Jun-2013.) |
⊢ 𝐴 = (𝑈 ∩ On) ⇒ ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝐴 ∈ Inacc) | ||
Theorem | grur1a 9829 | A characterization of Grothendieck universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ 𝐴 = (𝑈 ∩ On) ⇒ ⊢ (𝑈 ∈ Univ → (𝑅1‘𝐴) ⊆ 𝑈) | ||
Theorem | grur1 9830 | A characterization of Grothendieck universes, part 2. (Contributed by Mario Carneiro, 24-Jun-2013.) |
⊢ 𝐴 = (𝑈 ∩ On) ⇒ ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → 𝑈 = (𝑅1‘𝐴)) | ||
Theorem | grutsk1 9831 | Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 9793.) (Contributed by Mario Carneiro, 17-Jun-2013.) |
⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ) | ||
Theorem | grutsk 9832 | Grothendieck universes are the same as transitive Tarski classes. (The proof in the forward direction requires Foundation.) (Contributed by Mario Carneiro, 24-Jun-2013.) |
⊢ Univ = {𝑥 ∈ Tarski ∣ Tr 𝑥} | ||
Axiom | ax-groth 9833* | The Tarski-Grothendieck Axiom. For every set 𝑥 there is an inaccessible cardinal 𝑦 such that 𝑦 is not in 𝑥. The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics." This version of the axiom is used by the Mizar project (http://www.mizar.org/JFM/Axiomatics/tarski.html). Unlike the ZFC axioms, this axiom is very long when expressed in terms of primitive symbols - see grothprim 9844. An open problem is finding a shorter equivalent. (Contributed by NM, 18-Mar-2007.) |
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))) | ||
Theorem | axgroth5 9834* | The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.) |
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) | ||
Theorem | axgroth2 9835* | Alternate version of the Tarski-Grothendieck Axiom. (Contributed by NM, 18-Mar-2007.) |
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑦 ≼ 𝑧 ∨ 𝑧 ∈ 𝑦))) | ||
Theorem | grothpw 9836* | Derive the Axiom of Power Sets ax-pow 4988 from the Tarski-Grothendieck axiom ax-groth 9833. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4988 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.) |
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | ||
Theorem | grothpwex 9837 | Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 9833. Note that ax-pow 4988 is not used by the proof. Use axpweq 4987 to obtain ax-pow 4988. Use pwex 4993 or pwexg 4995 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.) |
⊢ 𝒫 𝑥 ∈ V | ||
Theorem | axgroth6 9838* | The Tarski-Grothendieck axiom using abbreviations. This version is called Tarski's axiom: given a set 𝑥, there exists a set 𝑦 containing 𝑥, the subsets of the members of 𝑦, the power sets of the members of 𝑦, and the subsets of 𝑦 of cardinality less than that of 𝑦. (Contributed by NM, 21-Jun-2009.) |
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦)) | ||
Theorem | grothomex 9839 | The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 8709). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.) |
⊢ ω ∈ V | ||
Theorem | grothac 9840 | The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 9479). This can be put in a more conventional form via ween 9044 and dfac8 9145. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.) |
⊢ dom card = V | ||
Theorem | axgroth3 9841* | Alternate version of the Tarski-Grothendieck Axiom. ax-cc 9445 is used to derive this version. (Contributed by NM, 26-Mar-2007.) |
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → ((𝑦 ∖ 𝑧) ≼ 𝑧 ∨ 𝑧 ∈ 𝑦))) | ||
Theorem | axgroth4 9842* | Alternate version of the Tarski-Grothendieck Axiom. ax-ac 9469 is used to derive this version. (Contributed by NM, 16-Apr-2007.) |
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∃𝑣 ∈ 𝑦 ∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ (𝑦 ∩ 𝑣)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → ((𝑦 ∖ 𝑧) ≼ 𝑧 ∨ 𝑧 ∈ 𝑦))) | ||
Theorem | grothprimlem 9843* | Lemma for grothprim 9844. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.) |
⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣)))) | ||
Theorem | grothprim 9844* | The Tarski-Grothendieck Axiom ax-groth 9833 expanded into set theory primitives using 163 symbols (allowing the defined symbols ∧, ∨, ↔, and ∃). An open problem is whether a shorter equivalent exists (when expanded to primitives). (Contributed by NM, 16-Apr-2007.) |
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧((𝑧 ∈ 𝑦 → ∃𝑣(𝑣 ∈ 𝑦 ∧ ∀𝑤(∀𝑢(𝑢 ∈ 𝑤 → 𝑢 ∈ 𝑧) → (𝑤 ∈ 𝑦 ∧ 𝑤 ∈ 𝑣)))) ∧ ∃𝑤((𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦) → (∀𝑣((𝑣 ∈ 𝑧 → ∃𝑡∀𝑢(∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑣 ∨ ℎ = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣 ∈ 𝑦 → (𝑣 ∈ 𝑧 ∨ ∃𝑢(𝑢 ∈ 𝑧 ∧ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))))) ∨ 𝑧 ∈ 𝑦)))) | ||
Theorem | grothtsk 9845 | The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.) |
⊢ ∪ Tarski = V | ||
Theorem | inaprc 9846 | An equivalent to the Tarski-Grothendieck Axiom: there is a proper class of inaccessible cardinals. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ Inacc ∉ V | ||
Syntax | ctskm 9847 | Extend class definition to include the map whose value is the smallest Tarski class. |
class tarskiMap | ||
Definition | df-tskm 9848* | A function that maps a set 𝑥 to the smallest Tarski class that contains the set. (Contributed by FL, 30-Dec-2010.) |
⊢ tarskiMap = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ Tarski ∣ 𝑥 ∈ 𝑦}) | ||
Theorem | tskmval 9849* | Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.) |
⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) | ||
Theorem | tskmid 9850 | The set 𝐴 is an element of the smallest Tarski class that contains 𝐴. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴)) | ||
Theorem | tskmcl 9851 | A Tarski class that contains 𝐴 is a Tarski class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.) |
⊢ (tarskiMap‘𝐴) ∈ Tarski | ||
Theorem | sstskm 9852* | Being a part of (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥))) | ||
Theorem | eltskm 9853* | Belonging to (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥))) | ||
This section derives the basics of real and complex numbers. We first construct and axiomatize real and complex numbers (e.g., ax-resscn 10181). After that, we derive their basic properties, various operations like addition (df-add 10135) and sine (df-sin 14995), and subsets such as the integers (df-z 11566) and natural numbers (df-nn 11209). | ||
Syntax | cnpi 9854 |
The set of positive integers, which is the set of natural numbers ω
with 0 removed.
