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Theorem List for Metamath Proof Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubsub3d 10301 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 − (𝐵𝐶)) = ((𝐴 + 𝐶) − 𝐵))

Theoremsubsub4d 10302 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) − 𝐶) = (𝐴 − (𝐵 + 𝐶)))

Theoremsub32d 10303 Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) − 𝐶) = ((𝐴𝐶) − 𝐵))

Theoremnnncand 10304 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 − (𝐵𝐶)) − 𝐶) = (𝐴𝐵))

Theoremnnncan1d 10305 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) − (𝐴𝐶)) = (𝐶𝐵))

Theoremnnncan2d 10306 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐶) − (𝐵𝐶)) = (𝐴𝐵))

Theoremnpncan3d 10307 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) + (𝐶𝐴)) = (𝐶𝐵))

Theorempnpcand 10308 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵𝐶))

Theorempnpcan2d 10309 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐶) − (𝐵 + 𝐶)) = (𝐴𝐵))

Theorempnncand 10310 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − (𝐴𝐶)) = (𝐵 + 𝐶))

Theoremppncand 10311 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) + (𝐶𝐵)) = (𝐴 + 𝐶))

Theoremsubcand 10312 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) = (𝐴𝐶))       (𝜑𝐵 = 𝐶)

Theoremsubcan2d 10313 Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → (𝐴𝐶) = (𝐵𝐶))       (𝜑𝐴 = 𝐵)

Theoremsubcanad 10314 Cancellation law for subtraction. Deduction form of subcan 10215. Generalization of subcand 10312. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) = (𝐴𝐶) ↔ 𝐵 = 𝐶))

Theoremsubneintrd 10315 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 10312. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐴𝐵) ≠ (𝐴𝐶))

Theoremsubcan2ad 10316 Cancellation law for subtraction. Deduction form of subcan2 10185. Generalization of subcan2d 10313. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐶) = (𝐵𝐶) ↔ 𝐴 = 𝐵))

Theoremsubneintr2d 10317 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 10313. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ≠ (𝐵𝐶))

Theoremaddsub4d 10318 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − (𝐶 + 𝐷)) = ((𝐴𝐶) + (𝐵𝐷)))

Theoremsubadd4d 10319 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴𝐵) − (𝐶𝐷)) = ((𝐴 + 𝐷) − (𝐵 + 𝐶)))

Theoremsub4d 10320 Rearrangement of 4 terms in a subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴𝐵) − (𝐶𝐷)) = ((𝐴𝐶) − (𝐵𝐷)))

Theorem2addsubd 10321 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → (((𝐴 + 𝐵) + 𝐶) − 𝐷) = (((𝐴 + 𝐶) − 𝐷) + 𝐵))

Theoremaddsubeq4d 10322 Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐶𝐴) = (𝐵𝐷)))

Theoremmvlraddd 10323 Move LHS right addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) = 𝐶)       (𝜑𝐴 = (𝐶𝐵))

Theoremmvrraddd 10324 Move RHS right addition to LHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 = (𝐵 + 𝐶))       (𝜑 → (𝐴𝐶) = 𝐵)

Theoremsubaddeqd 10325 Transfer two terms of a subtraction to an addition in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷))       (𝜑 → (𝐴𝐷) = (𝐶𝐵))

Theoremaddlsub 10326 Left-subtraction: Subtraction of the left summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) = 𝐶𝐴 = (𝐶𝐵)))

Theoremaddrsub 10327 Right-subtraction: Subtraction of the right summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) = 𝐶𝐵 = (𝐶𝐴)))

Theoremsubexsub 10328 A subtraction law: Exchanging the subtrahend and the result of the subtracion. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 = (𝐶𝐵) ↔ 𝐵 = (𝐶𝐴)))

