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Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsupval2ti 6401* Alternate expression for the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)

Theoremeqsupti 6402* Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)

Theoremeqsuptid 6403* Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 24-Nov-2021.)

Theoremsupclti 6404* A supremum belongs to its base class (closure law). See also supubti 6405 and suplubti 6406. (Contributed by Jim Kingdon, 24-Nov-2021.)

Theoremsupubti 6405* A supremum is an upper bound. See also supclti 6404 and suplubti 6406.

This proof demonstrates how to expand an iota-based definition (df-iota 4895) using riotacl2 5509.

(Contributed by Jim Kingdon, 24-Nov-2021.)

Theoremsuplubti 6406* A supremum is the least upper bound. See also supclti 6404 and supubti 6405. (Contributed by Jim Kingdon, 24-Nov-2021.)

Theoremsup00 6407 The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)

Theoremsupmaxti 6408* The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.)

Theoremsupsnti 6409* The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)

Theoremisotilem 6410* Lemma for isoti 6411. (Contributed by Jim Kingdon, 26-Nov-2021.)

Theoremisoti 6411* An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.)

Theoremsupisolem 6412* Lemma for supisoti 6414. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremsupisoex 6413* Lemma for supisoti 6414. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremsupisoti 6414* Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)

2.6.29  Ordinal isomorphism

Theoremordiso2 6415 Generalize ordiso 6416 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)

Theoremordiso 6416* Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)

2.6.30  Cardinal numbers

Syntaxccrd 6417 Extend class definition to include the cardinal size function.

Definitiondf-card 6418* Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)

Theoremcardcl 6419* The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)

Theoremisnumi 6420 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremfinnum 6421 Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremonenon 6422 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremcardval3ex 6423* The value of . (Contributed by Jim Kingdon, 30-Aug-2021.)

Theoremoncardval 6424* The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremcardonle 6425 The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)

Theoremcard0 6426 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)

Theoremcarden2bex 6427* If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)

PART 3  REAL AND COMPLEX NUMBERS

This section derives the basics of real and complex numbers.

To construct the real numbers constructively, we follow two main sources. The first is Metamath Proof Explorer, which has the advantage of being already formalized in metamath. Its disadvantage, for our purposes, is that it assumes the law of the excluded middle throughout. Since we have already developed natural numbers ( for example, nna0 6084 and similar theorems ), going from there to positive integers (df-ni 6460) and then positive rational numbers (df-nqqs 6504) does not involve a major change in approach compared with the Metamath Proof Explorer.

It is when we proceed to Dedekind cuts that we bring in more material from Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which are different without excluded middle. With excluded middle, it is natural to define the cut as the lower set only (as Metamath Proof Explorer does), but we define the cut as a pair of both the lower and upper sets, as [HoTT] does. There are also differences in how we handle order and replacing "not equal to zero" with "apart from zero".

3.1  Construction and axiomatization of real and complex numbers

3.1.1  Dedekind-cut construction of real and complex numbers

Syntaxcnpi 6428 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.

Syntaxcmi 6430 Positive integer multiplication.

Syntaxclti 6431 Positive integer ordering relation.

Syntaxcmpq 6433 Positive pre-fraction multiplication.

Syntaxcltpq 6434 Positive pre-fraction ordering relation.

Syntaxceq 6435 Equivalence class used to construct positive fractions.

Syntaxcnq 6436 Set of positive fractions.

Syntaxc1q 6437 The positive fraction constant 1.

Syntaxcmq 6439 Positive fraction multiplication.

Syntaxcrq 6440 Positive fraction reciprocal operation.

Syntaxcltq 6441 Positive fraction ordering relation.

Syntaxceq0 6442 Equivalence class used to construct non-negative fractions.
~Q0

Syntaxcnq0 6443 Set of non-negative fractions.
Q0

Syntaxc0q0 6444 The non-negative fraction constant 0.
0Q0

+Q0

Syntaxcmq0 6446 Non-negative fraction multiplication.
·Q0

Syntaxcnp 6447 Set of positive reals.

Syntaxc1p 6448 Positive real constant 1.

Syntaxcmp 6450 Positive real multiplication.

Syntaxcltp 6451 Positive real ordering relation.

Syntaxcer 6452 Equivalence class used to construct signed reals.

Syntaxcnr 6453 Set of signed reals.

Syntaxc0r 6454 The signed real constant 0.

Syntaxc1r 6455 The signed real constant 1.

Syntaxcm1r 6456 The signed real constant -1.

Syntaxcmr 6458 Signed real multiplication.

Syntaxcltr 6459 Signed real ordering relation.

Definitiondf-ni 6460 Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.)

Definitiondf-pli 6461 Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.)

Definitiondf-mi 6462 Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.)

Definitiondf-lti 6463 Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.)

Theoremelni 6464 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)

Theorempinn 6465 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)

Theorempion 6466 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)

Theorempiord 6467 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)

Theoremniex 6468 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)

Theorem0npi 6469 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)

Theoremelni2 6470 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)

Theorem1pi 6471 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)

Theoremmulpiord 6473 Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)

Theoremmulidpi 6474 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremltpiord 6475 Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremltsopi 6476 Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.)

Theorempitric 6477 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)

Theorempitri3or 6478 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)

Theoremltdcpi 6479 Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
DECID

Theoremltrelpi 6480 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)

Theoremdmaddpi 6481 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)

Theoremdmmulpi 6482 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)

Theoremaddclpi 6483 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)

Theoremmulclpi 6484 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)

Theoremaddcompig 6485 Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)

Theoremaddasspig 6486 Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)

Theoremmulcompig 6487 Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)

Theoremmulasspig 6488 Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)

Theoremdistrpig 6489 Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.)

Theoremaddcanpig 6490 Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.)

Theoremmulcanpig 6491 Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)

Theoremaddnidpig 6492 There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.)

Theoremltexpi 6493* Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)

Theoremltapig 6494 Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)

Theoremltmpig 6495 Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)

Theorem1lt2pi 6496 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)

Theoremnlt1pig 6497 No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)

Theoremindpi 6498* Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.)

Theoremnnppipi 6499 A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.)

Definitiondf-plpq 6500* Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers, 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-plqqs 6505) works with the equivalence classes of these ordered pairs determined by the equivalence relation (df-enq 6503). (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.)

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