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Theorem List for Intuitionistic Logic Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltrelpi 6301 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
<N ⊆ (N × N)
 
Theoremdmaddpi 6302 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
dom +N = (N × N)
 
Theoremdmmulpi 6303 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
dom ·N = (N × N)
 
Theoremaddclpi 6304 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
((A N B N) → (A +N B) N)
 
Theoremmulclpi 6305 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
((A N B N) → (A ·N B) N)
 
Theoremaddcompig 6306 Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((A N B N) → (A +N B) = (B +N A))
 
Theoremaddasspig 6307 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 6308 Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((A N B N) → (A ·N B) = (B ·N A))
 
Theoremmulasspig 6309 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 6310 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 6311 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 6312 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 6313 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 6314* 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 6315 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 6316 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 6317 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
1𝑜 <N (1𝑜 +N 1𝑜)
 
Theoremnlt1pig 6318 No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
(A N → ¬ A <N 1𝑜)
 
Theoremindpi 6319* Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.)
(x = 1𝑜 → (φψ))    &   (x = y → (φχ))    &   (x = (y +N 1𝑜) → (φθ))    &   (x = A → (φτ))    &   ψ    &   (y N → (χθ))       (A Nτ)
 
Theoremnnppipi 6320 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 6321* 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 6326) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 6324). (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 6322* 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 6323* 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 6324* 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 6325 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 6326* 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 6327* 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 6328 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 6329* 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 6330* 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 6331* 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 6332* 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)⟩))}
 
Theoremenqbreq 6333 Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.)
(((A N B N) (𝐶 N 𝐷 N)) → (⟨A, B⟩ ~Q𝐶, 𝐷⟩ ↔ (A ·N 𝐷) = (B ·N 𝐶)))
 
Theoremenqbreq2 6334 Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)
((A (N × N) B (N × N)) → (A ~Q B ↔ ((1stA) ·N (2ndB)) = ((1stB) ·N (2ndA))))
 
Theoremenqer 6335 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.)
~Q Er (N × N)
 
Theoremenqeceq 6336 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)
(((A N B N) (𝐶 N 𝐷 N)) → ([⟨A, B⟩] ~Q = [⟨𝐶, 𝐷⟩] ~Q ↔ (A ·N 𝐷) = (B ·N 𝐶)))
 
Theoremenqex 6337 The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
~Q V
 
Theoremenqdc 6338 The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
(((A N B N) (𝐶 N 𝐷 N)) → DECIDA, B⟩ ~Q𝐶, 𝐷⟩)
 
Theoremenqdc1 6339 The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
(((A N B N) 𝐶 (N × N)) → DECIDA, B⟩ ~Q 𝐶)
 
Theoremnqex 6340 The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
Q V
 
Theorem0nnq 6341 The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
¬ ∅ Q
 
Theoremltrelnq 6342 Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.)
<Q ⊆ (Q × Q)
 
Theorem1nq 6343 The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
1Q Q
 
Theoremaddcmpblnq 6344 Lemma showing compatibility of addition. (Contributed by NM, 27-Aug-1995.)
((((A N B N) (𝐶 N 𝐷 N)) ((𝐹 N 𝐺 N) (𝑅 N 𝑆 N))) → (((A ·N 𝐷) = (B ·N 𝐶) (𝐹 ·N 𝑆) = (𝐺 ·N 𝑅)) → ⟨((A ·N 𝐺) +N (B ·N 𝐹)), (B ·N 𝐺)⟩ ~Q ⟨((𝐶 ·N 𝑆) +N (𝐷 ·N 𝑅)), (𝐷 ·N 𝑆)⟩))
 
Theoremmulcmpblnq 6345 Lemma showing compatibility of multiplication. (Contributed by NM, 27-Aug-1995.)
((((A N B N) (𝐶 N 𝐷 N)) ((𝐹 N 𝐺 N) (𝑅 N 𝑆 N))) → (((A ·N 𝐷) = (B ·N 𝐶) (𝐹 ·N 𝑆) = (𝐺 ·N 𝑅)) → ⟨(A ·N 𝐹), (B ·N 𝐺)⟩ ~Q ⟨(𝐶 ·N 𝑅), (𝐷 ·N 𝑆)⟩))
 
Theoremaddpipqqslem 6346 Lemma for addpipqqs 6347. (Contributed by Jim Kingdon, 11-Sep-2019.)
(((A N B N) (𝐶 N 𝐷 N)) → ⟨((A ·N 𝐷) +N (B ·N 𝐶)), (B ·N 𝐷)⟩ (N × N))
 
Theoremaddpipqqs 6347 Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
(((A N B N) (𝐶 N 𝐷 N)) → ([⟨A, B⟩] ~Q +Q [⟨𝐶, 𝐷⟩] ~Q ) = [⟨((A ·N 𝐷) +N (B ·N 𝐶)), (B ·N 𝐷)⟩] ~Q )
 
