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Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubsub23i 7301 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = 𝐶 ↔ (𝐴𝐶) = 𝐵)

Theoremaddsubassi 7302 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 7304 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = (𝐴𝐶) ↔ 𝐵 = 𝐶)

Theoremsubcan2i 7305 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐶) = (𝐵𝐶) ↔ 𝐴 = 𝐵)

Theorempnncani 7306 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 7307 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 7308 Alternate proof of 0re 7027. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℝ

Theoremnegcld 7309 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → -𝐴 ∈ ℂ)

Theoremsubidd 7310 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴𝐴) = 0)

Theoremsubid1d 7311 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 − 0) = 𝐴)

Theoremnegidd 7312 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 + -𝐴) = 0)

Theoremnegnegd 7313 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 7314 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 = 0 ↔ -𝐴 = 0))

Theoremnegne0bd 7315 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 ≠ 0 ↔ -𝐴 ≠ 0))

Theoremnegcon1d 7316 Contraposition law for unary minus. Deduction form of negcon1 7263. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))

Theoremnegcon1ad 7317 Contraposition law for unary minus. One-way deduction form of negcon1 7263. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → -𝐴 = 𝐵)       (𝜑 → -𝐵 = 𝐴)

Theoremneg11ad 7318 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 7262. Generalization of neg11d 7334. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 = -𝐵𝐴 = 𝐵))

Theoremnegned 7319 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 7334. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → -𝐴 ≠ -𝐵)

Theoremnegne0d 7320 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → -𝐴 ≠ 0)

Theoremnegrebd 7321 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → -𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℝ)

Theoremsubcld 7322 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴𝐵) ∈ ℂ)

Theorempncand 7323 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴)

Theorempncan2d 7324 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − 𝐴) = 𝐵)

Theorempncan3d 7325 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 + (𝐵𝐴)) = 𝐵)

Theoremnpcand 7326 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴𝐵) + 𝐵) = 𝐴)

Theoremnncand 7327 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 − (𝐴𝐵)) = 𝐵)

Theoremnegsubd 7328 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 7329 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵))

Theoremsubeq0d 7330 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 7331 Two unequal numbers have nonzero difference. See also subap0d 7631 which is the same thing for apartness rather than negated equality. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴𝐵) ≠ 0)

Theoremsubeq0ad 7332 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 7237. Generalization of subeq0d 7330. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴𝐵) = 0 ↔ 𝐴 = 𝐵))

Theoremsubne0ad 7333 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 7331. Contrapositive of subeq0bd 7377. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) ≠ 0)       (𝜑𝐴𝐵)

Theoremneg11d 7334 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 7335 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))

Theoremnegdi2d 7336 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴 + 𝐵) = (-𝐴𝐵))

Theoremnegsubdid 7337 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴𝐵) = (-𝐴 + 𝐵))

Theoremnegsubdi2d 7338 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴𝐵) = (𝐵𝐴))

Theoremneg2subd 7339 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 − -𝐵) = (𝐵𝐴))

Theoremsubaddd 7340 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 7341 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 7342 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 7343 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 7344 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 7345 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)))

Theoremnpncand 7346 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) + (𝐵𝐶)) = (𝐴𝐶))

Theoremnppcand 7347 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (((𝐴𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶))

Theoremnppcan2d 7348 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 − (𝐵 + 𝐶)) + 𝐶) = (𝐴𝐵))

Theoremnppcan3d 7349 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) + (𝐶 + 𝐵)) = (𝐴 + 𝐶))

Theoremsubsubd 7350 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 − (𝐵𝐶)) = ((𝐴𝐵) + 𝐶))

Theoremsubsub2d 7351 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 − (𝐵𝐶)) = (𝐴 + (𝐶𝐵)))

Theoremsubsub3d 7352 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 − (𝐵𝐶)) = ((𝐴 + 𝐶) − 𝐵))

Theoremsubsub4d 7353 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) − 𝐶) = (𝐴 − (𝐵 + 𝐶)))

Theoremsub32d 7354 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 7355 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 − (𝐵𝐶)) − 𝐶) = (𝐴𝐵))

Theoremnnncan1d 7356 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) − (𝐴𝐶)) = (𝐶𝐵))

Theoremnnncan2d 7357 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐶) − (𝐵𝐶)) = (𝐴𝐵))

Theoremnpncan3d 7358 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) + (𝐶𝐴)) = (𝐶𝐵))

Theorempnpcand 7359 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 7360 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 7361 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 7362 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 7363 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) = (𝐴𝐶))       (𝜑𝐵 = 𝐶)

Theoremsubcan2d 7364 Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → (𝐴𝐶) = (𝐵𝐶))       (𝜑𝐴 = 𝐵)

Theoremsubcanad 7365 Cancellation law for subtraction. Deduction form of subcan 7266. Generalization of subcand 7363. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) = (𝐴𝐶) ↔ 𝐵 = 𝐶))

Theoremsubneintrd 7366 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 7363. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐴𝐵) ≠ (𝐴𝐶))

Theoremsubcan2ad 7367 Cancellation law for subtraction. Deduction form of subcan2 7236. Generalization of subcan2d 7364. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐶) = (𝐵𝐶) ↔ 𝐴 = 𝐵))

Theoremsubneintr2d 7368 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 7364. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ≠ (𝐵𝐶))

Theoremaddsub4d 7369 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 7370 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 7371 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 7372 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 7373 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)))

Theoremsubeqrev 7374 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 7375 Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(𝐴 ∈ ℂ → ((𝐴 + 1) − 1) = 𝐴)

Theoremnpcan1 7376 Cancellation law for subtraction and addition with 1. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
(𝐴 ∈ ℂ → ((𝐴 − 1) + 1) = 𝐴)

Theoremsubeq0bd 7377 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 7332. Converse of subeq0d 7330. Contrapositive of subne0ad 7333. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐵) = 0)

Theoremrenegcld 7378 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → -𝐴 ∈ ℝ)

Theoremresubcld 7379 Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝐵) ∈ ℝ)

3.3.3  Multiplication

Theoremkcnktkm1cn 7380 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 7381 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 7382 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 7383 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)))

Theoremmul02 7384 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 10-Aug-1999.)
(𝐴 ∈ ℂ → (0 · 𝐴) = 0)

Theoremmul02lem2 7385 Zero times a real is zero. Although we prove it as a corollary of mul02 7384, the name is for consistency with the Metamath Proof Explorer which proves it before mul02 7384. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℝ → (0 · 𝐴) = 0)

Theoremmul01 7386 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → (𝐴 · 0) = 0)

Theoremmul02i 7387 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ       (0 · 𝐴) = 0

Theoremmul01i 7388 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ       (𝐴 · 0) = 0

Theoremmul02d 7389 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (0 · 𝐴) = 0)

Theoremmul01d 7390 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 0) = 0)

Theoremine0 7391 The imaginary unit i is not zero. (Contributed by NM, 6-May-1999.)
i ≠ 0

Theoremmulneg1 7392 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 7393 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 7394 Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = (𝐴 · -𝐵))

Theoremmul2neg 7395 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 7396 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 7397 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)
(𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴)

Theoremmulsub 7398 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 7399 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 7400 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)
𝐴 ∈ ℂ       (-1 · 𝐴) = -𝐴

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