 Home Intuitionistic Logic ExplorerTheorem List (p. 75 of 95) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

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

Theoremmulap0d 7401 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 7402 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7401 and consequence of mulap0bd 7400. (Contributed by Jim Kingdon, 24-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ → (A · B) # 0)       (φA # 0)

Theoremmulap0bbd 7403 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7401 and consequence of mulap0bd 7400. (Contributed by Jim Kingdon, 24-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ → (A · B) # 0)       (φB # 0)

Theoremmulcanapd 7404 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 7405 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 7406 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcanapd 7404. (Contributed by Jim Kingdon, 21-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)    &   (φ → (𝐶 · A) = (𝐶 · B))       (φA = B)

Theoremmulcanap2ad 7407 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcanap2d 7405. (Contributed by Jim Kingdon, 21-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)    &   (φ → (A · 𝐶) = (B · 𝐶))       (φA = B)

Theoremmulcanap 7408 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 7409 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A · 𝐶) = (B · 𝐶) ↔ A = B))

Theoremmulcanapi 7410 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
A     &   B     &   𝐶     &   𝐶 # 0       ((𝐶 · A) = (𝐶 · B) ↔ A = B)

Theoremmuleqadd 7411 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 7412* 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 7413 Extend class notation to include division.
class /

Definitiondf-div 7414* Define division. Theorem divmulap 7416 relates it to multiplication, and divclap 7419 and redivclap 7469 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 7415* 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 7416 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A / 𝐶) = B ↔ (𝐶 · B) = A))

Theoremdivmulap2 7417 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A / 𝐶) = BA = (𝐶 · B)))

Theoremdivmulap3 7418 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A / 𝐶) = BA = (B · 𝐶)))

Theoremdivclap 7419 Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B B # 0) → (A / B) ℂ)

Theoremrecclap 7420 Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A A # 0) → (1 / A) ℂ)

Theoremdivcanap2 7421 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B B # 0) → (B · (A / B)) = A)

Theoremdivcanap1 7422 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B B # 0) → ((A / B) · B) = A)

Theoremdiveqap0 7423 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 7424 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 7425 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 7426 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 7427 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A A # 0) → (A · (1 / A)) = 1)

Theoremrecidap2 7428 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A A # 0) → ((1 / A) · A) = 1)

Theoremdivrecap 7429 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A B B # 0) → (A / B) = (A · (1 / B)))

Theoremdivrecap2 7430 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → (A / B) = ((1 / B) · A))

Theoremdivassap 7431 An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A · B) / 𝐶) = (A · (B / 𝐶)))

Theoremdiv23ap 7432 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 7433 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 7434 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 7435 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 7436 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 7437 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → ((B · A) / B) = A)

Theoremdivcanap4 7438 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → ((A · B) / B) = A)

Theoremdiv11ap 7439 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 7440 A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A A # 0) → (A / A) = 1)

Theoremdiv0ap 7441 Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A A # 0) → (0 / A) = 0)

Theoremdiv1 7442 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 7443 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
(1 / 1) = 1

Theoremdiveqap1 7444 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → ((A / B) = 1 ↔ A = B))

Theoremdivnegap 7445 Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → -(A / B) = (-A / B))

Theoremdivsubdirap 7446 Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
((A B (𝐶 𝐶 # 0)) → ((AB) / 𝐶) = ((A / 𝐶) − (B / 𝐶)))

Theoremrecrecap 7447 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 7448 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 7449 Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((A A # 0) (B B # 0)) → ((1 / A) = B ↔ (1 / B) = A))

Theoremdivmuldivap 7450 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 7451 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 7452 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 7453 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 7454 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 7455 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 7456 The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A A # 0) (B B # 0)) → (1 / (A / B)) = (B / A))

Theoremdivcanap6 7457 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A A # 0) (B B # 0)) → ((A / B) · (B / A)) = 1)

Theoremdivdiv32ap 7458 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 7459 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 7460 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 7461 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 7462 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 7463 Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A A # 0) (B B # 0)) → ((1 / A) / B) = (1 / (A · B)))

Theoremddcanap 7464 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 7466 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 7467 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 7468 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
((A A # 0) → (1 / A) ℝ)

Theoremredivclap 7469 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
((A B B # 0) → (A / B) ℝ)

Theoremeqneg 7470 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 7471 A complex number equals its negative iff it is zero. Deduction form of eqneg 7470. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)       (φ → (A = -AA = 0))

Theoremeqnegad 7472 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 7470. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φA = -A)       (φA = 0)

Theoremdiv2negap 7473 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
((A B B # 0) → (-A / -B) = (A / B))

Theoremdivneg2ap 7474 Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.)
((A B B # 0) → -(A / B) = (A / -B))

Theoremrecclapzi 7475 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
A        (A # 0 → (1 / A) ℂ)

Theoremrecap0apzi 7476 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 7477 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
A        (A # 0 → (A · (1 / A)) = 1)

Theoremdiv1i 7478 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
A        (A / 1) = A

Theoremeqnegi 7479 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
A        (A = -AA = 0)

Theoremrecclapi 7480 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
A     &   A # 0       (1 / A)

Theoremrecidapi 7481 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
A     &   A # 0       (A · (1 / A)) = 1

Theoremrecrecapi 7482 A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)
A     &   A # 0       (1 / (1 / A)) = A

Theoremdividapi 7483 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
A     &   A # 0       (A / A) = 1

Theoremdiv0api 7484 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
A     &   A # 0       (0 / A) = 0

Theoremdivclapzi 7485 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → (A / B) ℂ)

Theoremdivcanap1zi 7486 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → ((A / B) · B) = A)

Theoremdivcanap2zi 7487 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → (B · (A / B)) = A)

Theoremdivrecapzi 7488 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → (A / B) = (A · (1 / B)))

Theoremdivcanap3zi 7489 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → ((B · A) / B) = A)

Theoremdivcanap4zi 7490 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → ((A · B) / B) = A)

Theoremrec11api 7491 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B        ((A # 0 B # 0) → ((1 / A) = (1 / B) ↔ A = B))

Theoremdivclapi 7492 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       (A / B)

Theoremdivcanap2i 7493 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       (B · (A / B)) = A

Theoremdivcanap1i 7494 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       ((A / B) · B) = A

Theoremdivrecapi 7495 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       (A / B) = (A · (1 / B))

Theoremdivcanap3i 7496 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       ((B · A) / B) = A

Theoremdivcanap4i 7497 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       ((A · B) / B) = A

Theoremdivap0i 7498 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   A # 0    &   B # 0       (A / B) # 0

Theoremrec11apii 7499 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   A # 0    &   B # 0       ((1 / A) = (1 / B) ↔ A = B)

Theoremdivassapzi 7500 An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   𝐶        (𝐶 # 0 → ((A · B) / 𝐶) = (A · (B / 𝐶)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9427
 Copyright terms: Public domain < Previous  Next >