Type  Label  Description 
Statement 

Theorem  ltrelpi 6301 
Positive integer 'less than' is a relation on positive integers.
(Contributed by NM, 8Feb1996.)

⊢ <_{N} ⊆
(N × N) 

Theorem  dmaddpi 6302 
Domain of addition on positive integers. (Contributed by NM,
26Aug1995.)

⊢ dom +_{N} =
(N × N) 

Theorem  dmmulpi 6303 
Domain of multiplication on positive integers. (Contributed by NM,
26Aug1995.)

⊢ dom ·_{N} =
(N × N) 

Theorem  addclpi 6304 
Closure of addition of positive integers. (Contributed by NM,
18Oct1995.)

⊢ ((A ∈ N ∧ B ∈ N) → (A +_{N} B) ∈
N) 

Theorem  mulclpi 6305 
Closure of multiplication of positive integers. (Contributed by NM,
18Oct1995.)

⊢ ((A ∈ N ∧ B ∈ N) → (A ·_{N} B) ∈
N) 

Theorem  addcompig 6306 
Addition of positive integers is commutative. (Contributed by Jim
Kingdon, 26Aug2019.)

⊢ ((A ∈ N ∧ B ∈ N) → (A +_{N} B) = (B
+_{N} A)) 

Theorem  addasspig 6307 
Addition of positive integers is associative. (Contributed by Jim
Kingdon, 26Aug2019.)

⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈
N) → ((A
+_{N} B)
+_{N} 𝐶) = (A
+_{N} (B
+_{N} 𝐶))) 

Theorem  mulcompig 6308 
Multiplication of positive integers is commutative. (Contributed by Jim
Kingdon, 26Aug2019.)

⊢ ((A ∈ N ∧ B ∈ N) → (A ·_{N} B) = (B
·_{N} A)) 

Theorem  mulasspig 6309 
Multiplication of positive integers is associative. (Contributed by Jim
Kingdon, 26Aug2019.)

⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈
N) → ((A
·_{N} B)
·_{N} 𝐶) = (A
·_{N} (B
·_{N} 𝐶))) 

Theorem  distrpig 6310 
Multiplication of positive integers is distributive. (Contributed by Jim
Kingdon, 26Aug2019.)

⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈
N) → (A
·_{N} (B
+_{N} 𝐶)) = ((A ·_{N} B) +_{N} (A ·_{N} 𝐶))) 

Theorem  addcanpig 6311 
Addition cancellation law for positive integers. (Contributed by Jim
Kingdon, 27Aug2019.)

⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈
N) → ((A
+_{N} B) = (A +_{N} 𝐶) ↔ B = 𝐶)) 

Theorem  mulcanpig 6312 
Multiplication cancellation law for positive integers. (Contributed by
Jim Kingdon, 29Aug2019.)

⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈
N) → ((A
·_{N} B) =
(A ·_{N}
𝐶) ↔ B = 𝐶)) 

Theorem  addnidpig 6313 
There is no identity element for addition on positive integers.
(Contributed by NM, 28Nov1995.)

⊢ ((A ∈ N ∧ B ∈ N) → ¬ (A +_{N} B) = A) 

Theorem  ltexpi 6314* 
Ordering on positive integers in terms of existence of sum.
(Contributed by NM, 15Mar1996.) (Revised by Mario Carneiro,
14Jun2013.)

⊢ ((A ∈ N ∧ B ∈ N) → (A <_{N} B ↔ ∃x ∈ N (A +_{N} x) = B)) 

Theorem  ltapig 6315 
Ordering property of addition for positive integers. (Contributed by Jim
Kingdon, 31Aug2019.)

⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈
N) → (A
<_{N} B ↔
(𝐶
+_{N} A)
<_{N} (𝐶 +_{N} B))) 

Theorem  ltmpig 6316 
Ordering property of multiplication for positive integers. (Contributed
by Jim Kingdon, 31Aug2019.)

⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈
N) → (A
<_{N} B ↔
(𝐶
·_{N} A)
<_{N} (𝐶 ·_{N}
B))) 

Theorem  1lt2pi 6317 
One is less than two (one plus one). (Contributed by NM, 13Mar1996.)

⊢ 1_{𝑜}
<_{N} (1_{𝑜}
+_{N} 1_{𝑜}) 

Theorem  nlt1pig 6318 
No positive integer is less than one. (Contributed by Jim Kingdon,
31Aug2019.)

⊢ (A ∈ N → ¬ A <_{N}
1_{𝑜}) 

Theorem  indpi 6319* 
Principle of Finite Induction on positive integers. (Contributed by NM,
23Mar1996.)

⊢ (x =
1_{𝑜} → (φ ↔
ψ)) & ⊢ (x = y →
(φ ↔ χ)) & ⊢ (x = (y
+_{N} 1_{𝑜}) → (φ ↔ θ)) & ⊢ (x = A →
(φ ↔ τ)) & ⊢ ψ
& ⊢ (y ∈ N → (χ → θ)) ⇒ ⊢ (A ∈
N → τ) 

Theorem  nnppipi 6320 
A natural number plus a positive integer is a positive integer.
(Contributed by Jim Kingdon, 10Nov2019.)

⊢ ((A ∈ 𝜔 ∧
B ∈
N) → (A
+_{𝑜} B) ∈ N) 

Definition  dfplpq 6321* 
Define preaddition 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 "preaddition" operation works
directly
with ordered pairs of integers. The actual positive fraction addition
+_{Q} (dfplqqs 6326) works with the equivalence classes of these
ordered pairs determined by the equivalence relation ~_{Q}
(dfenq 6324). (Analogous remarks apply to the other
"pre" operations
in the complex number construction that follows.) From Proposition
92.3 of [Gleason] p. 117. (Contributed
by NM, 28Aug1995.)

⊢ +_{pQ} = (x ∈
(N × N), y ∈
(N × N) ↦ ⟨(((1^{st}
‘x)
·_{N} (2^{nd} ‘y)) +_{N} ((1^{st}
‘y)
·_{N} (2^{nd} ‘x))), ((2^{nd} ‘x) ·_{N}
(2^{nd} ‘y))⟩) 

Definition  dfmpq 6322* 
Define premultiplication 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 92.4 of [Gleason]
p. 119. (Contributed by NM, 28Aug1995.)

⊢ ·_{pQ} =
(x ∈
(N × N), y ∈
(N × N) ↦ ⟨((1^{st}
‘x)
·_{N} (1^{st} ‘y)), ((2^{nd} ‘x) ·_{N}
(2^{nd} ‘y))⟩) 

Definition  dfltpq 6323* 
Define preordering 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,
28Aug1995.)

⊢ <_{pQ} =
{⟨x, y⟩ ∣ ((x ∈
(N × N) ∧
y ∈
(N × N)) ∧
((1^{st} ‘x)
·_{N} (2^{nd} ‘y)) <_{N} ((1^{st}
‘y)
·_{N} (2^{nd} ‘x)))} 

Definition  dfenq 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 92.1 of
[Gleason] p. 117. (Contributed by NM,
27Aug1995.)

⊢ ~_{Q} = {⟨x, y⟩
∣ ((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ ∃z∃w∃v∃u((x =
⟨z, w⟩ ∧ y = ⟨v,
u⟩) ∧ (z
·_{N} u) =
(w ·_{N}
v)))} 

Definition  dfnqqs 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 92.2 of [Gleason] p. 117.
(Contributed by NM, 16Aug1995.)

⊢ Q = ((N ×
N) / ~_{Q} ) 

Definition  dfplqqs 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 92.3 of [Gleason] p. 117.
(Contributed by NM, 24Aug1995.)

