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Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremecopovsym 6101* Assuming the operation 𝐹 is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
= {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z + u) = (w + v)))}    &   (x + y) = (y + x)       (A BB A)
 
Theoremecopovtrn 6102* Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is transitive. (Contributed by NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
= {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z + u) = (w + v)))}    &   (x + y) = (y + x)    &   ((x 𝑆 y 𝑆) → (x + y) 𝑆)    &   ((x + y) + z) = (x + (y + z))    &   ((x 𝑆 y 𝑆) → ((x + y) = (x + z) → y = z))       ((A B B 𝐶) → A 𝐶)
 
Theoremecopover 6103* Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z + u) = (w + v)))}    &   (x + y) = (y + x)    &   ((x 𝑆 y 𝑆) → (x + y) 𝑆)    &   ((x + y) + z) = (x + (y + z))    &   ((x 𝑆 y 𝑆) → ((x + y) = (x + z) → y = z))        Er (𝑆 × 𝑆)
 
Theoremecopovsymg 6104* Assuming the operation 𝐹 is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.)
= {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z + u) = (w + v)))}    &   ((x 𝑆 y 𝑆) → (x + y) = (y + x))       (A BB A)
 
Theoremecopovtrng 6105* Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is transitive. (Contributed by Jim Kingdon, 1-Sep-2019.)
= {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z + u) = (w + v)))}    &   ((x 𝑆 y 𝑆) → (x + y) = (y + x))    &   ((x 𝑆 y 𝑆) → (x + y) 𝑆)    &   ((x 𝑆 y 𝑆 z 𝑆) → ((x + y) + z) = (x + (y + z)))    &   ((x 𝑆 y 𝑆 z 𝑆) → ((x + y) = (x + z) → y = z))       ((A B B 𝐶) → A 𝐶)
 
Theoremecopoverg 6106* Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is an equivalence relation. (Contributed by Jim Kingdon, 1-Sep-2019.)
= {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z + u) = (w + v)))}    &   ((x 𝑆 y 𝑆) → (x + y) = (y + x))    &   ((x 𝑆 y 𝑆) → (x + y) 𝑆)    &   ((x 𝑆 y 𝑆 z 𝑆) → ((x + y) + z) = (x + (y + z)))    &   ((x 𝑆 y 𝑆 z 𝑆) → ((x + y) = (x + z) → y = z))        Er (𝑆 × 𝑆)
 
Theoremth3qlem1 6107* Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption. (Contributed by NM, 3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Er 𝑆    &   (((y 𝑆 w 𝑆) (z 𝑆 v 𝑆)) → ((y w z v) → (y + z) (w + v)))       ((A (𝑆 / ) B (𝑆 / )) → ∃*xyz((A = [y] B = [z] ) x = [(y + z)] ))
 
Theoremth3qlem2 6108* Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
V    &    Er (𝑆 × 𝑆)    &   ((((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) ((𝑠 𝑆 f 𝑆) (g 𝑆 𝑆))) → ((⟨w, vu, 𝑡𝑠, fg, ⟩) → (⟨w, v+𝑠, f⟩) (⟨u, 𝑡+g, ⟩)))       ((A ((𝑆 × 𝑆) / ) B ((𝑆 × 𝑆) / )) → ∃*zwvu𝑡((A = [⟨w, v⟩] B = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))
 
Theoremth3qcor 6109* Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)
V    &    Er (𝑆 × 𝑆)    &   ((((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) ((𝑠 𝑆 f 𝑆) (g 𝑆 𝑆))) → ((⟨w, vu, 𝑡𝑠, fg, ⟩) → (⟨w, v+𝑠, f⟩) (⟨u, 𝑡+g, ⟩)))    &   𝐺 = {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) wvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))}       Fun 𝐺
 
Theoremth3q 6110* Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
V    &    Er (𝑆 × 𝑆)    &   ((((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) ((𝑠 𝑆 f 𝑆) (g 𝑆 𝑆))) → ((⟨w, vu, 𝑡𝑠, fg, ⟩) → (⟨w, v+𝑠, f⟩) (⟨u, 𝑡+g, ⟩)))    &   𝐺 = {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) wvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))}       (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → ([⟨A, B⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨A, B+𝐶, 𝐷⟩)] )
 
