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Theorem List for Intuitionistic Logic Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremqssre 8301 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
ℚ ⊆ ℝ
 
Theoremqsscn 8302 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℚ ⊆ ℂ
 
Theoremqex 8303 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
V
 
Theoremnnq 8304 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(A ℕ → A ℚ)
 
Theoremqcn 8305 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
(A ℚ → A ℂ)
 
Theoremqaddcl 8306 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
((A B ℚ) → (A + B) ℚ)
 
Theoremqnegcl 8307 Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)
(A ℚ → -A ℚ)
 
Theoremqmulcl 8308 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
((A B ℚ) → (A · B) ℚ)
 
Theoremqsubcl 8309 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
((A B ℚ) → (AB) ℚ)
 
Theoremqapne 8310 Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 20-Mar-2020.)
((A B ℚ) → (A # BAB))
 
Theoremqreccl 8311 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
((A A ≠ 0) → (1 / A) ℚ)
 
Theoremqdivcl 8312 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
((A B B ≠ 0) → (A / B) ℚ)
 
Theoremqrevaddcl 8313 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
(B ℚ → ((A (A + B) ℚ) ↔ A ℚ))
 
Theoremnnrecq 8314 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(A ℕ → (1 / A) ℚ)
 
Theoremirradd 8315 The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008.)
((A (ℝ ∖ ℚ) B ℚ) → (A + B) (ℝ ∖ ℚ))
 
Theoremirrmul 8316 The product of a real which is not rational with a nonzero rational is not rational. Note that by "not rational" we mean the negation of "is rational" (whereas "irrational" is often defined to mean apart from any rational number - given excluded middle these two definitions would be equivalent). (Contributed by NM, 7-Nov-2008.)
((A (ℝ ∖ ℚ) B B ≠ 0) → (A · B) (ℝ ∖ ℚ))
 
3.4.12  Complex numbers as pairs of reals
 
Theoremcnref1o 8317* There is a natural one-to-one mapping from (ℝ × ℝ) to , where we map x, y to (x + (i · y)). In our construction of the complex numbers, this is in fact our definition of (see df-c 6677), but in the axiomatic treatment we can only show that there is the expected mapping between these two sets. (Contributed by Mario Carneiro, 16-Jun-2013.) (Revised by Mario Carneiro, 17-Feb-2014.)
𝐹 = (x ℝ, y ℝ ↦ (x + (i · y)))       𝐹:(ℝ × ℝ)–1-1-onto→ℂ
 
3.5  Order sets
 
3.5.1  Positive reals (as a subset of complex numbers)
 
Syntaxcrp 8318 Extend class notation to include the class of positive reals.
class +
 
Definitiondf-rp 8319 Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
+ = {x ℝ ∣ 0 < x}
 
Theoremelrp 8320 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
(A + ↔ (A 0 < A))
 
Theoremelrpii 8321 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
A     &   0 < A       A +
 
Theorem1rp 8322 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
1 +
 
Theorem2rp 8323 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
2 +
 
Theoremrpre 8324 A positive real is a real. (Contributed by NM, 27-Oct-2007.)
(A +A ℝ)
 
Theoremrpxr 8325 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
(A +A *)
 
Theoremrpcn 8326 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
(A +A ℂ)
 
Theoremnnrp 8327 A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
(A ℕ → A +)
 
Theoremrpssre 8328 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
+ ⊆ ℝ
 
Theoremrpgt0 8329 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
(A + → 0 < A)
 
Theoremrpge0 8330 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
(A + → 0 ≤ A)
 
Theoremrpregt0 8331 A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
(A + → (A 0 < A))
 
Theoremrprege0 8332 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
(A + → (A 0 ≤ A))
 
Theoremrpne0 8333 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
(A +A ≠ 0)
 
Theoremrpap0 8334 A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
(A +A # 0)
 
Theoremrprene0 8335 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
(A + → (A A ≠ 0))
 
Theoremrpreap0 8336 A positive real is a real number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
(A + → (A A # 0))
 
Theoremrpcnne0 8337 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
(A + → (A A ≠ 0))
 
Theoremrpcnap0 8338 A positive real is a complex number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
(A + → (A A # 0))
 
Theoremralrp 8339 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
(x + φx ℝ (0 < xφ))
 
Theoremrexrp 8340 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
(x + φx ℝ (0 < x φ))
 
Theoremrpaddcl 8341 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
((A + B +) → (A + B) +)
 
Theoremrpmulcl 8342 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
((A + B +) → (A · B) +)
 
Theoremrpdivcl 8343 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
((A + B +) → (A / B) +)
 
Theoremrpreccl 8344 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
(A + → (1 / A) +)
 
Theoremrphalfcl 8345 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
(A + → (A / 2) +)
 
Theoremrpgecl 8346 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
((A + B AB) → B +)
 
Theoremrphalflt 8347 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)
(A + → (A / 2) < A)
 
Theoremrerpdivcl 8348 Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)
((A B +) → (A / B) ℝ)
 
Theoremge0p1rp 8349 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)
((A 0 ≤ A) → (A + 1) +)
 
