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

Theoremnnncan 10301 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵𝐶)) − 𝐶) = (𝐴𝐵))

Theoremnnncan1 10302 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) − (𝐴𝐶)) = (𝐶𝐵))

Theoremnnncan2 10303 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐶) − (𝐵𝐶)) = (𝐴𝐵))

Theoremnpncan3 10304 Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + (𝐶𝐴)) = (𝐶𝐵))

Theorempnpcan 10305 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵𝐶))

Theorempnpcan2 10306 Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶) − (𝐵 + 𝐶)) = (𝐴𝐵))

Theorempnncan 10307 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴𝐶)) = (𝐵 + 𝐶))

Theoremppncan 10308 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + (𝐶𝐵)) = (𝐴 + 𝐶))

Theoremaddsub4 10309 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) − (𝐶 + 𝐷)) = ((𝐴𝐶) + (𝐵𝐷)))

Theoremsubadd4 10310 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴𝐵) − (𝐶𝐷)) = ((𝐴 + 𝐷) − (𝐵 + 𝐶)))

Theoremsub4 10311 Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴𝐵) − (𝐶𝐷)) = ((𝐴𝐶) − (𝐵𝐷)))

Theoremneg0 10312 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
-0 = 0

Theoremnegid 10313 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
(𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0)

Theoremnegsub 10314 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴𝐵))

Theoremsubneg 10315 Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = (𝐴 + 𝐵))

Theoremnegneg 10316 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 12-Jan-2002.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → --𝐴 = 𝐴)

Theoremneg11 10317 Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = -𝐵𝐴 = 𝐵))

Theoremnegcon1 10318 Negative contraposition law. (Contributed by NM, 9-May-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))

Theoremnegcon2 10319 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = -𝐵𝐵 = -𝐴))

Theoremnegeq0 10320 A number is zero iff its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴 = 0 ↔ -𝐴 = 0))

Theoremsubcan 10321 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = (𝐴𝐶) ↔ 𝐵 = 𝐶))

Theoremnegsubdi 10322 Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴𝐵) = (-𝐴 + 𝐵))

Theoremnegdi 10323 Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))

Theoremnegdi2 10324 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴𝐵))

Theoremnegsubdi2 10325 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴𝐵) = (𝐵𝐴))

Theoremneg2sub 10326 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 − -𝐵) = (𝐵𝐴))

Theoremrenegcli 10327 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 10329 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝐴 ∈ ℝ       -𝐴 ∈ ℝ

Theoremresubcli 10328 Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴𝐵) ∈ ℝ

Theoremrenegcl 10329 Closure law for negative of reals. The weak deduction theorem dedth 4130 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 10327, to an antecedent. (Contributed by NM, 20-Jan-1997.) (Proof modification is discouraged.)
(𝐴 ∈ ℝ → -𝐴 ∈ ℝ)

Theoremresubcl 10330 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵) ∈ ℝ)

Theoremnegreb 10331 The negative of a real is real. (Contributed by NM, 11-Aug-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ))

Theorempeano2cnm 10332 "Reverse" second Peano postulate analogue for complex numbers: A complex number minus 1 is a complex number. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
(𝑁 ∈ ℂ → (𝑁 − 1) ∈ ℂ)

Theorempeano2rem 10333 "Reverse" second Peano postulate analogue for reals. (Contributed by NM, 6-Feb-2007.)
(𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)

Theoremnegcli 10334 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       -𝐴 ∈ ℂ

Theoremnegidi 10335 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       (𝐴 + -𝐴) = 0

Theoremnegnegi 10336 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 8-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝐴 ∈ ℂ       --𝐴 = 𝐴

Theoremsubidi 10337 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       (𝐴𝐴) = 0

Theoremsubid1i 10338 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 − 0) = 𝐴

Theoremnegne0bi 10339 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
𝐴 ∈ ℂ       (𝐴 ≠ 0 ↔ -𝐴 ≠ 0)

Theoremnegrebi 10340 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℂ       (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)

Theoremnegne0i 10341 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       -𝐴 ≠ 0

Theoremsubcli 10342 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴𝐵) ∈ ℂ

Theorempncan3i 10343 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + (𝐵𝐴)) = 𝐵

Theoremnegsubi 10344 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + -𝐵) = (𝐴𝐵)

Theoremsubnegi 10345 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 − -𝐵) = (𝐴 + 𝐵)

Theoremsubeq0i 10346 If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴𝐵) = 0 ↔ 𝐴 = 𝐵)

Theoremneg11i 10347 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (-𝐴 = -𝐵𝐴 = 𝐵)

Theoremnegcon1i 10348 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)