Theoremoviec 6111* Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See iset.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.)
(((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → 𝐻 (𝑆 × 𝑆))    &   (((𝑎 𝑆 𝑏 𝑆) (g 𝑆 𝑆)) → 𝐾 (𝑆 × 𝑆))    &   (((𝑐 𝑆 𝑑 𝑆) (𝑡 𝑆 𝑠 𝑆)) → 𝐿 (𝑆 × 𝑆))    &    V    &    Er (𝑆 × 𝑆)    &    = {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) φ))}    &   (((z = 𝑎 w = 𝑏) (v = 𝑐 u = 𝑑)) → (φψ))    &   (((z = g w = ) (v = 𝑡 u = 𝑠)) → (φχ))    &    + = {⟨⟨x, y⟩, z⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝐽))}    &   (((w = 𝑎 v = 𝑏) (u = g f = )) → 𝐽 = 𝐾)    &   (((w = 𝑐 v = 𝑑) (u = 𝑡 f = 𝑠)) → 𝐽 = 𝐿)    &   (((w = A v = B) (u = 𝐶 f = 𝐷)) → 𝐽 = 𝐻)    &    = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑄 y 𝑄) 𝑎𝑏𝑐𝑑((x = [⟨𝑎, 𝑏⟩] y = [⟨𝑐, 𝑑⟩] ) z = [(⟨𝑎, 𝑏+𝑐, 𝑑⟩)] ))}    &   𝑄 = ((𝑆 × 𝑆) / )    &   ((((𝑎 𝑆 𝑏 𝑆) (𝑐 𝑆 𝑑 𝑆)) ((g 𝑆 𝑆) (𝑡 𝑆 𝑠 𝑆))) → ((ψ χ) → 𝐾 𝐿))       (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → ([⟨A, B⟩] [⟨𝐶, 𝐷⟩] ) = [𝐻] )
 
Theoremecovcom 6112* Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6113 instead. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
𝐶 = ((𝑆 × 𝑆) / )    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] + [⟨z, w⟩] ) = [⟨𝐷, 𝐺⟩] )    &   (((z 𝑆 w 𝑆) (x 𝑆 y 𝑆)) → ([⟨z, w⟩] + [⟨x, y⟩] ) = [⟨𝐻, 𝐽⟩] )    &   𝐷 = 𝐻    &   𝐺 = 𝐽       ((A 𝐶 B 𝐶) → (A + B) = (B + A))
 
Theoremecovicom 6113* Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.)
𝐶 = ((𝑆 × 𝑆) / )    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] + [⟨z, w⟩] ) = [⟨𝐷, 𝐺⟩] )    &   (((z 𝑆 w 𝑆) (x 𝑆 y 𝑆)) → ([⟨z, w⟩] + [⟨x, y⟩] ) = [⟨𝐻, 𝐽⟩] )    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → 𝐷 = 𝐻)    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → 𝐺 = 𝐽)       ((A 𝐶 B 𝐶) → (A + B) = (B + A))
 
Theoremecovass 6114* Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6115 will be more useful. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
𝐷 = ((𝑆 × 𝑆) / )    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] + [⟨z, w⟩] ) = [⟨𝐺, 𝐻⟩] )    &   (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨z, w⟩] + [⟨v, u⟩] ) = [⟨𝑁, 𝑄⟩] )    &   (((𝐺 𝑆 𝐻 𝑆) (v 𝑆 u 𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨v, u⟩] ) = [⟨𝐽, 𝐾⟩] )    &   (((x 𝑆 y 𝑆) (𝑁 𝑆 𝑄 𝑆)) → ([⟨x, y⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → (𝐺 𝑆 𝐻 𝑆))    &   (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → (𝑁 𝑆 𝑄 𝑆))    &   𝐽 = 𝐿    &   𝐾 = 𝑀       ((A 𝐷 B 𝐷 𝐶 𝐷) → ((A + B) + 𝐶) = (A + (B + 𝐶)))
 
