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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mkvprop 6901* | Markov's Principle expressed in terms of propositions (or more precisely, the case is Markov's Principle). (Contributed by Jim Kingdon, 19-Mar-2023.) |
Markov DECID | ||
Theorem | fodjumkvlemres 6902* | Lemma for fodjumkv 6903. The final result with expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.) |
Markov ⊔ inl | ||
Theorem | fodjumkv 6903* | A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.) |
Markov ⊔ | ||
Syntax | ccrd 6904 | Extend class definition to include the cardinal size function. |
Definition | df-card 6905* | 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.) |
Theorem | cardcl 6906* | The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Theorem | isnumi 6907 | A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
Theorem | finnum 6908 | Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Theorem | onenon 6909 | Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
Theorem | cardval3ex 6910* | The value of . (Contributed by Jim Kingdon, 30-Aug-2021.) |
Theorem | oncardval 6911* | 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.) |
Theorem | cardonle 6912 | 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.) |
Theorem | card0 6913 | The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
Theorem | carden2bex 6914* | If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Theorem | pm54.43 6915 | Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.) |
Theorem | pr2nelem 6916 | Lemma for pr2ne 6917. (Contributed by FL, 17-Aug-2008.) |
Theorem | pr2ne 6917 | If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.) |
Theorem | en2eleq 6918 | Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
Theorem | en2other2 6919 | Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
Theorem | dju1p1e2 6920 | Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.) |
⊔ | ||
Theorem | infpwfidom 6921 | The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption because this theorem also implies that is a set if is.) (Contributed by Mario Carneiro, 17-May-2015.) |
Theorem | exmidfodomrlemeldju 6922 | Lemma for exmidfodomr 6927. A variant of djur 6837. (Contributed by Jim Kingdon, 2-Jul-2022.) |
⊔ inl inr | ||
Theorem | exmidfodomrlemreseldju 6923 | Lemma for exmidfodomrlemrALT 6926. A variant of eldju 6839. (Contributed by Jim Kingdon, 9-Jul-2022.) |
⊔ inl inr | ||
Theorem | exmidfodomrlemim 6924* | Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.) |
EXMID | ||
Theorem | exmidfodomrlemr 6925* | The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.) |
EXMID | ||
Theorem | exmidfodomrlemrALT 6926* | The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. An alternative proof of exmidfodomrlemr 6925. In particular, this proof uses eldju 6839 instead of djur 6837 and avoids djulclb 6827. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.) |
EXMID | ||
Theorem | exmidfodomr 6927* | Excluded middle is equivalent to the existence of a mapping from any set onto any inhabited set that it dominates. (Contributed by Jim Kingdon, 1-Jul-2022.) |
EXMID | ||
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 6275 and similar theorems ), going from there to positive integers (df-ni 6960) and then positive rational numbers (df-nqqs 7004) 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". | ||
Syntax | cnpi 6928 |
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. |
Syntax | cpli 6929 | Positive integer addition. |
Syntax | cmi 6930 | Positive integer multiplication. |
Syntax | clti 6931 | Positive integer ordering relation. |
Syntax | cplpq 6932 | Positive pre-fraction addition. |
Syntax | cmpq 6933 | Positive pre-fraction multiplication. |
Syntax | cltpq 6934 | Positive pre-fraction ordering relation. |
Syntax | ceq 6935 | Equivalence class used to construct positive fractions. |
Syntax | cnq 6936 | Set of positive fractions. |
Syntax | c1q 6937 | The positive fraction constant 1. |
Syntax | cplq 6938 | Positive fraction addition. |
Syntax | cmq 6939 | Positive fraction multiplication. |
Syntax | crq 6940 | Positive fraction reciprocal operation. |
Syntax | cltq 6941 | Positive fraction ordering relation. |
Syntax | ceq0 6942 | Equivalence class used to construct nonnegative fractions. |
~_{Q0} | ||
Syntax | cnq0 6943 | Set of nonnegative fractions. |
Q_{0} | ||
Syntax | c0q0 6944 | The nonnegative fraction constant 0. |
0_{Q0} | ||
Syntax | cplq0 6945 | Nonnegative fraction addition. |
+_{Q0} | ||
Syntax | cmq0 6946 | Nonnegative fraction multiplication. |
·_{Q0} | ||
Syntax | cnp 6947 | Set of positive reals. |
Syntax | c1p 6948 | Positive real constant 1. |
Syntax | cpp 6949 | Positive real addition. |
Syntax | cmp 6950 | Positive real multiplication. |
Syntax | cltp 6951 | Positive real ordering relation. |
Syntax | cer 6952 | Equivalence class used to construct signed reals. |
Syntax | cnr 6953 | Set of signed reals. |
Syntax | c0r 6954 | The signed real constant 0. |
Syntax | c1r 6955 | The signed real constant 1. |
Syntax | cm1r 6956 | The signed real constant -1. |
Syntax | cplr 6957 | Signed real addition. |
Syntax | cmr 6958 | Signed real multiplication. |
Syntax | cltr 6959 | Signed real ordering relation. |
Definition | df-ni 6960 | 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.) |
Definition | df-pli 6961 | 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.) |
Definition | df-mi 6962 | 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.) |
Definition | df-lti 6963 | 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.) |
Theorem | elni 6964 | Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) |
Theorem | pinn 6965 | A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) |
Theorem | pion 6966 | A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) |
Theorem | piord 6967 | A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) |
Theorem | niex 6968 | The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) |
Theorem | 0npi 6969 | The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) |
Theorem | elni2 6970 | Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) |
Theorem | 1pi 6971 | Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) |
Theorem | addpiord 6972 | Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) |
Theorem | mulpiord 6973 | Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) |
Theorem | mulidpi 6974 | 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Theorem | ltpiord 6975 | Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | ltsopi 6976 | Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) |
Theorem | pitric 6977 | Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
Theorem | pitri3or 6978 | Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
Theorem | ltdcpi 6979 | Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) |
DECID | ||
Theorem | ltrelpi 6980 | Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) |
Theorem | dmaddpi 6981 | Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) |
Theorem | dmmulpi 6982 | Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) |
Theorem | addclpi 6983 | Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) |
Theorem | mulclpi 6984 | Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) |
Theorem | addcompig 6985 | Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Theorem | addasspig 6986 | Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Theorem | mulcompig 6987 | Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Theorem | mulasspig 6988 | Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Theorem | distrpig 6989 | Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Theorem | addcanpig 6990 | Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.) |
Theorem | mulcanpig 6991 | Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.) |
Theorem | addnidpig 6992 | There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) |
Theorem | ltexpi 6993* | Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) |
Theorem | ltapig 6994 | Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Theorem | ltmpig 6995 | Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Theorem | 1lt2pi 6996 | One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) |
Theorem | nlt1pig 6997 | No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Theorem | indpi 6998* | Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.) |
Theorem | nnppipi 6999 | A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.) |
Definition | df-plpq 7000* | 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 7005) works with the equivalence classes of these ordered pairs determined by the equivalence relation (df-enq 7003). (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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