Theoremmulpipq2 6348 Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)
((A (N × N) B (N × N)) → (A ·pQ B) = ⟨((1stA) ·N (1stB)), ((2ndA) ·N (2ndB))⟩)
 
Theoremmulpipq 6349 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.)
(((A N B N) (𝐶 N 𝐷 N)) → (⟨A, B⟩ ·pQ𝐶, 𝐷⟩) = ⟨(A ·N 𝐶), (B ·N 𝐷)⟩)
 
Theoremmulpipqqs 6350 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
(((A N B N) (𝐶 N 𝐷 N)) → ([⟨A, B⟩] ~Q ·Q [⟨𝐶, 𝐷⟩] ~Q ) = [⟨(A ·N 𝐶), (B ·N 𝐷)⟩] ~Q )
 
Theoremordpipqqs 6351 Ordering of positive fractions in terms of positive integers. (Contributed by Jim Kingdon, 14-Sep-2019.)
(((A N B N) (𝐶 N 𝐷 N)) → ([⟨A, B⟩] ~Q <Q [⟨𝐶, 𝐷⟩] ~Q ↔ (A ·N 𝐷) <N (B ·N 𝐶)))
 
Theoremaddclnq 6352 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.)
((A Q B Q) → (A +Q B) Q)
 
Theoremmulclnq 6353 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.)
((A Q B Q) → (A ·Q B) Q)
 
Theoremdmaddpqlem 6354* Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6356. (Contributed by Jim Kingdon, 15-Sep-2019.)
(x Qwv x = [⟨w, v⟩] ~Q )
 
Theoremnqpi 6355* Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 6354 but also shows that the numerator and denominator are positive integers. (Contributed by Jim Kingdon, 20-Sep-2019.)
(A Qwv((w N v N) A = [⟨w, v⟩] ~Q ))
 
Theoremdmaddpq 6356 Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
dom +Q = (Q × Q)
 
Theoremdmmulpq 6357 Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
dom ·Q = (Q × Q)
 
Theoremaddcomnqg 6358 Addition of positive fractions is commutative. (Contributed by Jim Kingdon, 15-Sep-2019.)
((A Q B Q) → (A +Q B) = (B +Q A))
 
Theoremaddassnqg 6359 Addition of positive fractions is associative. (Contributed by Jim Kingdon, 16-Sep-2019.)
((A Q B Q 𝐶 Q) → ((A +Q B) +Q 𝐶) = (A +Q (B +Q 𝐶)))
 
Theoremmulcomnqg 6360 Multiplication of positive fractions is commutative. (Contributed by Jim Kingdon, 17-Sep-2019.)
((A Q B Q) → (A ·Q B) = (B ·Q A))
 
Theoremmulassnqg 6361 Multiplication of positive fractions is associative. (Contributed by Jim Kingdon, 17-Sep-2019.)
((A Q B Q 𝐶 Q) → ((A ·Q B) ·Q 𝐶) = (A ·Q (B ·Q 𝐶)))
 
Theoremmulcanenq 6362 Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.)
((A N B N 𝐶 N) → ⟨(A ·N B), (A ·N 𝐶)⟩ ~QB, 𝐶⟩)
 
Theoremmulcanenqec 6363 Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 17-Sep-2019.)
((A N B N 𝐶 N) → [⟨(A ·N B), (A ·N 𝐶)⟩] ~Q = [⟨B, 𝐶⟩] ~Q )
 
Theoremdistrnqg 6364 Multiplication of positive fractions is distributive. (Contributed by Jim Kingdon, 17-Sep-2019.)
((A Q B Q 𝐶 Q) → (A ·Q (B +Q 𝐶)) = ((A ·Q B) +Q (A ·Q 𝐶)))
 
Theorem1qec 6365 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
(A N → 1Q = [⟨A, A⟩] ~Q )
 
Theoremmulidnq 6366 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
(A Q → (A ·Q 1Q) = A)
 
Theoremrecexnq 6367* Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)
(A Qy(y Q (A ·Q y) = 1Q))
 
Theoremrecmulnqg 6368 Relationship between reciprocal and multiplication on positive fractions. (Contributed by Jim Kingdon, 19-Sep-2019.)
((A Q B Q) → ((*QA) = B ↔ (A ·Q B) = 1Q))
 
Theoremrecclnq 6369 Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
(A Q → (*QA) Q)
 
Theoremrecidnq 6370 A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
(A Q → (A ·Q (*QA)) = 1Q)
 
Theoremrecrecnq 6371 Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.)
(A Q → (*Q‘(*QA)) = A)
 
Theoremrec1nq 6372 Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
(*Q‘1Q) = 1Q
 
Theoremnqtri3or 6373 Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.)
((A Q B Q) → (A <Q B A = B B <Q A))
 