Theoremrpnegap 8350 Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.)
((A A # 0) → (A + ⊻ -A +))
 
Theorem0nrp 8351 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
¬ 0 +
 
Theoremltsubrp 8352 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
((A B +) → (AB) < A)
 
Theoremltaddrp 8353 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
((A B +) → A < (A + B))
 
Theoremdifrp 8354 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)
((A B ℝ) → (A < B ↔ (BA) +))
 
Theoremelrpd 8355 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φ → 0 < A)       (φA +)
 
Theoremnnrpd 8356 A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℕ)       (φA +)
 
Theoremrpred 8357 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φA ℝ)
 
Theoremrpxrd 8358 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φA *)
 
Theoremrpcnd 8359 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φA ℂ)
 
Theoremrpgt0d 8360 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → 0 < A)
 
Theoremrpge0d 8361 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → 0 ≤ A)
 
Theoremrpne0d 8362 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φA ≠ 0)
 
Theoremrpregt0d 8363 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (A 0 < A))
 
Theoremrprege0d 8364 A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (A 0 ≤ A))
 
Theoremrprene0d 8365 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (A A ≠ 0))
 
Theoremrpcnne0d 8366 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (A A ≠ 0))
 
Theoremrpreccld 8367 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (1 / A) +)
 
Theoremrprecred 8368 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (1 / A) ℝ)
 
Theoremrphalfcld 8369 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (A / 2) +)
 
Theoremreclt1d 8370 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (A < 1 ↔ 1 < (1 / A)))
 
Theoremrecgt1d 8371 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (1 < A ↔ (1 / A) < 1))
 
Theoremrpaddcld 8372 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φB +)       (φ → (A + B) +)
 
Theoremrpmulcld 8373 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φB +)       (φ → (A · B) +)
 
Theoremrpdivcld 8374 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φB +)       (φ → (A / B) +)
 
Theoremltrecd 8375 The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φB +)       (φ → (A < B ↔ (1 / B) < (1 / A)))
 
Theoremlerecd 8376 The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φB +)       (φ → (AB ↔ (1 / B) ≤ (1 / A)))
 
Theoremltrec1d 8377 Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φB +)    &   (φ → (1 / A) < B)       (φ → (1 / B) < A)
 
Theoremlerec2d 8378 Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φB +)    &   (φA ≤ (1 / B))       (φB ≤ (1 / A))
 
Theoremlediv2ad 8379 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φB +)    &   (φ𝐶 ℝ)    &   (φ → 0 ≤ 𝐶)    &   (φAB)       (φ → (𝐶 / B) ≤ (𝐶 / A))
 
Theoremltdiv2d 8380 Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φB +)    &   (φ𝐶 +)       (φ → (A < B ↔ (𝐶 / B) < (𝐶 / A)))
 
Theoremlediv2d 8381 Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φB +)    &   (φ𝐶 +)       (φ → (AB ↔ (𝐶 / B) ≤ (𝐶 / A)))
 
Theoremledivdivd 8382 Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φB +)    &   (φ𝐶 +)    &   (φ𝐷 +)    &   (φ → (A / B) ≤ (𝐶 / 𝐷))       (φ → (𝐷 / 𝐶) ≤ (B / A))
 
Theoremge0p1rpd 8383 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φ → 0 ≤ A)       (φ → (A + 1) +)
 
Theoremrerpdivcld 8384 Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φ → (A / B) ℝ)
 
Theoremltsubrpd 8385 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φ → (AB) < A)
 
Theoremltaddrpd 8386 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φA < (A + B))
 
Theoremltaddrp2d 8387 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φA < (B + A))
 
Theoremltmulgt11d 8388 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φ → (1 < AB < (B · A)))
 
Theoremltmulgt12d 8389 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φ → (1 < AB < (A · B)))
 
Theoremgt0divd 8390 Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φ → (0 < A ↔ 0 < (A / B)))
 
Theoremge0divd 8391 Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φ → (0 ≤ A ↔ 0 ≤ (A / B)))
 
Theoremrpgecld 8392 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)    &   (φBA)       (φA +)
 
Theoremdivge0d 8393 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)    &   (φ → 0 ≤ A)       (φ → 0 ≤ (A / B))
 
Theoremltmul1d 8394 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → (A < B ↔ (A · 𝐶) < (B · 𝐶)))
 
Theoremltmul2d 8395 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → (A < B ↔ (𝐶 · A) < (𝐶 · B)))
 
Theoremlemul1d 8396 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → (AB ↔ (A · 𝐶) ≤ (B · 𝐶)))
 
Theoremlemul2d 8397 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → (AB ↔ (𝐶 · A) ≤ (𝐶 · B)))
 
Theoremltdiv1d 8398 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → (A < B ↔ (A / 𝐶) < (B / 𝐶)))
 
Theoremlediv1d 8399 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → (AB ↔ (A / 𝐶) ≤ (B / 𝐶)))
 
Theoremltmuldivd 8400 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → ((A · 𝐶) < BA < (B / 𝐶)))
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