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Theorem List for Metamath Proof Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremle2addi 11201 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 11202 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑 → 0 < 𝐴)       (𝜑𝐴 ≠ 0)

Theoremlt0ne0d 11203 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 < 0)       (𝜑𝐴 ≠ 0)

Theoremleidd 11204 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴𝐴)

Theoremmsqgt0d 11205 A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → 0 < (𝐴 · 𝐴))

Theoremmsqge0d 11206 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → 0 ≤ (𝐴 · 𝐴))

Theoremlt0neg1d 11207 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 11208 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (0 < 𝐴 ↔ -𝐴 < 0))

Theoremle0neg1d 11209 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))

Theoremle0neg2d 11210 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0))

Theoremaddgegt0d 11211 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))

Theoremaddgtge0d 11212 Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → 0 < (𝐴 + 𝐵))

Theoremaddgt0d 11213 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 11214 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))

Theoremmulge0d 11215 The product of two 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 11216 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 11217 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 11218 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → -𝐴 < 𝐵)       (𝜑 → -𝐵 < 𝐴)

Theoremltnegcon2d 11219 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < -𝐵)       (𝜑𝐵 < -𝐴)

Theoremlenegcon1d 11220 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 11221 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 11222 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 11223 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 11224 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 11225 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 11226 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 11227 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 11228 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ (𝐴𝐵) ↔ 𝐵𝐴))

Theoremsuble0d 11229 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ((𝐴𝐵) ≤ 0 ↔ 𝐴𝐵))

Theoremsubge02d 11230 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ 𝐵 ↔ (𝐴𝐵) ≤ 𝐴))

Theoremltadd1d 11231 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 11232 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 11233 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 11234 '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 11235 '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 11236 '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 11237 '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 11238 '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 11239 '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 11240 '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 11241 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 11242 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ≤ (𝐵𝐶))       (𝜑𝐶 ≤ (𝐵𝐴))

Theoremltsub23d 11243 '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 11244 '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 11245 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 11246 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 11247 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 11248 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 11249 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 11250 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 11251 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 11252 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 11253 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 11254 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 11255 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))

Theoremlesub3d 11256 The result of subtracting a number less than or equal to an intermediate number from a number greater than or equal to a third number increased by the intermediate number is greater than or equal to the third number. (Contributed by AV, 13-Aug-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑 → (𝑋 + 𝐶) ≤ 𝐴)    &   (𝜑𝐵𝑋)       (𝜑𝐶 ≤ (𝐴𝐵))

Theoremle2addd 11257 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 11258 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 11259 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 11260 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 11261 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 11262 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 11263 Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 < (𝐴 + 𝐵) ↔ -𝐵 < 𝐴))

Theoremsublt0d 11264 When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ((𝐴𝐵) < 0 ↔ 𝐴 < 𝐵))

Theoremltaddsublt 11265 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 11266 One is less than or equal to one. (Contributed by David A. Wheeler, 16-Jul-2016.)
1 ≤ 1

5.3.5  Reciprocals

Theoremixi 11267 i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(i · i) = -1

Theoremrecextlem1 11268 Lemma for recex 11270. (Contributed by Eric Schmidt, 23-May-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (i · 𝐵)) · (𝐴 − (i · 𝐵))) = ((𝐴 · 𝐴) + (𝐵 · 𝐵)))

Theoremrecextlem2 11269 Lemma for recex 11270. (Contributed by Eric Schmidt, 23-May-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + (i · 𝐵)) ≠ 0) → ((𝐴 · 𝐴) + (𝐵 · 𝐵)) ≠ 0)

Theoremrecex 11270* Existence of reciprocal of nonzero complex number. (Contributed by Eric Schmidt, 22-May-2007.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1)

Theoremmulcand 11271 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))

Theoremmulcan2d 11272 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))

Theoremmulcanad 11273 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcand 11271. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)    &   (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵))       (𝜑𝐴 = 𝐵)