Theoremaddid0 10329 If adding a number to a another number yields the other number, the added number must be 0. This shows that 0 is the unique (right) identity of the complex numbers. (Contributed by AV, 17-Jan-2021.)
((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ) → ((𝑋 + 𝑌) = 𝑋𝑌 = 0))

Theoremaddn0nid 10330 Adding a nonzero number to a complex number does not yield the complex number. (Contributed by AV, 17-Jan-2021.)
((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ ∧ 𝑌 ≠ 0) → (𝑋 + 𝑌) ≠ 𝑋)

Theorempnpncand 10331 Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + (𝐵𝐶)) + (𝐶𝐵)) = 𝐴)

Theoremsubeqrev 10332 Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴𝐵) = (𝐶𝐷) ↔ (𝐵𝐴) = (𝐷𝐶)))

Theorempncan1 10333 Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(𝐴 ∈ ℂ → ((𝐴 + 1) − 1) = 𝐴)

Theoremnpcan1 10334 Cancellation law for subtraction and addition with 1. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
(𝐴 ∈ ℂ → ((𝐴 − 1) + 1) = 𝐴)

Theoremsubeq0bd 10335 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 10281. Converse of subeq0d 10279. Contrapositive of subne0ad 10282. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐵) = 0)

Theoremrenegcld 10336 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → -𝐴 ∈ ℝ)

Theoremresubcld 10337 Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝐵) ∈ ℝ)

Theoremnegn0 10338* The image under negation of a nonempty set of reals is nonempty. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → {𝑧 ∈ ℝ ∣ -𝑧𝐴} ≠ ∅)

Theoremnegf1o 10339* Negation is an isomorphism of a subset of the real numbers to the negated elements of the subset. (Contributed by AV, 9-Aug-2020.)
𝐹 = (𝑥𝐴 ↦ -𝑥)       (𝐴 ⊆ ℝ → 𝐹:𝐴1-1-onto→{𝑛 ∈ ℝ ∣ -𝑛𝐴})

5.3.3  Multiplication

Theoremkcnktkm1cn 10340 k times k minus 1 is a complex number if k is a complex number. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
(𝐾 ∈ ℂ → (𝐾 · (𝐾 − 1)) ∈ ℂ)

Theoremmuladd 10341 Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) + ((𝐴 · 𝐷) + (𝐶 · 𝐵))))

Theoremsubdi 10342 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶)))

Theoremsubdir 10343 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶)))

Theoremine0 10344 The imaginary unit i is not zero. (Contributed by NM, 6-May-1999.)
i ≠ 0

Theoremmulneg1 10345 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = -(𝐴 · 𝐵))

Theoremmulneg2 10346 The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · -𝐵) = -(𝐴 · 𝐵))

Theoremmulneg12 10347 Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = (𝐴 · -𝐵))

Theoremmul2neg 10348 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 30-Jul-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · -𝐵) = (𝐴 · 𝐵))

Theoremsubmul2 10349 Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 · 𝐶)) = (𝐴 + (𝐵 · -𝐶)))

Theoremmulm1 10350 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)
(𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴)

Theoremaddneg1mul 10351 Addition with product with minus one is a subtraction. (Contributed by AV, 18-Oct-2021.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (-1 · 𝐵)) = (𝐴𝐵))

Theoremmulsub 10352 Product of two differences. (Contributed by NM, 14-Jan-2006.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴𝐵) · (𝐶𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) − ((𝐴 · 𝐷) + (𝐶 · 𝐵))))

Theoremmulsub2 10353 Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴𝐵) · (𝐶𝐷)) = ((𝐵𝐴) · (𝐷𝐶)))

Theoremmulm1i 10354 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)
𝐴 ∈ ℂ       (-1 · 𝐴) = -𝐴

Theoremmulneg1i 10355 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (-𝐴 · 𝐵) = -(𝐴 · 𝐵)

Theoremmulneg2i 10356 Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 · -𝐵) = -(𝐴 · 𝐵)

Theoremmul2negi 10357 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (-𝐴 · -𝐵) = (𝐴 · 𝐵)