⊢ +_{Q} =
{⟨⟨x, y⟩, z⟩ ∣ ((x ∈
Q ∧ y ∈
Q) ∧ ∃w∃v∃u∃f((x = [⟨w,
v⟩] ~_{Q} ∧ y =
[⟨u, f⟩] ~_{Q} ) ∧ z =
[(⟨w, v⟩ +_{pQ} ⟨u, f⟩)]
~_{Q} ))} 

Definition  dfmqqs 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 92.4 of [Gleason] p. 119.
(Contributed by NM, 24Aug1995.)

⊢ ·_{Q} =
{⟨⟨x, y⟩, z⟩ ∣ ((x ∈
Q ∧ y ∈
Q) ∧ ∃w∃v∃u∃f((x = [⟨w,
v⟩] ~_{Q} ∧ y =
[⟨u, f⟩] ~_{Q} ) ∧ z =
[(⟨w, v⟩ ·_{pQ}
⟨u, f⟩)] ~_{Q}
))} 

Definition  df1nqqs 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 92.2 of [Gleason] p. 117.
(Contributed by NM, 29Oct1995.)

⊢ 1_{Q} =
[⟨1_{𝑜}, 1_{𝑜}⟩]
~_{Q} 

Definition  dfrq 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 92.5 of [Gleason] p. 119, who uses an
asterisk to denote this unary operation. (Contributed by Jim Kingdon,
20Sep2019.)

⊢ *_{Q} = {⟨x, y⟩
∣ (x ∈ Q ∧ y ∈ Q ∧ (x
·_{Q} y) =
1_{Q})} 

Definition  dfltnqqs 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, 13Feb1996.)

⊢ <_{Q} =
{⟨x, y⟩ ∣ ((x ∈
Q ∧ y ∈
Q) ∧ ∃z∃w∃v∃u((x = [⟨z,
w⟩] ~_{Q} ∧ y =
[⟨v, u⟩] ~_{Q} ) ∧ (z
·_{N} u)
<_{N} (w
·_{N} v)))} 

Theorem  dfplpq2 6331* 
Alternative definition of preaddition on positive fractions.
(Contributed by Jim Kingdon, 12Sep2019.)

⊢ +_{pQ} =
{⟨⟨x, y⟩, z⟩ ∣ ((x ∈
(N × N) ∧
y ∈
(N × N)) ∧
∃w∃v∃u∃f((x = ⟨w,
v⟩ ∧
y = ⟨u, f⟩)
∧ z =
⟨((w
·_{N} f)
+_{N} (v
·_{N} u)),
(v ·_{N}
f)⟩))} 

Theorem  dfmpq2 6332* 
Alternative definition of premultiplication on positive fractions.
(Contributed by Jim Kingdon, 13Sep2019.)

⊢ ·_{pQ} =
{⟨⟨x, y⟩, z⟩ ∣ ((x ∈
(N × N) ∧
y ∈
(N × N)) ∧
∃w∃v∃u∃f((x = ⟨w,
v⟩ ∧
y = ⟨u, f⟩)
∧ z =
⟨(w
·_{N} u),
(v ·_{N}
f)⟩))} 

Theorem  enqbreq 6333 
Equivalence relation for positive fractions in terms of positive
integers. (Contributed by NM, 27Aug1995.)

⊢ (((A ∈ N ∧ B ∈ N) ∧ (𝐶 ∈
N ∧ 𝐷 ∈
N)) → (⟨A,
B⟩ ~_{Q}
⟨𝐶, 𝐷⟩ ↔ (A ·_{N} 𝐷) = (B ·_{N} 𝐶))) 

Theorem  enqbreq2 6334 
Equivalence relation for positive fractions in terms of positive
integers. (Contributed by Mario Carneiro, 8May2013.)

⊢ ((A ∈ (N × N) ∧ B ∈ (N × N))
→ (A ~_{Q}
B ↔ ((1^{st} ‘A) ·_{N}
(2^{nd} ‘B)) =
((1^{st} ‘B)
·_{N} (2^{nd} ‘A)))) 

Theorem  enqer 6335 
The equivalence relation for positive fractions is an equivalence
relation. Proposition 92.1 of [Gleason] p. 117. (Contributed by NM,
27Aug1995.) (Revised by Mario Carneiro, 6Jul2015.)