Theoremmulcan2ad 11274 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcan2d 11272. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)    &   (𝜑 → (𝐴 · 𝐶) = (𝐵 · 𝐶))       (𝜑𝐴 = 𝐵)

Theoremmulcan 11275 Cancellation law for multiplication (full theorem form). Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))

Theoremmulcan2 11276 Cancellation law for multiplication. (Contributed by NM, 21-Jan-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))

Theoremmulcani 11277 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 ≠ 0       ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)

Theoremmul0or 11278 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0)))

Theoremmulne0b 11279 The product of two nonzero numbers is nonzero. (Contributed by NM, 1-Aug-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) ↔ (𝐴 · 𝐵) ≠ 0))

Theoremmulne0 11280 The product of two nonzero numbers is nonzero. (Contributed by NM, 30-Dec-2007.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 · 𝐵) ≠ 0)

Theoremmulne0i 11281 The product of two nonzero numbers is nonzero. (Contributed by NM, 15-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 ≠ 0    &   𝐵 ≠ 0       (𝐴 · 𝐵) ≠ 0

Theoremmuleqadd 11282 Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = (𝐴 + 𝐵) ↔ ((𝐴 − 1) · (𝐵 − 1)) = 1))

Theoremreceu 11283* Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)

Theoremmulnzcnopr 11284 Multiplication maps nonzero complex numbers to nonzero complex numbers. (Contributed by Steve Rodriguez, 23-Feb-2007.)
( · ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))):((ℂ ∖ {0}) × (ℂ ∖ {0}))⟶(ℂ ∖ {0})

Theoremmsq0i 11285 A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℂ       ((𝐴 · 𝐴) = 0 ↔ 𝐴 = 0)

Theoremmul0ori 11286 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 7-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0))

Theoremmsq0d 11287 A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐴) = 0 ↔ 𝐴 = 0))

Theoremmul0ord 11288 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0)))

Theoremmulne0bd 11289 The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) ↔ (𝐴 · 𝐵) ≠ 0))

Theoremmulne0d 11290 The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 · 𝐵) ≠ 0)

Theoremmulcan1g 11291 A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) = (𝐴 · 𝐶) ↔ (𝐴 = 0 ∨ 𝐵 = 𝐶)))

Theoremmulcan2g 11292 A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ (𝐴 = 𝐵𝐶 = 0)))

Theoremmulne0bad 11293 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 11290 and consequence of mulne0bd 11289. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 · 𝐵) ≠ 0)       (𝜑𝐴 ≠ 0)

Theoremmulne0bbd 11294 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 11290 and consequence of mulne0bd 11289. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 · 𝐵) ≠ 0)       (𝜑𝐵 ≠ 0)

5.3.6  Division

Syntaxcdiv 11295 Extend class notation to include division.
class /

Definitiondf-div 11296* Define division. Theorem divmuli 11392 relates it to multiplication, and divcli 11380 and redivcli 11405 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divval 11298 instead. (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
/ = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))

Theorem1div0 11297 You can't divide by zero, because division explicitly excludes zero from the domain of the function. Thus, by the definition of function value, it evaluates to the empty set. (This theorem is for information only and normally is not referenced by other proofs. To be meaningful, it assumes that is not a complex number, which depends on the particular complex number construction that is used.) (Contributed by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
(1 / 0) = ∅

Theoremdivval 11298* Value of division: if 𝐴 and 𝐵 are complex numbers with 𝐵 nonzero, then (𝐴 / 𝐵) is the (unique) complex number such that (𝐵 · 𝑥) = 𝐴. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))

Theoremdivmul 11299 Relationship between division and multiplication. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 17-Feb-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ (𝐶 · 𝐵) = 𝐴))

Theoremdivmul2 11300 Relationship between division and multiplication. (Contributed by NM, 7-Feb-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐶 · 𝐵)))

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