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Theorem List for Intuitionistic Logic Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlediv2d 8401 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 8402 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 8403 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 8404 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 8405 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φ → (AB) < A)

Theoremltaddrpd 8406 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φA < (A + B))

Theoremltaddrp2d 8407 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φA < (B + A))

Theoremltmulgt11d 8408 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φ → (1 < AB < (B · A)))

Theoremltmulgt12d 8409 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB +)       (φ → (1 < AB < (A · B)))

Theoremgt0divd 8410 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 8411 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 8412 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 8413 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 8414 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 8415 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 8416 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 8417 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 8418 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 8419 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 8420 '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 8421 '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 8422 '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 8423 '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 8424 '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 8425 '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 8426 '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 8427 '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 8428 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 8429 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 8430 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 8431 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 8432 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 8433 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 8434 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 8435 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 8436 Extend class notation to include the negative of an extended real.
class -𝑒A

class +𝑒

Syntaxcxmu 8438 Extend class notation to include multiplication of extended reals.
class ·e

Definitiondf-xneg 8439 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)
-𝑒A = if(A = +∞, -∞, if(A = -∞, +∞, -A))

Definitiondf-xadd 8440* 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 8441* 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 8442 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)
+∞ *

Theorempnfex 8443 Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
+∞ V

Theoremmnfxr 8444 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 8445 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 8446 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
(A * ↔ (A A = +∞ A = -∞))

Theorempnfnemnf 8447 Plus and minus infinity are different elements of *. (Contributed by NM, 14-Oct-2005.)
+∞ ≠ -∞

Theoremmnfnepnf 8448 Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-∞ ≠ +∞

Theoremxrnemnf 8449 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
((A * A ≠ -∞) ↔ (A A = +∞))

Theoremxrnepnf 8450 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
((A * A ≠ +∞) ↔ (A A = -∞))

Theoremxrltnr 8451 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
(A * → ¬ A < A)

Theoremltpnf 8452 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
(A ℝ → A < +∞)

Theorem0ltpnf 8453 Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 < +∞

Theoremmnflt 8454 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
(A ℝ → -∞ < A)

Theoremmnflt0 8455 Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-∞ < 0

Theoremmnfltpnf 8456 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
-∞ < +∞

Theoremmnfltxr 8457 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 8458 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
(A * → ¬ +∞ < A)

Theoremnltmnf 8459 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
(A * → ¬ A < -∞)

Theorempnfge 8460 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
(A *A ≤ +∞)

Theorem0lepnf 8461 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≤ +∞

Theoremnn0pnfge0 8462 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 8463 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
(A * → -∞ ≤ A)

Theoremxrltnsym 8464 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
((A * B *) → (A < B → ¬ B < A))

Theoremxrltnsym2 8465 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
((A * B *) → ¬ (A < B B < A))

Theoremxrlttr 8466 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
((A * B * 𝐶 *) → ((A < B B < 𝐶) → A < 𝐶))

Theoremxrltso 8467 'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
< Or ℝ*

Theoremxrlttri3 8468 Extended real version of lttri3 6875. (Contributed by NM, 9-Feb-2006.)
((A * B *) → (A = B ↔ (¬ A < B ¬ B < A)))

Theoremxrltle 8469 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)
((A * B *) → (A < BAB))

Theoremxrleid 8470 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
(A *AA)

Theoremxrletri3 8471 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
((A * B *) → (A = B ↔ (AB BA)))

Theoremxrlelttr 8472 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((A * B * 𝐶 *) → ((AB B < 𝐶) → A < 𝐶))

Theoremxrltletr 8473 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((A * B * 𝐶 *) → ((A < B B𝐶) → A < 𝐶))

Theoremxrletr 8474 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)
((A * B * 𝐶 *) → ((AB B𝐶) → A𝐶))

Theoremxrlttrd 8475 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(φA *)    &   (φB *)    &   (φ𝐶 *)    &   (φA < B)    &   (φB < 𝐶)       (φA < 𝐶)

Theoremxrlelttrd 8476 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(φA *)    &   (φB *)    &   (φ𝐶 *)    &   (φAB)    &   (φB < 𝐶)       (φA < 𝐶)

Theoremxrltletrd 8477 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(φA *)    &   (φB *)    &   (φ𝐶 *)    &   (φA < B)    &   (φB𝐶)       (φA < 𝐶)

Theoremxrletrd 8478 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(φA *)    &   (φB *)    &   (φ𝐶 *)    &   (φAB)    &   (φB𝐶)       (φA𝐶)

Theoremxrltne 8479 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
((A * B * A < B) → BA)

Theoremnltpnft 8480 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
(A * → (A = +∞ ↔ ¬ A < +∞))

Theoremngtmnft 8481 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)
(A * → (A = -∞ ↔ ¬ -∞ < A))

Theoremxrrebnd 8482 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)
(A * → (A ℝ ↔ (-∞ < A A < +∞)))

Theoremxrre 8483 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)
(((A * B ℝ) (-∞ < A AB)) → A ℝ)

Theoremxrre2 8484 An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)
(((A * B * 𝐶 *) (A < B B < 𝐶)) → B ℝ)

Theoremxrre3 8485 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
(((A * B ℝ) (BA A < +∞)) → A ℝ)

Theoremge0gtmnf 8486 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
((A * 0 ≤ A) → -∞ < A)

Theoremge0nemnf 8487 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
((A * 0 ≤ A) → A ≠ -∞)

Theoremxrrege0 8488 A nonnegative extended real that is less than a real bound is real. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((A * B ℝ) (0 ≤ A AB)) → A ℝ)

Theoremz2ge 8489* There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.)
((𝑀 𝑁 ℤ) → 𝑘 ℤ (𝑀𝑘 𝑁𝑘))

Theoremxnegeq 8490 Equality of two extended numbers with -𝑒 in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
(A = B → -𝑒A = -𝑒B)

Theoremxnegpnf 8491 Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
-𝑒+∞ = -∞

Theoremxnegmnf 8492 Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
-𝑒-∞ = +∞

Theoremrexneg 8493 Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
(A ℝ → -𝑒A = -A)

Theoremxneg0 8494 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
-𝑒0 = 0

Theoremxnegcl 8495 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
(A * → -𝑒A *)

Theoremxnegneg 8496 Extended real version of negneg 7037. (Contributed by Mario Carneiro, 20-Aug-2015.)
(A * → -𝑒-𝑒A = A)

Theoremxneg11 8497 Extended real version of neg11 7038. (Contributed by Mario Carneiro, 20-Aug-2015.)
((A * B *) → (-𝑒A = -𝑒BA = B))

Theoremxltnegi 8498 Forward direction of xltneg 8499. (Contributed by Mario Carneiro, 20-Aug-2015.)
((A * B * A < B) → -𝑒B < -𝑒A)

Theoremxltneg 8499 Extended real version of ltneg 7232. (Contributed by Mario Carneiro, 20-Aug-2015.)
((A * B *) → (A < B ↔ -𝑒B < -𝑒A))

Theoremxleneg 8500 Extended real version of leneg 7235. (Contributed by Mario Carneiro, 20-Aug-2015.)
((A * B *) → (AB ↔ -𝑒B ≤ -𝑒A))

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