⊢ ~_{Q} Er
(N × N) 

Theorem  enqeceq 6336 
Equivalence class equality of positive fractions in terms of positive
integers. (Contributed by NM, 29Nov1995.)

⊢ (((A ∈ N ∧ B ∈ N) ∧ (𝐶 ∈
N ∧ 𝐷 ∈
N)) → ([⟨A,
B⟩] ~_{Q} =
[⟨𝐶, 𝐷⟩]
~_{Q} ↔ (A
·_{N} 𝐷) = (B
·_{N} 𝐶))) 

Theorem  enqex 6337 
The equivalence relation for positive fractions exists. (Contributed by
NM, 3Sep1995.)

⊢ ~_{Q} ∈ V 

Theorem  enqdc 6338 
The equivalence relation for positive fractions is decidable.
(Contributed by Jim Kingdon, 7Sep2019.)

⊢ (((A ∈ N ∧ B ∈ N) ∧ (𝐶 ∈
N ∧ 𝐷 ∈
N)) → DECID ⟨A, B⟩
~_{Q} ⟨𝐶, 𝐷⟩) 

Theorem  enqdc1 6339 
The equivalence relation for positive fractions is decidable.
(Contributed by Jim Kingdon, 7Sep2019.)

⊢ (((A ∈ N ∧ B ∈ N) ∧ 𝐶 ∈
(N × N)) → DECID
⟨A, B⟩ ~_{Q} 𝐶) 

Theorem  nqex 6340 
The class of positive fractions exists. (Contributed by NM,
16Aug1995.) (Revised by Mario Carneiro, 27Apr2013.)

⊢ Q ∈ V 

Theorem  0nnq 6341 
The empty set is not a positive fraction. (Contributed by NM,
24Aug1995.) (Revised by Mario Carneiro, 27Apr2013.)

⊢ ¬ ∅ ∈ Q 

Theorem  ltrelnq 6342 
Positive fraction 'less than' is a relation on positive fractions.
(Contributed by NM, 14Feb1996.) (Revised by Mario Carneiro,
27Apr2013.)

⊢ <_{Q} ⊆
(Q × Q) 

Theorem  1nq 6343 
The positive fraction 'one'. (Contributed by NM, 29Oct1995.)

⊢ 1_{Q} ∈ Q 

Theorem  addcmpblnq 6344 
Lemma showing compatibility of addition. (Contributed by NM,
27Aug1995.)

⊢ ((((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)⟩)) 

Theorem  mulcmpblnq 6345 
Lemma showing compatibility of multiplication. (Contributed by NM,
27Aug1995.)

⊢ ((((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)⟩)) 

Theorem  addpipqqslem 6346 
Lemma for addpipqqs 6347. (Contributed by Jim Kingdon, 11Sep2019.)

⊢ (((A ∈ N ∧ B ∈ N) ∧ (𝐶 ∈
N ∧ 𝐷 ∈
N)) → ⟨((A
·_{N} 𝐷) +_{N} (B ·_{N} 𝐶)), (B ·_{N} 𝐷)⟩ ∈ (N ×
N)) 

Theorem  addpipqqs 6347 
Addition of positive fractions in terms of positive integers.
(Contributed by NM, 28Aug1995.)

⊢ (((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} ) 

Theorem  mulpipq2 6348 
Multiplication of positive fractions in terms of positive integers.
(Contributed by Mario Carneiro, 8May2013.)

⊢ ((A ∈ (N × N) ∧ B ∈ (N × N))
→ (A
·_{pQ} B) =
⟨((1^{st} ‘A)
·_{N} (1^{st} ‘B)), ((2^{nd} ‘A) ·_{N}
(2^{nd} ‘B))⟩) 

Theorem  mulpipq 6349 
Multiplication of positive fractions in terms of positive integers.
(Contributed by NM, 28Aug1995.) (Revised by Mario Carneiro,
8May2013.)

