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Theorem List for Intuitionistic Logic Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmsqge0i 7401 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A        0 ≤ (A · A)

Theoremmsqge0d 7402 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → 0 ≤ (A · A))

Theoremmulge0 7403 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((A 0 ≤ A) (B 0 ≤ B)) → 0 ≤ (A · B))

Theoremmulge0i 7404 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)
A     &   B        ((0 ≤ A 0 ≤ B) → 0 ≤ (A · B))

Theoremmulge0d 7405 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))

Theoremapti 7406 Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
((A B ℂ) → (A = B ↔ ¬ A # B))

Theoremapne 7407 Apartness implies negated equality. We cannot in general prove the converse, which is the whole point of having separate notations for apartness and negated equality. (Contributed by Jim Kingdon, 21-Feb-2020.)
((A B ℂ) → (A # BAB))

Theoremgt0ap0 7408 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
((A 0 < A) → A # 0)

Theoremgt0ap0i 7409 Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.)
A        (0 < AA # 0)

Theoremgt0ap0ii 7410 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   0 < A       A # 0

Theoremgt0ap0d 7411 Positive implies apart from zero. Because of the way we define #, A must be an element of , not just *. (Contributed by Jim Kingdon, 27-Feb-2020.)
(φA ℝ)    &   (φ → 0 < A)       (φA # 0)

Theoremnegap0 7412 A number is apart from zero iff its negative is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
(A ℂ → (A # 0 ↔ -A # 0))

Theoremltleap 7413 Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.)
((A B ℝ) → (A < B ↔ (AB A # B)))

3.3.7  Reciprocals

Theoremrecextlem1 7414 Lemma for recexap 7416. (Contributed by Eric Schmidt, 23-May-2007.)
((A B ℂ) → ((A + (i · B)) · (A − (i · B))) = ((A · A) + (B · B)))

Theoremrecexaplem2 7415 Lemma for recexap 7416. (Contributed by Jim Kingdon, 20-Feb-2020.)
((A B (A + (i · B)) # 0) → ((A · A) + (B · B)) # 0)

Theoremrecexap 7416* Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)
((A A # 0) → x ℂ (A · x) = 1)

Theoremmulap0 7417 The product of two numbers apart from zero is apart from zero. Lemma 2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.)
(((A A # 0) (B B # 0)) → (A · B) # 0)

Theoremmulap0b 7418 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A B ℂ) → ((A # 0 B # 0) ↔ (A · B) # 0))

Theoremmulap0i 7419 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
A     &   B     &   A # 0    &   B # 0       (A · B) # 0

Theoremmulap0bd 7420 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
(φA ℂ)    &   (φB ℂ)       (φ → ((A # 0 B # 0) ↔ (A · B) # 0))

Theoremmulap0d 7421 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)       (φ → (A · B) # 0)

Theoremmulap0bad 7422 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7421 and consequence of mulap0bd 7420. (Contributed by Jim Kingdon, 24-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ → (A · B) # 0)       (φA # 0)

Theoremmulap0bbd 7423 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7421 and consequence of mulap0bd 7420. (Contributed by Jim Kingdon, 24-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ → (A · B) # 0)       (φB # 0)

Theoremmulcanapd 7424 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((𝐶 · A) = (𝐶 · B) ↔ A = B))

Theoremmulcanap2d 7425 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((A · 𝐶) = (B · 𝐶) ↔ A = B))

Theoremmulcanapad 7426 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcanapd 7424. (Contributed by Jim Kingdon, 21-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)    &   (φ → (𝐶 · A) = (𝐶 · B))       (φA = B)

Theoremmulcanap2ad 7427 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcanap2d 7425. (Contributed by Jim Kingdon, 21-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)    &   (φ → (A · 𝐶) = (B · 𝐶))       (φA = B)

Theoremmulcanap 7428 Cancellation law for multiplication (full theorem form). (Contributed by Jim Kingdon, 21-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((𝐶 · A) = (𝐶 · B) ↔ A = B))

Theoremmulcanap2 7429 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A · 𝐶) = (B · 𝐶) ↔ A = B))

Theoremmulcanapi 7430 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
A     &   B     &   𝐶     &   𝐶 # 0       ((𝐶 · A) = (𝐶 · B) ↔ A = B)

Theoremmuleqadd 7431 Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)
((A B ℂ) → ((A · B) = (A + B) ↔ ((A − 1) · (B − 1)) = 1))

Theoremreceuap 7432* Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.)
((A B B # 0) → ∃!x ℂ (B · x) = A)

