HomeHome Intuitionistic Logic Explorer
Theorem List (p. 75 of 94)
< 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
 
Theoremdivcanap2 7401 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B B # 0) → (B · (A / B)) = A)
 
Theoremdivcanap1 7402 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B B # 0) → ((A / B) · B) = A)
 
Theoremdiveqap0 7403 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 7404 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 7405 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 7406 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 7407 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A A # 0) → (A · (1 / A)) = 1)
 
Theoremrecidap2 7408 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A A # 0) → ((1 / A) · A) = 1)
 
Theoremdivrecap 7409 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A B B # 0) → (A / B) = (A · (1 / B)))
 
Theoremdivrecap2 7410 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → (A / B) = ((1 / B) · A))
 
Theoremdivassap 7411 An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A · B) / 𝐶) = (A · (B / 𝐶)))
 
Theoremdiv23ap 7412 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 7413 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 7414 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 7415 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 7416 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 7417 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → ((B · A) / B) = A)
 
Theoremdivcanap4 7418 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → ((A · B) / B) = A)
 
Theoremdiv11ap 7419 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 7420 A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A A # 0) → (A / A) = 1)
 
Theoremdiv0ap 7421 Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A A # 0) → (0 / A) = 0)
 
Theoremdiv1 7422 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 7423 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
(1 / 1) = 1
 
Theoremdiveqap1 7424 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → ((A / B) = 1 ↔ A = B))
 
Theoremdivnegap 7425 Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((A B B # 0) → -(A / B) = (-A / B))
 
Theoremdivsubdirap 7426 Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
((A B (𝐶 𝐶 # 0)) → ((AB) / 𝐶) = ((A / 𝐶) − (B / 𝐶)))
 
Theoremrecrecap 7427 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 7428 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 7429 Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((A A # 0) (B B # 0)) → ((1 / A) = B ↔ (1 / B) = A))
 
Theoremdivmuldivap 7430 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 7431 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 7432 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 7433 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 7434 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 7435 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 7436 The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A A # 0) (B B # 0)) → (1 / (A / B)) = (B / A))
 
Theoremdivcanap6 7437 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A A # 0) (B B # 0)) → ((A / B) · (B / A)) = 1)
 
Theoremdivdiv32ap 7438 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 7439 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 7440 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 7441 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 7442 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 7443 Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A A # 0) (B B # 0)) → ((1 / A) / B) = (1 / (A · B)))
 
Theoremddcanap 7444 Cancellation in a double division. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((A A # 0) (B B # 0)) → (A / (A / B)) = B)
 
Theoremdivadddivap 7445 Addition 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)))
 
Theoremdivsubdivap 7446 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 7447 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 7448 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
((A A # 0) → (1 / A) ℝ)
 
Theoremredivclap 7449 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
((A B B # 0) → (A / B) ℝ)
 
Theoremeqneg 7450 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 7451 A complex number equals its negative iff it is zero. Deduction form of eqneg 7450. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)       (φ → (A = -AA = 0))
 
Theoremeqnegad 7452 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 7450. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φA = -A)       (φA = 0)
 
Theoremdiv2negap 7453 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
((A B B # 0) → (-A / -B) = (A / B))
 
Theoremdivneg2ap 7454 Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.)
((A B B # 0) → -(A / B) = (A / -B))
 
Theoremrecclapzi 7455 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
A        (A # 0 → (1 / A) ℂ)
 
Theoremrecap0apzi 7456 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 7457 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
A        (A # 0 → (A · (1 / A)) = 1)
 
Theoremdiv1i 7458 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
A        (A / 1) = A
 
Theoremeqnegi 7459 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
A        (A = -AA = 0)
 
Theoremrecclapi 7460 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
A     &   A # 0       (1 / A)
 
Theoremrecidapi 7461 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
A     &   A # 0       (A · (1 / A)) = 1
 
Theoremrecrecapi 7462 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 7463 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
A     &   A # 0       (A / A) = 1
 
Theoremdiv0api 7464 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
A     &   A # 0       (0 / A) = 0
 
Theoremdivclapzi 7465 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → (A / B) ℂ)
 
Theoremdivcanap1zi 7466 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → ((A / B) · B) = A)
 
Theoremdivcanap2zi 7467 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → (B · (A / B)) = A)
 