⊢ (((A ∈ N ∧ B ∈ N) ∧ (𝐶 ∈
N ∧ 𝐷 ∈
N)) → (⟨A,
B⟩
·_{pQ} ⟨𝐶, 𝐷⟩) = ⟨(A ·_{N} 𝐶), (B ·_{N} 𝐷)⟩) 

Theorem  mulpipqqs 6350 
Multiplication of positive fractions in terms of positive integers.
(Contributed by NM, 28Aug1995.)

⊢ (((A ∈ N ∧ B ∈ N) ∧ (𝐶 ∈
N ∧ 𝐷 ∈
N)) → ([⟨A,
B⟩] ~_{Q}
·_{Q} [⟨𝐶, 𝐷⟩] ~_{Q} ) =
[⟨(A
·_{N} 𝐶), (B
·_{N} 𝐷)⟩] ~_{Q}
) 

Theorem  ordpipqqs 6351 
Ordering of positive fractions in terms of positive integers.
(Contributed by Jim Kingdon, 14Sep2019.)

⊢ (((A ∈ N ∧ B ∈ N) ∧ (𝐶 ∈
N ∧ 𝐷 ∈
N)) → ([⟨A,
B⟩] ~_{Q}
<_{Q} [⟨𝐶, 𝐷⟩] ~_{Q} ↔
(A ·_{N}
𝐷)
<_{N} (B
·_{N} 𝐶))) 

Theorem  addclnq 6352 
Closure of addition on positive fractions. (Contributed by NM,
29Aug1995.)

⊢ ((A ∈ Q ∧ B ∈ Q) → (A +_{Q} B) ∈
Q) 

Theorem  mulclnq 6353 
Closure of multiplication on positive fractions. (Contributed by NM,
29Aug1995.)

⊢ ((A ∈ Q ∧ B ∈ Q) → (A ·_{Q} B) ∈
Q) 

Theorem  dmaddpqlem 6354* 
Decomposition of a positive fraction into numerator and denominator.
Lemma for dmaddpq 6356. (Contributed by Jim Kingdon, 15Sep2019.)

⊢ (x ∈ Q → ∃w∃v x = [⟨w,
v⟩] ~_{Q}
) 

Theorem  nqpi 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,
20Sep2019.)

⊢ (A ∈ Q → ∃w∃v((w ∈
N ∧ v ∈
N) ∧ A = [⟨w,
v⟩] ~_{Q}
)) 

Theorem  dmaddpq 6356 
Domain of addition on positive fractions. (Contributed by NM,
24Aug1995.)

⊢ dom +_{Q} =
(Q × Q) 

Theorem  dmmulpq 6357 
Domain of multiplication on positive fractions. (Contributed by NM,
24Aug1995.)

⊢ dom ·_{Q} =
(Q × Q) 

Theorem  addcomnqg 6358 
Addition of positive fractions is commutative. (Contributed by Jim
Kingdon, 15Sep2019.)

⊢ ((A ∈ Q ∧ B ∈ Q) → (A +_{Q} B) = (B
+_{Q} A)) 

Theorem  addassnqg 6359 
Addition of positive fractions is associative. (Contributed by Jim
Kingdon, 16Sep2019.)

⊢ ((A ∈ Q ∧ B ∈ Q ∧ 𝐶 ∈
Q) → ((A
+_{Q} B)
+_{Q} 𝐶) = (A
+_{Q} (B
+_{Q} 𝐶))) 

Theorem  mulcomnqg 6360 
Multiplication of positive fractions is commutative. (Contributed by
Jim Kingdon, 17Sep2019.)

⊢ ((A ∈ Q ∧ B ∈ Q) → (A ·_{Q} B) = (B
·_{Q} A)) 

Theorem  mulassnqg 6361 
Multiplication of positive fractions is associative. (Contributed by
Jim Kingdon, 17Sep2019.)

⊢ ((A ∈ Q ∧ B ∈ Q ∧ 𝐶 ∈
Q) → ((A
·_{Q} B)
·_{Q} 𝐶) = (A
·_{Q} (B
·_{Q} 𝐶))) 

Theorem  mulcanenq 6362 
Lemma for distributive law: cancellation of common factor. (Contributed
by NM, 2Sep1995.) (Revised by Mario Carneiro, 8May2013.)

⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈
N) → ⟨(A
·_{N} B),
(A ·_{N}
𝐶)⟩
~_{Q} ⟨B, 𝐶⟩) 

Theorem  mulcanenqec 6363 
Lemma for distributive law: cancellation of common factor. (Contributed
by Jim Kingdon, 17Sep2019.)

⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈
N) → [⟨(A
·_{N} B),
(A ·_{N}
𝐶)⟩]
~_{Q} = [⟨B,
𝐶⟩]
~_{Q} ) 

Theorem  distrnqg 6364 
Multiplication of positive fractions is distributive. (Contributed by
Jim Kingdon, 17Sep2019.)

⊢ ((A ∈ Q ∧ B ∈ Q ∧ 𝐶 ∈
Q) → (A
·_{Q} (B
+_{Q} 𝐶)) = ((A ·_{Q} B) +_{Q} (A ·_{Q} 𝐶))) 

Theorem  1qec 6365 
The equivalence class of ratio 1. (Contributed by NM, 4Mar1996.)

⊢ (A ∈ N →
1_{Q} = [⟨A,
A⟩] ~_{Q}
) 

Theorem  mulidnq 6366 
Multiplication identity element for positive fractions. (Contributed by
NM, 3Mar1996.)

⊢ (A ∈ Q → (A ·_{Q}
1_{Q}) = A) 

Theorem  recexnq 6367* 
Existence of positive fraction reciprocal. (Contributed by Jim Kingdon,
20Sep2019.)

⊢ (A ∈ Q → ∃y(y ∈
Q ∧ (A ·_{Q} y) = 1_{Q})) 

Theorem  recmulnqg 6368 
Relationship between reciprocal and multiplication on positive
fractions. (Contributed by Jim Kingdon, 19Sep2019.)

⊢ ((A ∈ Q ∧ B ∈ Q) →
((*_{Q}‘A) =
B ↔ (A ·_{Q} B) = 1_{Q})) 

Theorem  recclnq 6369 
Closure law for positive fraction reciprocal. (Contributed by NM,
6Mar1996.) (Revised by Mario Carneiro, 8May2013.)

⊢ (A ∈ Q →
(*_{Q}‘A)
∈ Q) 

Theorem  recidnq 6370 
A positive fraction times its reciprocal is 1. (Contributed by NM,
6Mar1996.) (Revised by Mario Carneiro, 8May2013.)

⊢ (A ∈ Q → (A ·_{Q}
(*_{Q}‘A)) =
1_{Q}) 

Theorem  recrecnq 6371 
Reciprocal of reciprocal of positive fraction. (Contributed by NM,
26Apr1996.) (Revised by Mario Carneiro, 29Apr2013.)

⊢ (A ∈ Q →
(*_{Q}‘(*_{Q}‘A)) = A) 

Theorem  rec1nq 6372 
Reciprocal of positive fraction one. (Contributed by Jim Kingdon,
29Dec2019.)

⊢
(*_{Q}‘1_{Q}) =
1_{Q} 

Theorem  nqtri3or 6373 
Trichotomy for positive fractions. (Contributed by Jim Kingdon,
21Sep2019.)

⊢ ((A ∈ Q ∧ B ∈ Q) → (A <_{Q} B ∨ A = B ∨ B
<_{Q} A)) 

Theorem  ltdcnq 6374 
Lessthan for positive fractions is decidable. (Contributed by Jim
Kingdon, 12Dec2019.)

⊢ ((A ∈ Q ∧ B ∈ Q) → DECID
A <_{Q} B) 

Theorem  ltsonq 6375 
'Less than' is a strict ordering on positive fractions. (Contributed by
NM, 19Feb1996.) (Revised by Mario Carneiro, 4May2013.)