Theoremecoviass 6115* Lemma used to transfer an associative law via an equivalence relation. (Contributed by Jim Kingdon, 16-Sep-2019.)
𝐷 = ((𝑆 × 𝑆) / )    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] + [⟨z, w⟩] ) = [⟨𝐺, 𝐻⟩] )    &   (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨z, w⟩] + [⟨v, u⟩] ) = [⟨𝑁, 𝑄⟩] )    &   (((𝐺 𝑆 𝐻 𝑆) (v 𝑆 u 𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨v, u⟩] ) = [⟨𝐽, 𝐾⟩] )    &   (((x 𝑆 y 𝑆) (𝑁 𝑆 𝑄 𝑆)) → ([⟨x, y⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → (𝐺 𝑆 𝐻 𝑆))    &   (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → (𝑁 𝑆 𝑄 𝑆))    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → 𝐽 = 𝐿)    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → 𝐾 = 𝑀)       ((A 𝐷 B 𝐷 𝐶 𝐷) → ((A + B) + 𝐶) = (A + (B + 𝐶)))
 
Theoremecovdi 6116* Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6117 will be more helpful. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)
𝐷 = ((𝑆 × 𝑆) / )    &   (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨z, w⟩] + [⟨v, u⟩] ) = [⟨𝑀, 𝑁⟩] )    &   (((x 𝑆 y 𝑆) (𝑀 𝑆 𝑁 𝑆)) → ([⟨x, y⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] · [⟨z, w⟩] ) = [⟨𝑊, 𝑋⟩] )    &   (((x 𝑆 y 𝑆) (v 𝑆 u 𝑆)) → ([⟨x, y⟩] · [⟨v, u⟩] ) = [⟨𝑌, 𝑍⟩] )    &   (((𝑊 𝑆 𝑋 𝑆) (𝑌 𝑆 𝑍 𝑆)) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )    &   (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → (𝑀 𝑆 𝑁 𝑆))    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → (𝑊 𝑆 𝑋 𝑆))    &   (((x 𝑆 y 𝑆) (v 𝑆 u 𝑆)) → (𝑌 𝑆 𝑍 𝑆))    &   𝐻 = 𝐾    &   𝐽 = 𝐿       ((A 𝐷 B 𝐷 𝐶 𝐷) → (A · (B + 𝐶)) = ((A · B) + (A · 𝐶)))
 
Theoremecovidi 6117* Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.)
𝐷 = ((𝑆 × 𝑆) / )    &   (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨z, w⟩] + [⟨v, u⟩] ) = [⟨𝑀, 𝑁⟩] )    &   (((x 𝑆 y 𝑆) (𝑀 𝑆 𝑁 𝑆)) → ([⟨x, y⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] · [⟨z, w⟩] ) = [⟨𝑊, 𝑋⟩] )    &   (((x 𝑆 y 𝑆) (v 𝑆 u 𝑆)) → ([⟨x, y⟩] · [⟨v, u⟩] ) = [⟨𝑌, 𝑍⟩] )    &   (((𝑊 𝑆 𝑋 𝑆) (𝑌 𝑆 𝑍 𝑆)) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )    &   (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → (𝑀 𝑆 𝑁 𝑆))    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → (𝑊 𝑆 𝑋 𝑆))    &   (((x 𝑆 y 𝑆) (v 𝑆 u 𝑆)) → (𝑌 𝑆 𝑍 𝑆))    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → 𝐻 = 𝐾)    &   (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → 𝐽 = 𝐿)       ((A 𝐷 B 𝐷 𝐶 𝐷) → (A · (B + 𝐶)) = ((A · B) + (A · 𝐶)))
 
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 5956 and similar theorems ), going from there to positive integers (df-ni 6150) and then positive rational numbers (df-nqqs 6193) 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 6118 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.

