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Theorem List for Intuitionistic Logic Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulpiord 6301 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 6302 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 6303 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 6304 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 6305 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((A N B N) → (A <N B ↔ ¬ (A = B B <N A)))

Theorempitri3or 6306 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((A N B N) → (A <N B A = B B <N A))

Theoremltdcpi 6307 Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
((A N B N) → DECID A <N B)

Theoremltrelpi 6308 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
<N ⊆ (N × N)

Theoremdmaddpi 6309 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
dom +N = (N × N)

Theoremdmmulpi 6310 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
dom ·N = (N × N)

Theoremaddclpi 6311 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
((A N B N) → (A +N B) N)

Theoremmulclpi 6312 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
((A N B N) → (A ·N B) N)

Theoremaddcompig 6313 Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((A N B N) → (A +N B) = (B +N A))

Theoremaddasspig 6314 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 6315 Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((A N B N) → (A ·N B) = (B ·N A))

Theoremmulasspig 6316 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 6317 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 6318 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 6319 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 6320 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 6321* 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 6322 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 6323 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 6324 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
1𝑜 <N (1𝑜 +N 1𝑜)

Theoremnlt1pig 6325 No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
(A N → ¬ A <N 1𝑜)

Theoremindpi 6326* 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 6327 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 6328* 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 6333) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 6331). (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 6329* 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 6330* 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 6331* 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 6332 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 6333* 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 6334* 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 6335 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 6336* 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 6337* 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 6338* 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 6339* 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 6340 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 6341 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 6342 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 6343 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 6344 The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
~Q V

Theoremenqdc 6345 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 6346 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 6347 The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
Q V

Theorem0nnq 6348 The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
¬ ∅ Q

Theoremltrelnq 6349 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 6350 The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
1Q Q

((((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 6352 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)⟩))

(((A N B N) (𝐶 N 𝐷 N)) → ⟨((A ·N 𝐷) +N (B ·N 𝐶)), (B ·N 𝐷)⟩ (N × N))

Theoremaddpipqqs 6354 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 6355 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 6356 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 6357 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 6358 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 6359 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.)
((A Q B Q) → (A +Q B) Q)

Theoremmulclnq 6360 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.)
((A Q B Q) → (A ·Q B) Q)

Theoremdmaddpqlem 6361* Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6363. (Contributed by Jim Kingdon, 15-Sep-2019.)
(x Qwv x = [⟨w, v⟩] ~Q )

Theoremnqpi 6362* Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 6361 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 6363 Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
dom +Q = (Q × Q)

Theoremdmmulpq 6364 Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
dom ·Q = (Q × Q)

Theoremaddcomnqg 6365 Addition of positive fractions is commutative. (Contributed by Jim Kingdon, 15-Sep-2019.)
((A Q B Q) → (A +Q B) = (B +Q A))

Theoremaddassnqg 6366 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 6367 Multiplication of positive fractions is commutative. (Contributed by Jim Kingdon, 17-Sep-2019.)
((A Q B Q) → (A ·Q B) = (B ·Q A))

Theoremmulassnqg 6368 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 6369 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 6370 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 6371 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 6372 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
(A N → 1Q = [⟨A, A⟩] ~Q )

Theoremmulidnq 6373 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
(A Q → (A ·Q 1Q) = A)

Theoremrecexnq 6374* Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)
(A Qy(y Q (A ·Q y) = 1Q))

Theoremrecmulnqg 6375 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 6376 Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
(A Q → (*QA) Q)

Theoremrecidnq 6377 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 6378 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 6379 Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
(*Q‘1Q) = 1Q

Theoremnqtri3or 6380 Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.)
((A Q B Q) → (A <Q B A = B B <Q A))

Theoremltdcnq 6381 Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
((A Q B Q) → DECID A <Q B)

Theoremltsonq 6382 '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 6383 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((A Q B Q) → (A <Q B ↔ ¬ (A = B B <Q A)))

Theoremltanqg 6384 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 6385 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 6386 Ordering property of addition for positive fractions. One direction of ltanqg 6384. (Contributed by Jim Kingdon, 9-Dec-2019.)
((A <Q B 𝐶 Q) → (𝐶 +Q A) <Q (𝐶 +Q B))

Theoremltmnqi 6387 Ordering property of multiplication for positive fractions. One direction of ltmnqg 6385. (Contributed by Jim Kingdon, 9-Dec-2019.)
((A <Q B 𝐶 Q) → (𝐶 ·Q A) <Q (𝐶 ·Q B))

Theoremlt2addnq 6388 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 6389 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 6390 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 6391* 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 6392* 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 6393* One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.)
(A Qx Q (x +Q x) = A)

Theoremhalfnq 6394* 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 6395* There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
(A Qx Q x <Q A)

Theoremnsmallnq 6396* 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 6397* 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 6393). (Contributed by Jim Kingdon, 25-Nov-2019.)
(A Qx Q (x +Q x) <Q A)

Theoremltbtwnnqq 6398* 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 6399* 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 6400* 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 )

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