⊢ <_{Q} Or
Q 

Theorem  nqtric 6376 
Trichotomy for positive integers. (Contributed by Jim Kingdon,
21Sep2019.)

⊢ ((A ∈ Q ∧ B ∈ Q) → (A <_{Q} B ↔ ¬ (A = B ∨ B
<_{Q} A))) 

Theorem  ltanqg 6377 
Ordering property of addition for positive fractions. Proposition
92.6(ii) of [Gleason] p. 120.
(Contributed by Jim Kingdon,
22Sep2019.)

⊢ ((A ∈ Q ∧ B ∈ Q ∧ 𝐶 ∈
Q) → (A
<_{Q} B ↔
(𝐶
+_{Q} A)
<_{Q} (𝐶 +_{Q} B))) 

Theorem  ltmnqg 6378 
Ordering property of multiplication for positive fractions. Proposition
92.6(iii) of [Gleason] p. 120.
(Contributed by Jim Kingdon,
22Sep2019.)

⊢ ((A ∈ Q ∧ B ∈ Q ∧ 𝐶 ∈
Q) → (A
<_{Q} B ↔
(𝐶
·_{Q} A)
<_{Q} (𝐶 ·_{Q}
B))) 

Theorem  ltanqi 6379 
Ordering property of addition for positive fractions. One direction of
ltanqg 6377. (Contributed by Jim Kingdon, 9Dec2019.)

⊢ ((A
<_{Q} B ∧ 𝐶 ∈
Q) → (𝐶
+_{Q} A)
<_{Q} (𝐶 +_{Q} B)) 

Theorem  ltmnqi 6380 
Ordering property of multiplication for positive fractions. One direction
of ltmnqg 6378. (Contributed by Jim Kingdon, 9Dec2019.)

⊢ ((A
<_{Q} B ∧ 𝐶 ∈
Q) → (𝐶
·_{Q} A)
<_{Q} (𝐶 ·_{Q}
B)) 

Theorem  lt2addnq 6381 
Ordering property of addition for positive fractions. (Contributed by Jim
Kingdon, 7Dec2019.)

⊢ (((A ∈ Q ∧ B ∈ Q) ∧ (𝐶 ∈
Q ∧ 𝐷 ∈
Q)) → ((A
<_{Q} B ∧ 𝐶 <_{Q} 𝐷) → (A +_{Q} 𝐶) <_{Q} (B +_{Q} 𝐷))) 

Theorem  1lt2nq 6382 
One is less than two (one plus one). (Contributed by NM, 13Mar1996.)
(Revised by Mario Carneiro, 10May2013.)

⊢ 1_{Q}
<_{Q} (1_{Q}
+_{Q} 1_{Q}) 

Theorem  ltaddnq 6383 
The sum of two fractions is greater than one of them. (Contributed by
NM, 14Mar1996.) (Revised by Mario Carneiro, 10May2013.)

⊢ ((A ∈ Q ∧ B ∈ Q) → A <_{Q} (A +_{Q} B)) 

Theorem  ltexnqq 6384* 
Ordering on positive fractions in terms of existence of sum. Definition
in Proposition 92.6 of [Gleason] p.
119. (Contributed by Jim Kingdon,
23Sep2019.)

⊢ ((A ∈ Q ∧ B ∈ Q) → (A <_{Q} B ↔ ∃x ∈ Q (A +_{Q} x) = B)) 

Theorem  ltexnqi 6385* 
Ordering on positive fractions in terms of existence of sum.
(Contributed by Jim Kingdon, 30Apr2020.)

⊢ (A
<_{Q} B →
∃x
∈ Q (A +_{Q} x) = B) 

Theorem  halfnqq 6386* 
Onehalf of any positive fraction is a fraction. (Contributed by Jim
Kingdon, 23Sep2019.)

⊢ (A ∈ Q → ∃x ∈ Q (x +_{Q} x) = A) 

Theorem  halfnq 6387* 
Onehalf of any positive fraction exists. Lemma for Proposition
92.6(i) of [Gleason] p. 120.
(Contributed by NM, 16Mar1996.)
(Revised by Mario Carneiro, 10May2013.)

