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Theorem List for Metamath Proof Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaxgroth5 8701* The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.)

Theoremaxgroth2 8702* Alternate version of the Tarski-Grothendieck Axiom. (Contributed by NM, 18-Mar-2007.)

4.2.2  Derive the Power Set, Infinity and Choice Axioms

Theoremgrothpw 8703* Derive the Axiom of Power Sets ax-pow 4379 from the Tarski-Grothendieck axiom ax-groth 8700. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4379 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.)

Theoremgrothpwex 8704 Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 8700. Note that ax-pow 4379 is not used by the proof. Use axpweq 4378 to obtain ax-pow 4379. (Contributed by Gérard Lang, 22-Jun-2009.)

Theoremaxgroth6 8705* The Tarski-Grothendieck axiom using abbreviations. This version is called Tarski's axiom: given a set , there exists a set containing , the subsets of the members of , the power sets of the members of , and the subsets of of cardinality less than that of . (Contributed by NM, 21-Jun-2009.)

Theoremgrothomex 8706 The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 7600). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.)

Theoremgrothac 8707 The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 8351). This can be put in a more conventional form via ween 7918 and dfac8 8017. 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.)

Theoremaxgroth3 8708* Alternate version of the Tarski-Grothendieck Axiom. ax-cc 8317 is used to derive this version. (Contributed by NM, 26-Mar-2007.)

Theoremaxgroth4 8709* Alternate version of the Tarski-Grothendieck Axiom. ax-ac 8341 is used to derive this version. (Contributed by NM, 16-Apr-2007.)

Theoremgrothprimlem 8710* Lemma for grothprim 8711. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)

Theoremgrothprim 8711* The Tarski-Grothendieck Axiom ax-groth 8700 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.)

Theoremgrothtsk 8712 The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)

Theoreminaprc 8713 An equivalent to the Tarski-Grothendieck Axiom: there is a proper class of inaccessible cardinals. (Contributed by Mario Carneiro, 9-Jun-2013.)

4.2.3  Tarski map function

Syntaxctskm 8714 Extend class definition to include the map whose value is the smallest Tarski's class.

Definitiondf-tskm 8715* A function that maps a set to the smallest Tarski's class that contains the set. (Contributed by FL, 30-Dec-2010.)

Theoremtskmval 8716* Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)

Theoremtskmid 8717 The set is an element of the smallest Tarski's class that contains . CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)

Theoremtskmcl 8718 A Tarski's class that contains is a Tarski's class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)

Theoremsstskm 8719* Being a part of . (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Theoremeltskm 8720* Belonging to . (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)

PART 5  REAL AND COMPLEX NUMBERS

This section derives the basics of real and complex numbers. We first construct and axiomitize real and complex numbers (e.g., ax-resscn 9049). After that we derive their basic properties, various operations like addition (df-add 9003) and sine (df-sin 12674), and subsets such as the integers (df-z 10285) and natural numbers (df-nn 10003).

5.1  Construction and axiomatization of real and complex numbers

5.1.1  Dedekind-cut construction of real and complex numbers

Syntaxcnpi 8721 The set of positive integers, which is the set of natural numbers 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 9021. The actual set of Dedekind cuts is defined by df-np 8860.

Syntaxcpli 8722 Positive integer addition.

Syntaxcmi 8723 Positive integer multiplication.

Syntaxclti 8724 Positive integer ordering relation.

Syntaxcplpq 8725 Positive pre-fraction addition.

Syntaxcmpq 8726 Positive pre-fraction multiplication.

Syntaxcltpq 8727 Positive pre-fraction ordering relation.

Syntaxceq 8728 Equivalence class used to construct positive fractions.

Syntaxcnq 8729 Set of positive fractions.

Syntaxc1q 8730 The positive fraction constant 1.

Syntaxcerq 8731 Positive fraction equivalence class.

Syntaxcplq 8732 Positive fraction addition.

Syntaxcmq 8733 Positive fraction multiplication.

Syntaxcrq 8734 Positive fraction reciprocal operation.

Syntaxcltq 8735 Positive fraction ordering relation.

Syntaxcnp 8736 Set of positive reals.

