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Theorem List for Intuitionistic Logic Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrecidapi 7501 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
A     &   A # 0       (A · (1 / A)) = 1
 
Theoremrecrecapi 7502 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 7503 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
A     &   A # 0       (A / A) = 1
 
Theoremdiv0api 7504 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
A     &   A # 0       (0 / A) = 0
 
Theoremdivclapzi 7505 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → (A / B) ℂ)
 
Theoremdivcanap1zi 7506 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → ((A / B) · B) = A)
 
Theoremdivcanap2zi 7507 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → (B · (A / B)) = A)
 
Theoremdivrecapzi 7508 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → (A / B) = (A · (1 / B)))
 
Theoremdivcanap3zi 7509 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → ((B · A) / B) = A)
 
Theoremdivcanap4zi 7510 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   B        (B # 0 → ((A · B) / B) = A)
 
Theoremrec11api 7511 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 7512 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       (A / B)
 
Theoremdivcanap2i 7513 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       (B · (A / B)) = A
 
Theoremdivcanap1i 7514 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       ((A / B) · B) = A
 
Theoremdivrecapi 7515 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       (A / B) = (A · (1 / B))
 
Theoremdivcanap3i 7516 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       ((B · A) / B) = A
 
Theoremdivcanap4i 7517 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   B # 0       ((A · B) / B) = A
 
Theoremdivap0i 7518 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 7519 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 7520 An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   𝐶        (𝐶 # 0 → ((A · B) / 𝐶) = (A · (B / 𝐶)))
 
Theoremdivmulapzi 7521 Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.)
A     &   B     &   𝐶        (B # 0 → ((A / B) = 𝐶 ↔ (B · 𝐶) = A))
 
Theoremdivdirapzi 7522 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 7523 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 7524 Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.)
A     &   B     &   𝐶     &   B # 0       ((A / B) = 𝐶 ↔ (B · 𝐶) = A)
 
Theoremdivdiv32api 7525 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 7526 An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B     &   𝐶     &   𝐶 # 0       ((A · B) / 𝐶) = (A · (B / 𝐶))
 
Theoremdivdirapi 7527 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 7528 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 7529 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 7530 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 7531 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 7532 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 7533 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 7534 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
A        (A # 0 → (1 / A) ℝ)
 
Theoremrerecclapi 7535 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   A # 0       (1 / A)
 
Theoremredivclapzi 7536 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B        (B # 0 → (A / B) ℝ)
 
Theoremredivclapi 7537 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
A     &   B     &   B # 0       (A / B)
 
Theoremdiv1d 7538 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A / 1) = A)
 
Theoremrecclapd 7539 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → (1 / A) ℂ)
 
Theoremrecap0d 7540 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)
 
Theoremrecidapd 7541 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → (A · (1 / A)) = 1)
 
Theoremrecidap2d 7542 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → ((1 / A) · A) = 1)
 
Theoremrecrecapd 7543 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → (1 / (1 / A)) = A)
 
Theoremdividapd 7544 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → (A / A) = 1)
 
Theoremdiv0apd 7545 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → (0 / A) = 0)
 
Theoremapmul1 7546 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
((A B (𝐶 𝐶 # 0)) → (A # B ↔ (A · 𝐶) # (B · 𝐶)))
 
Theoremdivclapd 7547 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (A / B) ℂ)
 
Theoremdivcanap1d 7548 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → ((A / B) · B) = A)
 
Theoremdivcanap2d 7549 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (B · (A / B)) = A)
 
Theoremdivrecapd 7550 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (A / B) = (A · (1 / B)))
 
Theoremdivrecap2d 7551 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (A / B) = ((1 / B) · A))
 
Theoremdivcanap3d 7552 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → ((B · A) / B) = A)
 
Theoremdivcanap4d 7553 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → ((A · B) / B) = A)
 
Theoremdiveqap0d 7554 If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)    &   (φ → (A / B) = 0)       (φA = 0)
 
Theoremdiveqap1d 7555 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)    &   (φ → (A / B) = 1)       (φA = B)
 
Theoremdiveqap1ad 7556 The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 7464. Generalization of diveqap1d 7555. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → ((A / B) = 1 ↔ A = B))
 
Theoremdiveqap0ad 7557 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 7443. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → ((A / B) = 0 ↔ A = 0))
 
Theoremdivap1d 7558 If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)    &   (φA # B)       (φ → (A / B) # 1)
 
Theoremdivap0bd 7559 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (A # 0 ↔ (A / B) # 0))
 
Theoremdivnegapd 7560 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → -(A / B) = (-A / B))
 
Theoremdivneg2apd 7561 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → -(A / B) = (A / -B))
 