3.3.8  Division

Syntaxcdiv 7433 Extend class notation to include division.
class /

Definitiondf-div 7434* Define division. Theorem divmulap 7436 relates it to multiplication, and divclap 7439 and redivclap 7489 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
/ = (x ℂ, y (ℂ ∖ {0}) ↦ (z ℂ (y · z) = x))

Theoremdivvalap 7435* Value of division: the (unique) element x such that (B · x) = A. This is meaningful only when B is apart from zero. (Contributed by Jim Kingdon, 21-Feb-2020.)
((A B B # 0) → (A / B) = (x ℂ (B · x) = A))

Theoremdivmulap 7436 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A / 𝐶) = B ↔ (𝐶 · B) = A))

Theoremdivmulap2 7437 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A / 𝐶) = BA = (𝐶 · B)))

Theoremdivmulap3 7438 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A / 𝐶) = BA = (B · 𝐶)))

Theoremdivclap 7439 Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B B # 0) → (A / B) ℂ)

Theoremrecclap 7440 Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A A # 0) → (1 / A) ℂ)

Theoremdivcanap2 7441 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B B # 0) → (B · (A / B)) = A)

Theoremdivcanap1 7442 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B B # 0) → ((A / B) · B) = A)

Theoremdiveqap0 7443 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B B # 0) → ((A / B) = 0 ↔ A = 0))

Theoremdivap0b 7444 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B B # 0) → (A # 0 ↔ (A / B) # 0))

Theoremdivap0 7445 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
(((A A # 0) (B B # 0)) → (A / B) # 0)

Theoremrecap0 7446 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A A # 0) → (1 / A) # 0)

Theoremrecidap 7447 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A A # 0) → (A · (1 / A)) = 1)

Theoremrecidap2 7448 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A A # 0) → ((1 / A) · A) = 1)

Theoremdivrecap 7449 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A B B # 0) → (A / B) = (A · (1 / B)))

Theoremdivrecap2 7450 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → (A / B) = ((1 / B) · A))

Theoremdivassap 7451 An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A · B) / 𝐶) = (A · (B / 𝐶)))

Theoremdiv23ap 7452 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A · B) / 𝐶) = ((A / 𝐶) · B))

Theoremdiv32ap 7453 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A (B B # 0) 𝐶 ℂ) → ((A / B) · 𝐶) = (A · (𝐶 / B)))

Theoremdiv13ap 7454 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A (B B # 0) 𝐶 ℂ) → ((A / B) · 𝐶) = ((𝐶 / B) · A))

Theoremdiv12ap 7455 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → (A · (B / 𝐶)) = (B · (A / 𝐶)))

Theoremdivdirap 7456 Distribution of division over addition. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A + B) / 𝐶) = ((A / 𝐶) + (B / 𝐶)))

Theoremdivcanap3 7457 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → ((B · A) / B) = A)

Theoremdivcanap4 7458 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → ((A · B) / B) = A)

Theoremdiv11ap 7459 One-to-one relationship for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A / 𝐶) = (B / 𝐶) ↔ A = B))

Theoremdividap 7460 A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A A # 0) → (A / A) = 1)

Theoremdiv0ap 7461 Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A A # 0) → (0 / A) = 0)

Theoremdiv1 7462 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(A ℂ → (A / 1) = A)

Theorem1div1e1 7463 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
(1 / 1) = 1

Theoremdiveqap1 7464 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → ((A / B) = 1 ↔ A = B))

Theoremdivnegap 7465 Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → -(A / B) = (-A / B))

Theoremdivsubdirap 7466 Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
((A B (𝐶 𝐶 # 0)) → ((AB) / 𝐶) = ((A / 𝐶) − (B / 𝐶)))

Theoremrecrecap 7467 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A A # 0) → (1 / (1 / A)) = A)

Theoremrec11ap 7468 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((A A # 0) (B B # 0)) → ((1 / A) = (1 / B) ↔ A = B))

Theoremrec11rap 7469 Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((A A # 0) (B B # 0)) → ((1 / A) = B ↔ (1 / B) = A))

Theoremdivmuldivap 7470 Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((A B ℂ) ((𝐶 𝐶 # 0) (𝐷 𝐷 # 0))) → ((A / 𝐶) · (B / 𝐷)) = ((A · B) / (𝐶 · 𝐷)))

Theoremdivdivdivap 7471 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((A (B B # 0)) ((𝐶 𝐶 # 0) (𝐷 𝐷 # 0))) → ((A / B) / (𝐶 / 𝐷)) = ((A · 𝐷) / (B · 𝐶)))