Theoremltdcnq 6374 Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
((A Q B Q) → DECID A <Q B)
 
Theoremltsonq 6375 'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.)
<Q Or Q
 
Theoremnqtric 6376 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((A Q B Q) → (A <Q B ↔ ¬ (A = B B <Q A)))
 
Theoremltanqg 6377 Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.)
((A Q B Q 𝐶 Q) → (A <Q B ↔ (𝐶 +Q A) <Q (𝐶 +Q B)))
 
Theoremltmnqg 6378 Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.)
((A Q B Q 𝐶 Q) → (A <Q B ↔ (𝐶 ·Q A) <Q (𝐶 ·Q B)))
 
Theoremltanqi 6379 Ordering property of addition for positive fractions. One direction of ltanqg 6377. (Contributed by Jim Kingdon, 9-Dec-2019.)
((A <Q B 𝐶 Q) → (𝐶 +Q A) <Q (𝐶 +Q B))
 
Theoremltmnqi 6380 Ordering property of multiplication for positive fractions. One direction of ltmnqg 6378. (Contributed by Jim Kingdon, 9-Dec-2019.)
((A <Q B 𝐶 Q) → (𝐶 ·Q A) <Q (𝐶 ·Q B))
 
Theoremlt2addnq 6381 Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.)
(((A Q B Q) (𝐶 Q 𝐷 Q)) → ((A <Q B 𝐶 <Q 𝐷) → (A +Q 𝐶) <Q (B +Q 𝐷)))
 
Theorem1lt2nq 6382 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
1Q <Q (1Q +Q 1Q)
 
Theoremltaddnq 6383 The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
((A Q B Q) → A <Q (A +Q B))
 
Theoremltexnqq 6384* Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by Jim Kingdon, 23-Sep-2019.)
((A Q B Q) → (A <Q Bx Q (A +Q x) = B))
 
Theoremltexnqi 6385* Ordering on positive fractions in terms of existence of sum. (Contributed by Jim Kingdon, 30-Apr-2020.)
(A <Q Bx Q (A +Q x) = B)
 
Theoremhalfnqq 6386* One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.)
(A Qx Q (x +Q x) = A)
 
Theoremhalfnq 6387* One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 16-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
(A Qx(x +Q x) = A)
 
Theoremnsmallnqq 6388* There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
(A Qx Q x <Q A)
 
Theoremnsmallnq 6389* There is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.)
(A Qx x <Q A)
 
Theoremsubhalfnqq 6390* There is a number which is less than half of any positive fraction. The case where A is one is Lemma 11.4 of [BauerTaylor], p. 50, and they use the word "approximate half" for such a number (since there may be constructions, for some structures other than the rationals themselves, which rely on such an approximate half but do not require division by two as seen at halfnqq 6386). (Contributed by Jim Kingdon, 25-Nov-2019.)
(A Qx Q (x +Q x) <Q A)
 
Theoremltbtwnnqq 6391* There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by Jim Kingdon, 24-Sep-2019.)
(A <Q Bx Q (A <Q x x <Q B))
 
Theoremltbtwnnq 6392* There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
(A <Q Bx(A <Q x x <Q B))
 
Theoremarchnqq 6393* For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Jim Kingdon, 1-Dec-2019.)
(A Qx N A <Q [⟨x, 1𝑜⟩] ~Q )
 
Theoremprarloclemarch 6394* A version of the Archimedean property. This variation is "stronger" than archnqq 6393 in the sense that we provide an integer which is larger than a given rational A even after being multiplied by a second rational B. (Contributed by Jim Kingdon, 30-Nov-2019.)
((A Q B Q) → x N A <Q ([⟨x, 1𝑜⟩] ~Q ·Q B))
 
Theoremprarloclemarch2 6395* Like prarloclemarch 6394 but the integer must be at least two, and there is also B added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 6478. (Contributed by Jim Kingdon, 25-Nov-2019.)
((A Q B Q 𝐶 Q) → x N (1𝑜 <N x A <Q (B +Q ([⟨x, 1𝑜⟩] ~Q ·Q 𝐶))))
 
Theoremltrnqg 6396 Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 6397. (Contributed by Jim Kingdon, 29-Dec-2019.)
((A Q B Q) → (A <Q B ↔ (*QB) <Q (*QA)))
 
Theoremltrnqi 6397 Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 6396. (Contributed by Jim Kingdon, 24-Sep-2019.)
(A <Q B → (*QB) <Q (*QA))
 
Theoremnnnq 6398 The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
(A N → [⟨A, 1𝑜⟩] ~Q Q)
 
Definitiondf-enq0 6399* Define equivalence relation for non-negative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
~Q0 = {⟨x, y⟩ ∣ ((x (𝜔 × N) y (𝜔 × N)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v)))}
 
Definitiondf-nq0 6400 Define class of non-negative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
Q0 = ((𝜔 × N) / ~Q0 )
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