Theoremnegcon2i 10349 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 = -𝐵𝐵 = -𝐴)

Theoremnegdii 10350 Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by OpenAI, 25-Mar-2011.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       -(𝐴 + 𝐵) = (-𝐴 + -𝐵)

Theoremnegsubdii 10351 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       -(𝐴𝐵) = (-𝐴 + 𝐵)

Theoremnegsubdi2i 10352 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       -(𝐴𝐵) = (𝐵𝐴)

Theoremsubaddi 10353 Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴)

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴)

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   (𝐵 + 𝐶) = 𝐴       (𝐴𝐵) = 𝐶

Theoremsubsub23i 10356 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = 𝐶 ↔ (𝐴𝐶) = 𝐵)

Theoremaddsubassi 10357 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵𝐶))

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) − 𝐶) = ((𝐴𝐶) + 𝐵)

Theoremsubcani 10359 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = (𝐴𝐶) ↔ 𝐵 = 𝐶)

Theoremsubcan2i 10360 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐶) = (𝐵𝐶) ↔ 𝐴 = 𝐵)

Theorempnncani 10361 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) − (𝐴𝐶)) = (𝐵 + 𝐶)

Theoremaddsub4i 10362 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       ((𝐴 + 𝐵) − (𝐶 + 𝐷)) = ((𝐴𝐶) + (𝐵𝐷))

Theorem0reALT 10363 Alternate proof of 0re 10025. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℝ

Theoremnegcld 10364 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → -𝐴 ∈ ℂ)

Theoremsubidd 10365 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴𝐴) = 0)

Theoremsubid1d 10366 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 − 0) = 𝐴)

Theoremnegidd 10367 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 + -𝐴) = 0)

Theoremnegnegd 10368 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → --𝐴 = 𝐴)

Theoremnegeq0d 10369 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 = 0 ↔ -𝐴 = 0))

Theoremnegne0bd 10370 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 ≠ 0 ↔ -𝐴 ≠ 0))

Theoremnegcon1d 10371 Contraposition law for unary minus. Deduction form of negcon1 10318. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))

Theoremnegcon1ad 10372 Contraposition law for unary minus. One-way deduction form of negcon1 10318. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → -𝐴 = 𝐵)       (𝜑 → -𝐵 = 𝐴)

Theoremneg11ad 10373 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 10317. Generalization of neg11d 10389. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 = -𝐵𝐴 = 𝐵))

Theoremnegned 10374 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 10389. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → -𝐴 ≠ -𝐵)

Theoremnegne0d 10375 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → -𝐴 ≠ 0)

Theoremnegrebd 10376 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → -𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℝ)

Theoremsubcld 10377 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴𝐵) ∈ ℂ)

Theorempncand 10378 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴)

Theorempncan2d 10379 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − 𝐴) = 𝐵)

Theorempncan3d 10380 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 + (𝐵𝐴)) = 𝐵)

Theoremnpcand 10381 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴𝐵) + 𝐵) = 𝐴)

Theoremnncand 10382 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 − (𝐴𝐵)) = 𝐵)

Theoremnegsubd 10383 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 + -𝐵) = (𝐴𝐵))

Theoremsubnegd 10384 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵))

Theoremsubeq0d 10385 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) = 0)       (𝜑𝐴 = 𝐵)

Theoremsubne0d 10386 Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴𝐵) ≠ 0)

Theoremsubeq0ad 10387 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 10292. Generalization of subeq0d 10385. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴𝐵) = 0 ↔ 𝐴 = 𝐵))

Theoremsubne0ad 10388 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 10386. Contrapositive of subeq0bd 10441. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) ≠ 0)       (𝜑𝐴𝐵)

Theoremneg11d 10389 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → -𝐴 = -𝐵)       (𝜑𝐴 = 𝐵)

Theoremnegdid 10390 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))

Theoremnegdi2d 10391 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴 + 𝐵) = (-𝐴𝐵))

Theoremnegsubdid 10392 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴𝐵) = (-𝐴 + 𝐵))

Theoremnegsubdi2d 10393 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴𝐵) = (𝐵𝐴))

Theoremneg2subd 10394 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 − -𝐵) = (𝐵𝐴))

Theoremsubaddd 10395 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))

Theoremsubadd2d 10396 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴))

Theoremaddsubassd 10397 Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵𝐶)))

Theoremaddsubd 10398 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − 𝐶) = ((𝐴𝐶) + 𝐵))

Theoremsubadd23d 10399 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) + 𝐶) = (𝐴 + (𝐶𝐵)))

Theoremaddsub12d 10400 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 + (𝐵𝐶)) = (𝐵 + (𝐴𝐶)))

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