Syntaxc1p 8737 Positive real constant 1.

Syntaxcpp 8738 Positive real addition.

Syntaxcmp 8739 Positive real multiplication.

Syntaxcltp 8740 Positive real ordering relation.

Syntaxcplpr 8741 Signed real pre-addition.

Syntaxcmpr 8742 Signed real pre-multiplication.

Syntaxcer 8743 Equivalence class used to construct signed reals.

Syntaxcnr 8744 Set of signed reals.

Syntaxc0r 8745 The signed real constant 0.

Syntaxc1r 8746 The signed real constant 1.

Syntaxcm1r 8747 The signed real constant -1.

Syntaxcplr 8748 Signed real addition.

Syntaxcmr 8749 Signed real multiplication.

Syntaxcltr 8750 Signed real ordering relation.

Definitiondf-ni 8751 Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)

Definitiondf-pli 8752 Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)

Definitiondf-mi 8753 Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)

Definitiondf-lti 8754 Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.) (New usage is discouraged.)

Theoremelni 8755 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)

Theoremelni2 8756 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) (New usage is discouraged.)

Theorempinn 8757 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)

Theorempion 8758 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)

Theorempiord 8759 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) (New usage is discouraged.)

Theoremniex 8760 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)

Theorem0npi 8761 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)

Theorem1pi 8762 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.)

Theoremaddpiord 8763 Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Theoremmulpiord 8764 Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Theoremmulidpi 8765 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.)

Theoremltpiord 8766 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.)

Theoremltsopi 8767 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.)

Theoremltrelpi 8768 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.)

Theoremdmaddpi 8769 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)

Theoremdmmulpi 8770 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)

Theoremaddclpi 8771 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)

Theoremmulclpi 8772 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)

Theoremaddcompi 8773 Addition of positive integers is commutative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Theoremaddasspi 8774 Addition of positive integers is associative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Theoremmulcompi 8775 Multiplication of positive integers is commutative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)

Theoremmulasspi 8776 Multiplication of positive integers is associative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)

Theoremdistrpi 8777 Multiplication of positive integers is distributive. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)

Theoremaddcanpi 8778 Addition cancellation law for positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremmulcanpi 8779 Multiplication cancellation law for positive integers. (Contributed by NM, 4-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Theoremaddnidpi 8780 There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) (New usage is discouraged.)

Theoremltexpi 8781* Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)

Theoremltapi 8782 Ordering property of addition for positive integers. (Contributed by NM, 7-Mar-1996.) (New usage is discouraged.)

Theoremltmpi 8783 Ordering property of multiplication for positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.)

Theorem1lt2pi 8784 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)

Theoremnlt1pi 8785 No positive integer is less than one. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)

Theoremindpi 8786* Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)

Definitiondf-plpq 8787* Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition (df-plq 8793) works with the equivalence classes of these ordered pairs determined by the equivalence relation (df-enq 8790). (Analogous remarks apply to the other "pre-" operations in the complex number construction that follows.) From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)

Definitiondf-mpq 8788* Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)

Definitiondf-ltpq 8789* Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)

Definitiondf-enq 8790* Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. From Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Definitiondf-nq 8791* Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 16-Aug-1995.) (New usage is discouraged.)

Definitiondf-erq 8792 Define a convenience function that "reduces" a fraction to lowest terms. Note that in this form, it is not obviously a function; we prove this in nqerf 8809. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Definitiondf-plq 8793 Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 24-Aug-1995.) (New usage is discouraged.)

Definitiondf-mq 8794 Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 24-Aug-1995.) (New usage is discouraged.)

Definitiondf-1nq 8795 Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.)

Definitiondf-rq 8796 Define reciprocal on positive fractions. It means the same thing as one divided by the argument (although we don't define full division since we will never need it). This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. From Proposition 9-2.5 of [Gleason] p. 119, who uses an asterisk to denote this unary operation. (Contributed by NM, 6-Mar-1996.) (New usage is discouraged.)

Definitiondf-ltnq 8797 Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.) (New usage is discouraged.)

Theoremenqbreq 8798 Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Theoremenqbreq2 8799 Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremenqer 8800 The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.)

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