Theoremdivrecapzi 7468 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → (A / B) = (A · (1 / B)))
 
Theoremdivcanap3zi 7469 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → ((B · A) / B) = A)
 
Theoremdivcanap4zi 7470 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → ((A · B) / B) = A)
 
Theoremrec11api 7471 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 7472 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       (A / B)
 
Theoremdivcanap2i 7473 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       (B · (A / B)) = A
 
Theoremdivcanap1i 7474 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       ((A / B) · B) = A
 
Theoremdivrecapi 7475 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       (A / B) = (A · (1 / B))
 
Theoremdivcanap3i 7476 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       ((B · A) / B) = A
 
Theoremdivcanap4i 7477 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       ((A · B) / B) = A
 
Theoremdivap0i 7478 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 7479 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 7480 An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   𝐶        (𝐶 # 0 → ((A · B) / 𝐶) = (A · (B / 𝐶)))
 
Theoremdivmulapzi 7481 Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   𝐶        (B # 0 → ((A / B) = 𝐶 ↔ (B · 𝐶) = A))
 
Theoremdivdirapzi 7482 Distribution of division over addition. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   𝐶        (𝐶 # 0 → ((A + B) / 𝐶) = ((A / 𝐶) + (B / 𝐶)))
 
Theoremdivdiv23apzi 7483 Swap denominators in a division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   𝐶        ((B # 0 𝐶 # 0) → ((A / B) / 𝐶) = ((A / 𝐶) / B))
 
Theoremdivmulapi 7484 Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.)
A     &   B     &   𝐶     &   B # 0       ((A / B) = 𝐶 ↔ (B · 𝐶) = A)
 
Theoremdivdiv32api 7485 Swap denominators in a division. (Contributed by Jim Kingdon, 29-Feb-2020.)
A     &   B     &   𝐶     &   B # 0    &   𝐶 # 0       ((A / B) / 𝐶) = ((A / 𝐶) / B)
 
Theoremdivassapi 7486 An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B     &   𝐶     &   𝐶 # 0       ((A · B) / 𝐶) = (A · (B / 𝐶))
 
Theoremdivdirapi 7487 Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B     &   𝐶     &   𝐶 # 0       ((A + B) / 𝐶) = ((A / 𝐶) + (B / 𝐶))
 
Theoremdiv23api 7488 A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B     &   𝐶     &   𝐶 # 0       ((A · B) / 𝐶) = ((A / 𝐶) · B)
 
Theoremdiv11api 7489 One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B     &   𝐶     &   𝐶 # 0       ((A / 𝐶) = (B / 𝐶) ↔ A = B)
 
Theoremdivmuldivapi 7490 Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B     &   𝐶     &   𝐷     &   B # 0    &   𝐷 # 0       ((A / B) · (𝐶 / 𝐷)) = ((A · 𝐶) / (B · 𝐷))
 
Theoremdivmul13api 7491 Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B     &   𝐶     &   𝐷     &   B # 0    &   𝐷 # 0       ((A / B) · (𝐶 / 𝐷)) = ((𝐶 / B) · (A / 𝐷))
 
Theoremdivadddivapi 7492 Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B     &   𝐶     &   𝐷     &   B # 0    &   𝐷 # 0       ((A / B) + (𝐶 / 𝐷)) = (((A · 𝐷) + (𝐶 · B)) / (B · 𝐷))
 
Theoremdivdivdivapi 7493 Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B     &   𝐶     &   𝐷     &   B # 0    &   𝐷 # 0    &   𝐶 # 0       ((A / B) / (𝐶 / 𝐷)) = ((A · 𝐷) / (B · 𝐶))
 
Theoremrerecclapzi 7494 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
A        (A # 0 → (1 / A) ℝ)
 
Theoremrerecclapi 7495 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   A # 0       (1 / A)
 
Theoremredivclapzi 7496 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B        (B # 0 → (A / B) ℝ)
 
Theoremredivclapi 7497 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B     &   B # 0       (A / B)
 
Theoremdiv1d 7498 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A / 1) = A)
 
Theoremrecclapd 7499 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → (1 / A) ℂ)
 
Theoremrecap0d 7500 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → (1 / A) # 0)
    < Previous  Next >

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 9301-9381
  Copyright terms: Public domain < Previous  Next >