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Theorem List for Intuitionistic Logic Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrpge0d 8401 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → 0 ≤ A)
 
Theoremrpne0d 8402 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φA ≠ 0)
 
Theoremrpregt0d 8403 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (A 0 < A))
 
Theoremrprege0d 8404 A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (A 0 ≤ A))
 
Theoremrprene0d 8405 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (A A ≠ 0))
 
Theoremrpcnne0d 8406 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (A A ≠ 0))
 
Theoremrpreccld 8407 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (1 / A) +)
 
Theoremrprecred 8408 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (1 / A) ℝ)
 
Theoremrphalfcld 8409 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)       (φ → (A / 2) +)
 
Theoremreclt1d 8410 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 8411 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 8412 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 8413 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 8414 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φB +)       (φ → (A / B) +)
 
Theoremltrecd 8415 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 8416 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 8417 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 8418 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 8419 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 8420 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 8421 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 8422 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 8423 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 8424 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 8425 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φ → (AB) < A)
 
Theoremltaddrpd 8426 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φA < (A + B))
 
Theoremltaddrp2d 8427 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φA < (B + A))
 
Theoremltmulgt11d 8428 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φ → (1 < AB < (B · A)))
 
Theoremltmulgt12d 8429 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φ → (1 < AB < (A · B)))
 
Theoremgt0divd 8430 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 8431 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 8432 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 8433 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 8434 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 8435 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 8436 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 8437 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 8438 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 8439 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 8440 '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)))
 
Theoremltmuldiv2d 8441 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → ((𝐶 · A) < BA < (B / 𝐶)))
 
Theoremlemuldivd 8442 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → ((A · 𝐶) ≤ BA ≤ (B / 𝐶)))
 
Theoremlemuldiv2d 8443 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → ((𝐶 · A) ≤ BA ≤ (B / 𝐶)))
 
Theoremltdivmuld 8444 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → ((A / 𝐶) < BA < (𝐶 · B)))
 
Theoremltdivmul2d 8445 '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)))
 
Theoremledivmuld 8446 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → ((A / 𝐶) ≤ BA ≤ (𝐶 · B)))
 
Theoremledivmul2d 8447 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)       (φ → ((A / 𝐶) ≤ BA ≤ (B · 𝐶)))
 
Theoremltmul1dd 8448 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)    &   (φA < B)       (φ → (A · 𝐶) < (B · 𝐶))
 
Theoremltmul2dd 8449 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)    &   (φA < B)       (φ → (𝐶 · A) < (𝐶 · B))
 
Theoremltdiv1dd 8450 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)    &   (φA < B)       (φ → (A / 𝐶) < (B / 𝐶))
 
Theoremlediv1dd 8451 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)    &   (φAB)       (φ → (A / 𝐶) ≤ (B / 𝐶))
 
Theoremlediv12ad 8452 Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 +)    &   (φ𝐷 ℝ)    &   (φ → 0 ≤ A)    &   (φAB)    &   (φ𝐶𝐷)       (φ → (A / 𝐷) ≤ (B / 𝐶))
 
Theoremltdiv23d 8453 Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)    &   (φ𝐶 +)    &   (φ → (A / B) < 𝐶)       (φ → (A / 𝐶) < B)
 
Theoremlediv23d 8454 Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)    &   (φ𝐶 +)    &   (φ → (A / B) ≤ 𝐶)       (φ → (A / 𝐶) ≤ B)
 
Theoremlt2mul2divd 8455 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)    &   (φ𝐶 ℝ)    &   (φ𝐷 +)       (φ → ((A · B) < (𝐶 · 𝐷) ↔ (A / 𝐷) < (𝐶 / B)))
 
3.5.2  Infinity and the extended real number system (cont.)
 
Syntaxcxne 8456 Extend class notation to include the negative of an extended real.
class -𝑒A
 
Syntaxcxad 8457 Extend class notation to include addition of extended reals.
class +𝑒
 
Syntaxcxmu 8458 Extend class notation to include multiplication of extended reals.
class ·e
 
Definitiondf-xneg 8459 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)
-𝑒A = if(A = +∞, -∞, if(A = -∞, +∞, -A))
 
Definitiondf-xadd 8460* Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
+𝑒 = (x *, y * ↦ if(x = +∞, if(y = -∞, 0, +∞), if(x = -∞, if(y = +∞, 0, -∞), if(y = +∞, +∞, if(y = -∞, -∞, (x + y))))))
 