Note: This is the start of the Dedekind-cut construction of real and complex numbers. The last lemma of the construction is mulcnsrec 10153. The actual set of Dedekind cuts is defined by df-np 9991. |
class N | ||
Syntax | cpli 9855 | Positive integer addition. |
class +N | ||
Syntax | cmi 9856 | Positive integer multiplication. |
class ·N | ||
Syntax | clti 9857 | Positive integer ordering relation. |
class <N | ||
Syntax | cplpq 9858 | Positive pre-fraction addition. |
class +pQ | ||
Syntax | cmpq 9859 | Positive pre-fraction multiplication. |
class ·pQ | ||
Syntax | cltpq 9860 | Positive pre-fraction ordering relation. |
class <pQ | ||
Syntax | ceq 9861 | Equivalence class used to construct positive fractions. |
class ~Q | ||
Syntax | cnq 9862 | Set of positive fractions. |
class Q | ||
Syntax | c1q 9863 | The positive fraction constant 1. |
class 1Q | ||
Syntax | cerq 9864 | Positive fraction equivalence class. |
class [Q] | ||
Syntax | cplq 9865 | Positive fraction addition. |
class +Q | ||
Syntax | cmq 9866 | Positive fraction multiplication. |
class ·Q | ||
Syntax | crq 9867 | Positive fraction reciprocal operation. |
class *Q | ||
Syntax | cltq 9868 | Positive fraction ordering relation. |
class <Q | ||
Syntax | cnp 9869 | Set of positive reals. |
class P | ||
Syntax | c1p 9870 | Positive real constant 1. |
class 1P | ||
Syntax | cpp 9871 | Positive real addition. |
class +P | ||
Syntax | cmp 9872 | Positive real multiplication. |
class ·P | ||
Syntax | cltp 9873 | Positive real ordering relation. |
class <P | ||
Syntax | cer 9874 | Equivalence class used to construct signed reals. |
class ~R | ||
Syntax | cnr 9875 | Set of signed reals. |
class R | ||
Syntax | c0r 9876 | The signed real constant 0. |
class 0R | ||
Syntax | c1r 9877 | The signed real constant 1. |
class 1R | ||
Syntax | cm1r 9878 | The signed real constant -1. |
class -1R | ||
Syntax | cplr 9879 | Signed real addition. |
class +R | ||
Syntax | cmr 9880 | Signed real multiplication. |
class ·R | ||
Syntax | cltr 9881 | Signed real ordering relation. |
class <R | ||
Definition | df-ni 9882 | Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10130, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.) |
⊢ N = (ω ∖ {∅}) | ||
Definition | df-pli 9883 | Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10130, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.) |
⊢ +N = ( +𝑜 ↾ (N × N)) | ||
Definition | df-mi 9884 | Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10130, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.) |
⊢ ·N = ( ·𝑜 ↾ (N × N)) | ||
Definition | df-lti 9885 | Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10130, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.) (New usage is discouraged.) |
⊢ <N = ( E ∩ (N × N)) | ||
Theorem | elni 9886 | Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.) |
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | ||
Theorem | elni2 9887 | Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) (New usage is discouraged.) |
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) | ||
Theorem | pinn 9888 | A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.) |
⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | ||
Theorem | pion 9889 | A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ N → 𝐴 ∈ On) | ||
Theorem | piord 9890 | A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ N → Ord 𝐴) | ||
Theorem | niex 9891 | The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.) |
⊢ N ∈ V | ||
Theorem | 0npi 9892 | The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.) |
⊢ ¬ ∅ ∈ N | ||
Theorem | 1pi 9893 | Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.) |
⊢ 1𝑜 ∈ N | ||
Theorem | addpiord 9894 | Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵)) | ||
Theorem | mulpiord 9895 | Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵)) | ||
Theorem | mulidpi 9896 | 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
⊢ (𝐴 ∈ N → (𝐴 ·N 1𝑜) = 𝐴) | ||
Theorem | ltpiord 9897 | Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵)) | ||
Theorem | ltsopi 9898 | Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.) |
⊢ <N Or N | ||
Theorem | ltrelpi 9899 | Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.) |
⊢ <N ⊆ (N × N) | ||
Theorem | dmaddpi 9900 | Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.) |
⊢ dom +N = (N × N) |
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