Theoremsubdii 10358 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐴 · (𝐵𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))

Theoremsubdiri 10359 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶))

Theoremmuladdi 10360 Product of two sums. (Contributed by NM, 17-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) + ((𝐴 · 𝐷) + (𝐶 · 𝐵)))

Theoremmulm1d 10361 Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (-1 · 𝐴) = -𝐴)

Theoremmulneg1d 10362 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 · 𝐵) = -(𝐴 · 𝐵))

Theoremmulneg2d 10363 Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · -𝐵) = -(𝐴 · 𝐵))

Theoremmul2negd 10364 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 · -𝐵) = (𝐴 · 𝐵))

Theoremsubdid 10365 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 · (𝐵𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶)))

Theoremsubdird 10366 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶)))

Theoremsubdir2d 10367 Distribution of multiplication over subtraction. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐶 · (𝐴𝐵)) = ((𝐶 · 𝐴) − (𝐶 · 𝐵)))

Theoremmuladdd 10368 Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) + ((𝐴 · 𝐷) + (𝐶 · 𝐵))))

Theoremmulsubd 10369 Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴𝐵) · (𝐶𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) − ((𝐴 · 𝐷) + (𝐶 · 𝐵))))

Theoremmuls1d 10370 Multiplication by one minus a number. (Contributed by Scott Fenton, 23-Dec-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · (𝐵 − 1)) = ((𝐴 · 𝐵) − 𝐴))

Theoremmulsubfacd 10371 Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) − 𝐵) = ((𝐴 − 1) · 𝐵))

5.3.4  Ordering on reals (cont.)

Theoremgt0ne0 10372 Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0)

Theoremlt0ne0 10373 A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ≠ 0)

Theoremltadd1 10374 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶)))

Theoremleadd1 10375 Addition to both sides of 'less than or equal to'. (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)))

Theoremleadd2 10376 Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)))

Theoremltsubadd 10377 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) < 𝐶𝐴 < (𝐶 + 𝐵)))

Theoremltsubadd2 10378 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) < 𝐶𝐴 < (𝐵 + 𝐶)))

Theoremlesubadd 10379 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐶 + 𝐵)))

Theoremlesubadd2 10380 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐵 + 𝐶)))

Theoremltaddsub 10381 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶𝐴 < (𝐶𝐵)))

Theoremltaddsub2 10382 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶𝐵 < (𝐶𝐴)))

Theoremleaddsub 10383 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶𝐴 ≤ (𝐶𝐵)))

Theoremleaddsub2 10384 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶𝐵 ≤ (𝐶𝐴)))

Theoremsuble 10385 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) ≤ 𝐶 ↔ (𝐴𝐶) ≤ 𝐵))

Theoremlesub 10386 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ (𝐵𝐶) ↔ 𝐶 ≤ (𝐵𝐴)))

Theoremltsub23 10387 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) < 𝐶 ↔ (𝐴𝐶) < 𝐵))

Theoremltsub13 10388 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < (𝐵𝐶) ↔ 𝐶 < (𝐵𝐴)))

Theoremle2add 10389 Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴𝐶𝐵𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)))

Theoremltleadd 10390 Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶𝐵𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷)))

Theoremleltadd 10391 Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴𝐶𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷)))

Theoremlt2add 10392 Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷)))

Theoremaddgt0 10393 The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 + 𝐵))

Theoremaddgegt0 10394 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 + 𝐵))

Theoremaddgtge0 10395 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 ≤ 𝐵)) → 0 < (𝐴 + 𝐵))

Theoremaddge0 10396 The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 + 𝐵))

Theoremltaddpos 10397 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴𝐵 < (𝐵 + 𝐴)))

Theoremltaddpos2 10398 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴𝐵 < (𝐴 + 𝐵)))

Theoremltsubpos 10399 Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ (𝐵𝐴) < 𝐵))

Theoremposdif 10400 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴)))

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