Theoremdiv2negapd 7562 Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (-A / -B) = (A / B))
 
Theoremdivap0d 7563 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)       (φ → (A / B) # 0)
 
Theoremrecdivapd 7564 The reciprocal of a ratio. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)       (φ → (1 / (A / B)) = (B / A))
 
Theoremrecdivap2d 7565 Division into a reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)       (φ → ((1 / A) / B) = (1 / (A · B)))
 
Theoremdivcanap6d 7566 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)       (φ → ((A / B) · (B / A)) = 1)
 
Theoremddcanapd 7567 Cancellation in a double division. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)       (φ → (A / (A / B)) = B)
 
Theoremrec11apd 7568 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)    &   (φ → (1 / A) = (1 / B))       (φA = B)
 
Theoremdivmulapd 7569 Relationship between division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)       (φ → ((A / B) = 𝐶 ↔ (B · 𝐶) = A))
 
Theoremdiv32apd 7570 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)       (φ → ((A / B) · 𝐶) = (A · (𝐶 / B)))
 
Theoremdiv13apd 7571 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)       (φ → ((A / B) · 𝐶) = ((𝐶 / B) · A))
 
Theoremdivdiv32apd 7572 Swap denominators in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((A / B) / 𝐶) = ((A / 𝐶) / B))
 
Theoremdivcanap5d 7573 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((𝐶 · A) / (𝐶 · B)) = (A / B))
 
Theoremdivcanap5rd 7574 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((A · 𝐶) / (B · 𝐶)) = (A / B))
 
Theoremdivcanap7d 7575 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((A / 𝐶) / (B / 𝐶)) = (A / B))
 
Theoremdmdcanapd 7576 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((B / 𝐶) · (A / B)) = (A / 𝐶))
 
Theoremdmdcanap2d 7577 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((A / B) · (B / 𝐶)) = (A / 𝐶))
 
Theoremdivdivap1d 7578 Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((A / B) / 𝐶) = (A / (B · 𝐶)))
 
Theoremdivdivap2d 7579 Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → (A / (B / 𝐶)) = ((A · 𝐶) / B))
 
Theoremdivmulap2d 7580 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((A / 𝐶) = BA = (𝐶 · B)))
 
Theoremdivmulap3d 7581 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((A / 𝐶) = BA = (B · 𝐶)))
 
Theoremdivassapd 7582 An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((A · B) / 𝐶) = (A · (B / 𝐶)))
 
Theoremdiv12apd 7583 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → (A · (B / 𝐶)) = (B · (A / 𝐶)))
 
Theoremdiv23apd 7584 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((A · B) / 𝐶) = ((A / 𝐶) · B))
 
Theoremdivdirapd 7585 Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((A + B) / 𝐶) = ((A / 𝐶) + (B / 𝐶)))
 
Theoremdivsubdirapd 7586 Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((AB) / 𝐶) = ((A / 𝐶) − (B / 𝐶)))
 
Theoremdiv11apd 7587 One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)    &   (φ → (A / 𝐶) = (B / 𝐶))       (φA = B)
 
Theoremrerecclapd 7588 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℝ)    &   (φA # 0)       (φ → (1 / A) ℝ)
 
Theoremredivclapd 7589 Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℝ)    &   (φB ℝ)    &   (φB # 0)       (φ → (A / B) ℝ)
 
Theoremmvllmulapd 7590 Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φ → (A · B) = 𝐶)       (φB = (𝐶 / A))
 
3.3.9  Ordering on reals (cont.)
 
Theoremltp1 7591 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
(A ℝ → A < (A + 1))
 
Theoremlep1 7592 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
(A ℝ → A ≤ (A + 1))
 
Theoremltm1 7593 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
(A ℝ → (A − 1) < A)
 
Theoremlem1 7594 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
(A ℝ → (A − 1) ≤ A)
 
Theoremletrp1 7595 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)
((A B AB) → A ≤ (B + 1))
 
Theoremp1le 7596 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)
((A B (A + 1) ≤ B) → AB)
 
Theoremrecgt0 7597 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)
((A 0 < A) → 0 < (1 / A))
 
Theoremprodgt0gt0 7598 Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 7599 for the same theorem with 0 < A replaced by the weaker condition 0 ≤ A. (Contributed by Jim Kingdon, 29-Feb-2020.)
(((A B ℝ) (0 < A 0 < (A · B))) → 0 < B)
 
Theoremprodgt0 7599 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(((A B ℝ) (0 ≤ A 0 < (A · B))) → 0 < B)
 
Theoremprodgt02 7600 Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)
(((A B ℝ) (0 ≤ B 0 < (A · B))) → 0 < A)
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