class N
 
Syntaxcpli 6119 Positive integer addition.
class +N
 
Syntaxcmi 6120 Positive integer multiplication.
class ·N
 
Syntaxclti 6121 Positive integer ordering relation.
class <N
 
Syntaxcplpq 6122 Positive pre-fraction addition.
class +pQ
 
Syntaxcmpq 6123 Positive pre-fraction multiplication.
class ·pQ
 
Syntaxcltpq 6124 Positive pre-fraction ordering relation.
class <pQ
 
Syntaxceq 6125 Equivalence class used to construct positive fractions.
class ~Q
 
Syntaxcnq 6126 Set of positive fractions.
class Q
 
Syntaxc1q 6127 The positive fraction constant 1.
class 1Q
 
Syntaxcplq 6128 Positive fraction addition.
class +Q
 
Syntaxcmq 6129 Positive fraction multiplication.
class ·Q
 
Syntaxcrq 6130 Positive fraction reciprocal operation.
class *Q
 
Syntaxcltq 6131 Positive fraction ordering relation.
class <Q
 
Syntaxceq0 6132 Equivalence class used to construct non-negative fractions.
class ~Q0
 
Syntaxcnq0 6133 Set of non-negative fractions.
class Q0
 
Syntaxc0q0 6134 The non-negative fraction constant 0.
class 0Q0
 
Syntaxcplq0 6135 Non-negative fraction addition.
class +Q0
 
Syntaxcmq0 6136 Non-negative fraction multiplication.
class ·Q0
 
Syntaxcnp 6137 Set of positive reals.
class P
 
Syntaxc1p 6138 Positive real constant 1.
class 1P
 
Syntaxcpp 6139 Positive real addition.
class +P
 
Syntaxcmp 6140 Positive real multiplication.
class ·P
 
Syntaxcltp 6141 Positive real ordering relation.
class <P
 
Syntaxcer 6142 Equivalence class used to construct signed reals.
class ~R
 
Syntaxcnr 6143 Set of signed reals.
class R
 
Syntaxc0r 6144 The signed real constant 0.
class 0R
 
Syntaxc1r 6145 The signed real constant 1.
class 1R
 
Syntaxcm1r 6146 The signed real constant -1.
class -1R
 
Syntaxcplr 6147 Signed real addition.
class +R
 
Syntaxcmr 6148 Signed real multiplication.
class ·R
 
Syntaxcltr 6149 Signed real ordering relation.
class <R
 
Definitiondf-ni 6150 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.)
N = (𝜔 ∖ {∅})
 
Definitiondf-pli 6151 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.)
+N = ( +𝑜 ↾ (N × N))
 
Definitiondf-mi 6152 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.)
·N = ( ·𝑜 ↾ (N × N))
 
Definitiondf-lti 6153 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.)
<N = ( E ∩ (N × N))
 
Theoremelni 6154 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
(A N ↔ (A 𝜔 A ≠ ∅))
 
Theorempinn 6155 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
(A NA 𝜔)
 
Theorempion 6156 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
(A NA On)
 
Theorempiord 6157 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
(A N → Ord A)
 
Theoremniex 6158 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
N V
 
Theorem0npi 6159 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
¬ ∅ N
 
Theoremelni2 6160 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
(A N ↔ (A 𝜔 A))
 
Theorem1pi 6161 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
1𝑜 N
 
Theoremaddpiord 6162 Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.)
((A N B N) → (A +N B) = (A +𝑜 B))
 
Theoremmulpiord 6163 Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
((A N B N) → (A ·N B) = (A ·𝑜 B))
 
Theoremmulidpi 6164 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.)
(A N → (A ·N 1𝑜) = A)
 
Theoremltpiord 6165 Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
((A N B N) → (A <N BA B))
 
Theoremltsopi 6166 Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.)
<N Or N
 
Theorempitric 6167 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((A N B N) → (A <N B ↔ ¬ (A = B B <N A)))
 
Theorempitri3or 6168 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((A N B N) → (A <N B A = B B <N A))
 
Theoremltdcpi 6169 Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
((A N B N) → DECID A <N B)
 
Theoremltrelpi 6170 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
<N ⊆ (N × N)
 
Theoremdmaddpi 6171 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
dom +N = (N × N)
 
Theoremdmmulpi 6172 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
dom ·N = (N × N)
 
Theoremaddclpi 6173 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
((A N B N) → (A +N B) N)
 
