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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | en2other2 6801 | Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
Theorem | dju1p1e2 6802 | Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.) |
⊔ | ||
Theorem | infpwfidom 6803 | 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 6804 | Lemma for exmidfodomr 6809. A variant of djur 6736. (Contributed by Jim Kingdon, 2-Jul-2022.) |
⊔ inl inr | ||
Theorem | exmidfodomrlemreseldju 6805 | Lemma for exmidfodomrlemrALT 6808. A variant of eldju 6738. (Contributed by Jim Kingdon, 9-Jul-2022.) |
⊔ inl inr | ||
Theorem | exmidfodomrlemim 6806* | 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 6807* | 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 6808* | 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 6807. In particular, this proof uses eldju 6738 instead of djur 6736 and avoids djulclb 6726. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.) |
EXMID | ||
Theorem | exmidfodomr 6809* | 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 6217 and similar theorems ), going from there to positive integers (df-ni 6842) and then positive rational numbers (df-nqqs 6886) 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 6810 |
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 6811 | Positive integer addition. |
Syntax | cmi 6812 | Positive integer multiplication. |
Syntax | clti 6813 | Positive integer ordering relation. |
Syntax | cplpq 6814 | Positive pre-fraction addition. |
Syntax | cmpq 6815 | Positive pre-fraction multiplication. |
Syntax | cltpq 6816 | Positive pre-fraction ordering relation. |
Syntax | ceq 6817 | Equivalence class used to construct positive fractions. |
Syntax | cnq 6818 | Set of positive fractions. |
Syntax | c1q 6819 | The positive fraction constant 1. |
Syntax | cplq 6820 | Positive fraction addition. |
Syntax | cmq 6821 | Positive fraction multiplication. |
Syntax | crq 6822 | Positive fraction reciprocal operation. |
Syntax | cltq 6823 | Positive fraction ordering relation. |
Syntax | ceq0 6824 | Equivalence class used to construct nonnegative fractions. |
~_{Q0} | ||
Syntax | cnq0 6825 | Set of nonnegative fractions. |
Q_{0} | ||
Syntax | c0q0 6826 | The nonnegative fraction constant 0. |
0_{Q0} | ||
Syntax | cplq0 6827 | Nonnegative fraction addition. |
+_{Q0} | ||
Syntax | cmq0 6828 | Nonnegative fraction multiplication. |
·_{Q0} | ||
Syntax | cnp 6829 | Set of positive reals. |
Syntax | c1p 6830 | Positive real constant 1. |
Syntax | cpp 6831 | Positive real addition. |
Syntax | cmp 6832 | Positive real multiplication. |
Syntax | cltp 6833 | Positive real ordering relation. |
Syntax | cer 6834 | Equivalence class used to construct signed reals. |
Syntax | cnr 6835 | Set of signed reals. |
Syntax | c0r 6836 | The signed real constant 0. |
Syntax | c1r 6837 | The signed real constant 1. |
Syntax | cm1r 6838 | The signed real constant -1. |
Syntax | cplr 6839 | Signed real addition. |
Syntax | cmr 6840 | Signed real multiplication. |
Syntax | cltr 6841 | Signed real ordering relation. |
Definition | df-ni 6842 | 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 6843 | 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 6844 | 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 6845 | 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 6846 | Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) |
Theorem | pinn 6847 | A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) |
Theorem | pion 6848 | A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) |
Theorem | piord 6849 | A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) |
Theorem | niex 6850 | The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) |
Theorem | 0npi 6851 | The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) |
Theorem | elni2 6852 | Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) |
Theorem | 1pi 6853 | Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) |
Theorem | addpiord 6854 | Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) |
Theorem | mulpiord 6855 | Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) |
Theorem | mulidpi 6856 | 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 6857 | Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | ltsopi 6858 | Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) |
Theorem | pitric 6859 | Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
Theorem | pitri3or 6860 | Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
Theorem | ltdcpi 6861 | Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) |
DECID | ||
Theorem | ltrelpi 6862 | Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) |
Theorem | dmaddpi 6863 | Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) |
Theorem | dmmulpi 6864 | Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) |
Theorem | addclpi 6865 | Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) |
Theorem | mulclpi 6866 | Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) |
Theorem | addcompig 6867 | Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Theorem | addasspig 6868 | Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Theorem | mulcompig 6869 | Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Theorem | mulasspig 6870 | Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Theorem | distrpig 6871 | Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Theorem | addcanpig 6872 | Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.) |
Theorem | mulcanpig 6873 | Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.) |
Theorem | addnidpig 6874 | There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) |
Theorem | ltexpi 6875* | 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 6876 | Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Theorem | ltmpig 6877 | Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Theorem | 1lt2pi 6878 | One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) |
Theorem | nlt1pig 6879 | No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Theorem | indpi 6880* | Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.) |
Theorem | nnppipi 6881 | A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.) |
Definition | df-plpq 6882* | 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 6887) works with the equivalence classes of these ordered pairs determined by the equivalence relation (df-enq 6885). (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.) |
Definition | df-mpq 6883* | Define pre-multiplication 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. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 28-Aug-1995.) |
Definition | df-ltpq 6884* | Define pre-ordering relation 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. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.) |
Definition | df-enq 6885* | Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers, 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.) |
Definition | df-nqqs 6886 | Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers, 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.) |
Definition | df-plqqs 6887* | Define 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. From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 24-Aug-1995.) |
Definition | df-mqqs 6888* | Define multiplication 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. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 24-Aug-1995.) |
Definition | df-1nqqs 6889 | Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers, 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.) |
Definition | df-rq 6890* | 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, 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 Jim Kingdon, 20-Sep-2019.) |
Definition | df-ltnqqs 6891* | Define ordering relation 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. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.) |
Theorem | dfplpq2 6892* | Alternate definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.) |
Theorem | dfmpq2 6893* | Alternate definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.) |
Theorem | enqbreq 6894 | Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.) |
Theorem | enqbreq2 6895 | Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) |
Theorem | enqer 6896 | 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.) |
Theorem | enqeceq 6897 | Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) |
Theorem | enqex 6898 | The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) |
Theorem | enqdc 6899 | The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.) |
DECID | ||
Theorem | enqdc1 6900 | The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.) |
DECID |
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