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Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlesub0 7901 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 ≤ 𝐴𝐵 ≤ (𝐵𝐴)) ↔ 𝐴 = 0))
 
Theoremmullt0 7902 The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
(((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (𝐴 · 𝐵))
 
Theorem0le1 7903 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
0 ≤ 1
 
Theoremleidi 7904 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
𝐴 ∈ ℝ       𝐴𝐴
 
Theoremgt0ne0i 7905 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℝ       (0 < 𝐴𝐴 ≠ 0)
 
Theoremgt0ne0ii 7906 Positive implies nonzero. (Contributed by NM, 15-May-1999.)
𝐴 ∈ ℝ    &   0 < 𝐴       𝐴 ≠ 0
 
Theoremaddgt0i 7907 Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 + 𝐵))
 
Theoremaddge0i 7908 Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → 0 ≤ (𝐴 + 𝐵))
 
Theoremaddgegt0i 7909 Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 + 𝐵))
 
Theoremaddgt0ii 7910 Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   0 < 𝐴    &   0 < 𝐵       0 < (𝐴 + 𝐵)
 
Theoremadd20i 7911 Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
 
Theoremltnegi 7912 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵 ↔ -𝐵 < -𝐴)
 
Theoremlenegi 7913 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴𝐵 ↔ -𝐵 ≤ -𝐴)
 
Theoremltnegcon2i 7914 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < -𝐵𝐵 < -𝐴)
 
Theoremlesub0i 7915 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴𝐵 ≤ (𝐵𝐴)) ↔ 𝐴 = 0)
 
Theoremltaddposi 7916 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (0 < 𝐴𝐵 < (𝐵 + 𝐴))
 
Theoremposdifi 7917 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴))
 
Theoremltnegcon1i 7918 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴)
 
Theoremlenegcon1i 7919 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (-𝐴𝐵 ↔ -𝐵𝐴)
 
Theoremsubge0i 7920 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (0 ≤ (𝐴𝐵) ↔ 𝐵𝐴)
 
Theoremltadd1i 7921 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶))
 
Theoremleadd1i 7922 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (𝐴𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))
 
Theoremleadd2i 7923 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (𝐴𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))
 
Theoremltsubaddi 7924 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵) < 𝐶𝐴 < (𝐶 + 𝐵))
 
Theoremlesubaddi 7925 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐶 + 𝐵))
 
Theoremltsubadd2i 7926 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵) < 𝐶𝐴 < (𝐵 + 𝐶))
 
Theoremlesubadd2i 7927 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐵 + 𝐶))
 
Theoremltaddsubi 7928 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴 + 𝐵) < 𝐶𝐴 < (𝐶𝐵))
 
Theoremlt2addi 7929 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   𝐷 ∈ ℝ       ((𝐴 < 𝐶𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷))
 
Theoremle2addi 7930 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   𝐷 ∈ ℝ       ((𝐴𝐶𝐵𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))
 
Theoremgt0ne0d 7931 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑 → 0 < 𝐴)       (𝜑𝐴 ≠ 0)
 
Theoremlt0ne0d 7932 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 < 0)       (𝜑𝐴 ≠ 0)
 
Theoremleidd 7933 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴𝐴)
 
Theoremlt0neg1d 7934 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 < 0 ↔ 0 < -𝐴))
 
Theoremlt0neg2d 7935 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (0 < 𝐴 ↔ -𝐴 < 0))
 
Theoremle0neg1d 7936 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
 
Theoremle0neg2d 7937 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0))
 
Theoremaddgegt0d 7938 Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 < 𝐵)       (𝜑 → 0 < (𝐴 + 𝐵))
 
Theoremaddgt0d 7939 Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 0 < 𝐵)       (𝜑 → 0 < (𝐴 + 𝐵))
 
Theoremaddge0d 7940 Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → 0 ≤ (𝐴 + 𝐵))
 
Theoremltnegd 7941 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴))
 
Theoremlenegd 7942 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ -𝐵 ≤ -𝐴))
 
Theoremltnegcon1d 7943 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → -𝐴 < 𝐵)       (𝜑 → -𝐵 < 𝐴)
 
Theoremltnegcon2d 7944 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < -𝐵)       (𝜑𝐵 < -𝐴)
 
Theoremlenegcon1d 7945 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → -𝐴𝐵)       (𝜑 → -𝐵𝐴)
 
Theoremlenegcon2d 7946 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 ≤ -𝐵)       (𝜑𝐵 ≤ -𝐴)
 
Theoremltaddposd 7947 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 < 𝐴𝐵 < (𝐵 + 𝐴)))
 
Theoremltaddpos2d 7948 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 < 𝐴𝐵 < (𝐴 + 𝐵)))
 
Theoremltsubposd 7949 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 < 𝐴 ↔ (𝐵𝐴) < 𝐵))
 
Theoremposdifd 7950 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴)))
 
Theoremaddge01d 7951 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ 𝐵𝐴 ≤ (𝐴 + 𝐵)))
 
Theoremaddge02d 7952 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ 𝐵𝐴 ≤ (𝐵 + 𝐴)))
 
Theoremsubge0d 7953 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ (𝐴𝐵) ↔ 𝐵𝐴))
 
Theoremsuble0d 7954 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ((𝐴𝐵) ≤ 0 ↔ 𝐴𝐵))
 