⊢ (A ∈ Q → ∃x(x +_{Q} x) = A) 

Theorem  nsmallnqq 6388* 
There is no smallest positive fraction. (Contributed by Jim Kingdon,
24Sep2019.)

⊢ (A ∈ Q → ∃x ∈ Q x <_{Q} A) 

Theorem  nsmallnq 6389* 
There is no smallest positive fraction. (Contributed by NM,
26Apr1996.) (Revised by Mario Carneiro, 10May2013.)

⊢ (A ∈ Q → ∃x x <_{Q} A) 

Theorem  subhalfnqq 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,
25Nov2019.)

⊢ (A ∈ Q → ∃x ∈ Q (x +_{Q} x) <_{Q} A) 

Theorem  ltbtwnnqq 6391* 
There exists a number between any two positive fractions. Proposition
92.6(i) of [Gleason] p. 120.
(Contributed by Jim Kingdon,
24Sep2019.)

⊢ (A
<_{Q} B ↔
∃x
∈ Q (A <_{Q} x ∧ x <_{Q} B)) 

Theorem  ltbtwnnq 6392* 
There exists a number between any two positive fractions. Proposition
92.6(i) of [Gleason] p. 120.
(Contributed by NM, 17Mar1996.)
(Revised by Mario Carneiro, 10May2013.)

⊢ (A
<_{Q} B ↔
∃x(A
<_{Q} x ∧ x
<_{Q} B)) 

Theorem  archnqq 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,
1Dec2019.)

⊢ (A ∈ Q → ∃x ∈ N A <_{Q} [⟨x, 1_{𝑜}⟩]
~_{Q} ) 

Theorem  prarloclemarch 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, 30Nov2019.)

⊢ ((A ∈ Q ∧ B ∈ Q) → ∃x ∈ N A <_{Q} ([⟨x, 1_{𝑜}⟩]
~_{Q} ·_{Q} B)) 

Theorem  prarloclemarch2 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, 25Nov2019.)

⊢ ((A ∈ Q ∧ B ∈ Q ∧ 𝐶 ∈
Q) → ∃x ∈
N (1_{𝑜} <_{N} x ∧ A <_{Q} (B +_{Q} ([⟨x, 1_{𝑜}⟩]
~_{Q} ·_{Q} 𝐶)))) 

Theorem  ltrnqg 6396 
Ordering property of reciprocal for positive fractions. For a
simplified version of the forward implication, see ltrnqi 6397.
(Contributed by Jim Kingdon, 29Dec2019.)

⊢ ((A ∈ Q ∧ B ∈ Q) → (A <_{Q} B ↔
(*_{Q}‘B)
<_{Q} (*_{Q}‘A))) 

Theorem  ltrnqi 6397 
Ordering property of reciprocal for positive fractions. For the
converse, see ltrnqg 6396. (Contributed by Jim Kingdon, 24Sep2019.)

⊢ (A
<_{Q} B →
(*_{Q}‘B)
<_{Q} (*_{Q}‘A)) 

Theorem  nnnq 6398 
The canonical embedding of positive integers into positive fractions.
(Contributed by Jim Kingdon, 26Apr2020.)

⊢ (A ∈ N → [⟨A, 1_{𝑜}⟩]
~_{Q} ∈
Q) 

Definition  dfenq0 6399* 
Define equivalence relation for nonnegative 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, 2Nov2019.)

⊢ ~_{Q0} = {⟨x, y⟩
∣ ((x ∈ (𝜔 × N) ∧ y ∈ (𝜔 × N)) ∧ ∃z∃w∃v∃u((x =
⟨z, w⟩ ∧ y = ⟨v,
u⟩) ∧ (z
·_{𝑜} u) =
(w ·_{𝑜} v)))} 

Definition  dfnq0 6400 
Define class of nonnegative 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, 2Nov2019.)

⊢ Q_{0} = ((𝜔
× N) / ~_{Q0}
) 