Theoremmulclpi 6174 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
((A N B N) → (A ·N B) N)
 
Theoremaddcompig 6175 Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((A N B N) → (A +N B) = (B +N A))
 
Theoremaddasspig 6176 Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((A N B N 𝐶 N) → ((A +N B) +N 𝐶) = (A +N (B +N 𝐶)))
 
Theoremmulcompig 6177 Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((A N B N) → (A ·N B) = (B ·N A))
 
Theoremmulasspig 6178 Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((A N B N 𝐶 N) → ((A ·N B) ·N 𝐶) = (A ·N (B ·N 𝐶)))
 
Theoremdistrpig 6179 Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.)
((A N B N 𝐶 N) → (A ·N (B +N 𝐶)) = ((A ·N B) +N (A ·N 𝐶)))
 
Theoremaddcanpig 6180 Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.)
((A N B N 𝐶 N) → ((A +N B) = (A +N 𝐶) ↔ B = 𝐶))
 
Theoremmulcanpig 6181 Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)
((A N B N 𝐶 N) → ((A ·N B) = (A ·N 𝐶) ↔ B = 𝐶))
 
Theoremaddnidpig 6182 There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.)
((A N B N) → ¬ (A +N B) = A)
 
Theoremltexpi 6183* Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
((A N B N) → (A <N Bx N (A +N x) = B))
 
Theoremltapig 6184 Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
((A N B N 𝐶 N) → (A <N B ↔ (𝐶 +N A) <N (𝐶 +N B)))
 
Theoremltmpig 6185 Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
((A N B N 𝐶 N) → (A <N B ↔ (𝐶 ·N A) <N (𝐶 ·N B)))
 
Theorem1lt2pi 6186 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
1𝑜 <N (1𝑜 +N 1𝑜)
 
Theoremnlt1pig 6187 No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
(A N → ¬ A <N 1𝑜)
 
Theoremnnppipi 6188 A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.)
((A 𝜔 B N) → (A +𝑜 B) N)
 
Definitiondf-plpq 6189* 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 +Q (df-plqqs 6194) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 6192). (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.)
+pQ = (x (N × N), y (N × N) ↦ ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩)
 
Definitiondf-mpq 6190* 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.)
·pQ = (x (N × N), y (N × N) ↦ ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩)
 
Definitiondf-ltpq 6191* 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.)
<pQ = {⟨x, y⟩ ∣ ((x (N × N) y (N × N)) ((1stx) ·N (2ndy)) <N ((1sty) ·N (2ndx)))}
 
Definitiondf-enq 6192* 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.)
~Q = {⟨x, y⟩ ∣ ((x (N × N) y (N × N)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z ·N u) = (w ·N v)))}
 
Definitiondf-nqqs 6193 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.)
Q = ((N × N) / ~Q )
 
Definitiondf-plqqs 6194* 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.)
+Q = {⟨⟨x, y⟩, z⟩ ∣ ((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ +pQu, f⟩)] ~Q ))}
 
Definitiondf-mqqs 6195* 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.)
·Q = {⟨⟨x, y⟩, z⟩ ∣ ((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ ·pQu, f⟩)] ~Q ))}
 
Definitiondf-1nqqs 6196 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.)
1Q = [⟨1𝑜, 1𝑜⟩] ~Q
 
Definitiondf-rq 6197* 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.)
*Q = {⟨x, y⟩ ∣ (x Q y Q (x ·Q y) = 1Q)}
 
Definitiondf-ltnqqs 6198* 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.)
<Q = {⟨x, y⟩ ∣ ((x Q y Q) zwvu((x = [⟨z, w⟩] ~Q y = [⟨v, u⟩] ~Q ) (z ·N u) <N (w ·N v)))}
 
Theoremdfplpq2 6199* Alternative definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.)
+pQ = {⟨⟨x, y⟩, z⟩ ∣ ((x (N × N) y (N × N)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩))}
 
Theoremdfmpq2 6200* Alternative definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.)
·pQ = {⟨⟨x, y⟩, z⟩ ∣ ((x (N × N) y (N × N)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w ·N u), (v ·N f)⟩))}
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