Theoremsubge02d 7955 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ 𝐵 ↔ (𝐴𝐵) ≤ 𝐴))
 
Theoremltadd1d 7956 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶)))
 
Theoremleadd1d 7957 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)))
 
Theoremleadd2d 7958 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)))
 
Theoremltsubaddd 7959 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴𝐵) < 𝐶𝐴 < (𝐶 + 𝐵)))
 
Theoremlesubaddd 7960 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐶 + 𝐵)))
 
Theoremltsubadd2d 7961 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴𝐵) < 𝐶𝐴 < (𝐵 + 𝐶)))
 
Theoremlesubadd2d 7962 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐵 + 𝐶)))
 
Theoremltaddsubd 7963 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴 + 𝐵) < 𝐶𝐴 < (𝐶𝐵)))
 
Theoremltaddsub2d 7964 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴 + 𝐵) < 𝐶𝐵 < (𝐶𝐴)))
 
Theoremleaddsub2d 7965 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴 + 𝐵) ≤ 𝐶𝐵 ≤ (𝐶𝐴)))
 
Theoremsubled 7966 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝐴𝐵) ≤ 𝐶)       (𝜑 → (𝐴𝐶) ≤ 𝐵)
 
Theoremlesubd 7967 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ≤ (𝐵𝐶))       (𝜑𝐶 ≤ (𝐵𝐴))
 
Theoremltsub23d 7968 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝐴𝐵) < 𝐶)       (𝜑 → (𝐴𝐶) < 𝐵)
 
Theoremltsub13d 7969 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < (𝐵𝐶))       (𝜑𝐶 < (𝐵𝐴))
 
Theoremlesub1d 7970 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ (𝐴𝐶) ≤ (𝐵𝐶)))
 
Theoremlesub2d 7971 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ (𝐶𝐵) ≤ (𝐶𝐴)))
 
Theoremltsub1d 7972 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴𝐶) < (𝐵𝐶)))
 
Theoremltsub2d 7973 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐶𝐵) < (𝐶𝐴)))
 
Theoremltadd1dd 7974 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴 + 𝐶) < (𝐵 + 𝐶))
 
Theoremltsub1dd 7975 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴𝐶) < (𝐵𝐶))
 
Theoremltsub2dd 7976 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐶𝐵) < (𝐶𝐴))
 
Theoremleadd1dd 7977 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))
 
Theoremleadd2dd 7978 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))
 
Theoremlesub1dd 7979 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ≤ (𝐵𝐶))
 
Theoremlesub2dd 7980 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐶𝐵) ≤ (𝐶𝐴))
 
Theoremle2addd 7981 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))
 
Theoremle2subd 7982 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → (𝐴𝐷) ≤ (𝐶𝐵))
 
Theoremltleaddd 7983 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))
 
Theoremleltaddd 7984 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵 < 𝐷)       (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))
 
Theoremlt2addd 7985 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐵 < 𝐷)       (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))
 
Theoremlt2subd 7986 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐵 < 𝐷)       (𝜑 → (𝐴𝐷) < (𝐶𝐵))
 
Theorempossumd 7987 Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 < (𝐴 + 𝐵) ↔ -𝐵 < 𝐴))
 
Theoremsublt0d 7988 When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ((𝐴𝐵) < 0 ↔ 𝐴 < 𝐵))
 
Theoremltaddsublt 7989 Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐶 ↔ ((𝐴 + 𝐵) − 𝐶) < 𝐴))
 
Theorem1le1 7990 1 ≤ 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
1 ≤ 1
 
Theoremgt0add 7991 A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < (𝐴 + 𝐵)) → (0 < 𝐴 ∨ 0 < 𝐵))
 
3.3.5  Real Apartness
 
Syntaxcreap 7992 Class of real apartness relation.
class #
 
Definitiondf-reap 7993* Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although # is an apartness relation on the reals (see df-ap 8000 for more discussion of apartness relations), for our purposes it is just a stepping stone to defining # which is an apartness relation on complex numbers. On the reals, # and # agree (apreap 8005). (Contributed by Jim Kingdon, 26-Jan-2020.)
# = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦𝑦 < 𝑥))}
 
Theoremreapval 7994 Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8006 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
 
Theoremreapirr 7995 Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8023 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
(𝐴 ∈ ℝ → ¬ 𝐴 # 𝐴)
 
Theoremrecexre 7996* Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
 
Theoremreapti 7997 Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8040. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵))
 
Theoremrecexgt0 7998* Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))
 
3.3.6  Complex Apartness
 
Syntaxcap 7999 Class of complex apartness relation.
class #
 
Definitiondf-ap 8000* Define complex apartness. Definition 6.1 of [Geuvers], p. 17.

Two numbers are considered apart if it is possible to separate them. One common usage is that we can divide by a number if it is apart from zero (see for example recclap 8085 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8023), symmetry (apsym 8024), and cotransitivity (apcotr 8025). Apartness implies negated equality, as seen at apne 8041, and the converse would also follow if we assumed excluded middle.

In addition, apartness of complex numbers is tight, which means that two numbers which are not apart are equal (apti 8040).

(Contributed by Jim Kingdon, 26-Jan-2020.)

# = {⟨𝑥, 𝑦⟩ ∣ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 # 𝑡𝑠 # 𝑢))}
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