Theoremdivcanap5 7472 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A (B B # 0) (𝐶 𝐶 # 0)) → ((𝐶 · A) / (𝐶 · B)) = (A / B))

Theoremdivmul13ap 7473 Swap the denominators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A B ℂ) ((𝐶 𝐶 # 0) (𝐷 𝐷 # 0))) → ((A / 𝐶) · (B / 𝐷)) = ((B / 𝐶) · (A / 𝐷)))

Theoremdivmul24ap 7474 Swap the numerators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A B ℂ) ((𝐶 𝐶 # 0) (𝐷 𝐷 # 0))) → ((A / 𝐶) · (B / 𝐷)) = ((A / 𝐷) · (B / 𝐶)))

Theoremdivmuleqap 7475 Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A B ℂ) ((𝐶 𝐶 # 0) (𝐷 𝐷 # 0))) → ((A / 𝐶) = (B / 𝐷) ↔ (A · 𝐷) = (B · 𝐶)))

Theoremrecdivap 7476 The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A A # 0) (B B # 0)) → (1 / (A / B)) = (B / A))

Theoremdivcanap6 7477 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A A # 0) (B B # 0)) → ((A / B) · (B / A)) = 1)

Theoremdivdiv32ap 7478 Swap denominators in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
((A (B B # 0) (𝐶 𝐶 # 0)) → ((A / B) / 𝐶) = ((A / 𝐶) / B))

Theoremdivcanap7 7479 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
((A (B B # 0) (𝐶 𝐶 # 0)) → ((A / 𝐶) / (B / 𝐶)) = (A / B))

Theoremdmdcanap 7480 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A A # 0) (B B # 0) 𝐶 ℂ) → ((A / B) · (𝐶 / A)) = (𝐶 / B))

Theoremdivdivap1 7481 Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
((A (B B # 0) (𝐶 𝐶 # 0)) → ((A / B) / 𝐶) = (A / (B · 𝐶)))

Theoremdivdivap2 7482 Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
((A (B B # 0) (𝐶 𝐶 # 0)) → (A / (B / 𝐶)) = ((A · 𝐶) / B))

Theoremrecdivap2 7483 Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A A # 0) (B B # 0)) → ((1 / A) / B) = (1 / (A · B)))

Theoremddcanap 7484 Cancellation in a double division. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A A # 0) (B B # 0)) → (A / (A / B)) = B)

(((A B ℂ) ((𝐶 𝐶 # 0) (𝐷 𝐷 # 0))) → ((A / 𝐶) + (B / 𝐷)) = (((A · 𝐷) + (B · 𝐶)) / (𝐶 · 𝐷)))

Theoremdivsubdivap 7486 Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A B ℂ) ((𝐶 𝐶 # 0) (𝐷 𝐷 # 0))) → ((A / 𝐶) − (B / 𝐷)) = (((A · 𝐷) − (B · 𝐶)) / (𝐶 · 𝐷)))

Theoremconjmulap 7487 Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝑃 𝑃 # 0) (𝑄 𝑄 # 0)) → (((1 / 𝑃) + (1 / 𝑄)) = 1 ↔ ((𝑃 − 1) · (𝑄 − 1)) = 1))

Theoremrerecclap 7488 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
((A A # 0) → (1 / A) ℝ)

Theoremredivclap 7489 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
((A B B # 0) → (A / B) ℝ)

Theoremeqneg 7490 A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(A ℂ → (A = -AA = 0))

Theoremeqnegd 7491 A complex number equals its negative iff it is zero. Deduction form of eqneg 7490. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)       (φ → (A = -AA = 0))

Theoremeqnegad 7492 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 7490. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φA = -A)       (φA = 0)

Theoremdiv2negap 7493 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
((A B B # 0) → (-A / -B) = (A / B))

Theoremdivneg2ap 7494 Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.)
((A B B # 0) → -(A / B) = (A / -B))

Theoremrecclapzi 7495 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
A        (A # 0 → (1 / A) ℂ)

Theoremrecap0apzi 7496 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
A        (A # 0 → (1 / A) # 0)

Theoremrecidapzi 7497 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
A        (A # 0 → (A · (1 / A)) = 1)

Theoremdiv1i 7498 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
A        (A / 1) = A

Theoremeqnegi 7499 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
A        (A = -AA = 0)

Theoremrecclapi 7500 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
A     &   A # 0       (1 / A)

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