Definitiondf-xmul 8461* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
·e = (x *, y * ↦ if((x = 0 y = 0), 0, if((((0 < y x = +∞) (y < 0 x = -∞)) ((0 < x y = +∞) (x < 0 y = -∞))), +∞, if((((0 < y x = -∞) (y < 0 x = +∞)) ((0 < x y = -∞) (x < 0 y = +∞))), -∞, (x · y)))))
 
Theorempnfxr 8462 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)
+∞ *
 
Theorempnfex 8463 Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
+∞ V
 
Theoremmnfxr 8464 Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
-∞ *
 
Theoremltxr 8465 The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)
((A * B *) → (A < B ↔ ((((A B ℝ) A < B) (A = -∞ B = +∞)) ((A B = +∞) (A = -∞ B ℝ)))))
 
Theoremelxr 8466 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
(A * ↔ (A A = +∞ A = -∞))
 
Theorempnfnemnf 8467 Plus and minus infinity are different elements of *. (Contributed by NM, 14-Oct-2005.)
+∞ ≠ -∞
 
Theoremmnfnepnf 8468 Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-∞ ≠ +∞
 
Theoremxrnemnf 8469 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
((A * A ≠ -∞) ↔ (A A = +∞))
 
Theoremxrnepnf 8470 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
((A * A ≠ +∞) ↔ (A A = -∞))
 
Theoremxrltnr 8471 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
(A * → ¬ A < A)
 
Theoremltpnf 8472 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
(A ℝ → A < +∞)
 
Theorem0ltpnf 8473 Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 < +∞
 
Theoremmnflt 8474 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
(A ℝ → -∞ < A)
 
Theoremmnflt0 8475 Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-∞ < 0
 
Theoremmnfltpnf 8476 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
-∞ < +∞
 
Theoremmnfltxr 8477 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)
((A A = +∞) → -∞ < A)
 
Theorempnfnlt 8478 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
(A * → ¬ +∞ < A)
 
Theoremnltmnf 8479 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
(A * → ¬ A < -∞)
 
Theorempnfge 8480 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
(A *A ≤ +∞)
 
Theorem0lepnf 8481 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≤ +∞
 
Theoremnn0pnfge0 8482 If a number is a nonnegative integer or positive infinity, it is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
((𝑁 0 𝑁 = +∞) → 0 ≤ 𝑁)
 
Theoremmnfle 8483 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
(A * → -∞ ≤ A)
 
Theoremxrltnsym 8484 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
((A * B *) → (A < B → ¬ B < A))
 
Theoremxrltnsym2 8485 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
((A * B *) → ¬ (A < B B < A))
 
Theoremxrlttr 8486 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
((A * B * 𝐶 *) → ((A < B B < 𝐶) → A < 𝐶))
 
Theoremxrltso 8487 'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
< Or ℝ*
 
Theoremxrlttri3 8488 Extended real version of lttri3 6895. (Contributed by NM, 9-Feb-2006.)
((A * B *) → (A = B ↔ (¬ A < B ¬ B < A)))
 
Theoremxrltle 8489 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)
((A * B *) → (A < BAB))
 
Theoremxrleid 8490 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
(A *AA)
 
Theoremxrletri3 8491 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
((A * B *) → (A = B ↔ (AB BA)))
 
Theoremxrlelttr 8492 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((A * B * 𝐶 *) → ((AB B < 𝐶) → A < 𝐶))
 
Theoremxrltletr 8493 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((A * B * 𝐶 *) → ((A < B B𝐶) → A < 𝐶))
 
Theoremxrletr 8494 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)
((A * B * 𝐶 *) → ((AB B𝐶) → A𝐶))
 
Theoremxrlttrd 8495 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(φA *)    &   (φB *)    &   (φ𝐶 *)    &   (φA < B)    &   (φB < 𝐶)       (φA < 𝐶)
 
Theoremxrlelttrd 8496 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(φA *)    &   (φB *)    &   (φ𝐶 *)    &   (φAB)    &   (φB < 𝐶)       (φA < 𝐶)
 
Theoremxrltletrd 8497 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(φA *)    &   (φB *)    &   (φ𝐶 *)    &   (φA < B)    &   (φB𝐶)       (φA < 𝐶)
 
Theoremxrletrd 8498 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(φA *)    &   (φB *)    &   (φ𝐶 *)    &   (φAB)    &   (φB𝐶)       (φA𝐶)
 
Theoremxrltne 8499 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
((A * B * A < B) → BA)
 
Theoremnltpnft 8500 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
(A * → (A = +∞ ↔ ¬ A < +∞))
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