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Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltmuldiv 7601 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B (𝐶 0 < 𝐶)) → ((A · 𝐶) < BA < (B / 𝐶)))
 
Theoremltmuldiv2 7602 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
((A B (𝐶 0 < 𝐶)) → ((𝐶 · A) < BA < (B / 𝐶)))
 
Theoremltdivmul 7603 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
((A B (𝐶 0 < 𝐶)) → ((A / 𝐶) < BA < (𝐶 · B)))
 
Theoremledivmul 7604 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
((A B (𝐶 0 < 𝐶)) → ((A / 𝐶) ≤ BA ≤ (𝐶 · B)))
 
Theoremltdivmul2 7605 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)
((A B (𝐶 0 < 𝐶)) → ((A / 𝐶) < BA < (B · 𝐶)))
 
Theoremlt2mul2div 7606 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)
(((A (B 0 < B)) (𝐶 (𝐷 0 < 𝐷))) → ((A · B) < (𝐶 · 𝐷) ↔ (A / 𝐷) < (𝐶 / B)))
 
Theoremledivmul2 7607 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
((A B (𝐶 0 < 𝐶)) → ((A / 𝐶) ≤ BA ≤ (B · 𝐶)))
 
Theoremlemuldiv 7608 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
((A B (𝐶 0 < 𝐶)) → ((A · 𝐶) ≤ BA ≤ (B / 𝐶)))
 
Theoremlemuldiv2 7609 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
((A B (𝐶 0 < 𝐶)) → ((𝐶 · A) ≤ BA ≤ (B / 𝐶)))
 
Theoremltrec 7610 The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((A 0 < A) (B 0 < B)) → (A < B ↔ (1 / B) < (1 / A)))
 
Theoremlerec 7611 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((A 0 < A) (B 0 < B)) → (AB ↔ (1 / B) ≤ (1 / A)))
 
Theoremlt2msq1 7612 Lemma for lt2msq 7613. (Contributed by Mario Carneiro, 27-May-2016.)
(((A 0 ≤ A) B A < B) → (A · A) < (B · B))
 
Theoremlt2msq 7613 Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(((A 0 ≤ A) (B 0 ≤ B)) → (A < B ↔ (A · A) < (B · B)))
 
Theoremltdiv2 7614 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)
(((A 0 < A) (B 0 < B) (𝐶 0 < 𝐶)) → (A < B ↔ (𝐶 / B) < (𝐶 / A)))
 
Theoremltrec1 7615 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)
(((A 0 < A) (B 0 < B)) → ((1 / A) < B ↔ (1 / B) < A))
 
Theoremlerec2 7616 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)
(((A 0 < A) (B 0 < B)) → (A ≤ (1 / B) ↔ B ≤ (1 / A)))
 
Theoremledivdiv 7617 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)
((((A 0 < A) (B 0 < B)) ((𝐶 0 < 𝐶) (𝐷 0 < 𝐷))) → ((A / B) ≤ (𝐶 / 𝐷) ↔ (𝐷 / 𝐶) ≤ (B / A)))
 
Theoremlediv2 7618 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)
(((A 0 < A) (B 0 < B) (𝐶 0 < 𝐶)) → (AB ↔ (𝐶 / B) ≤ (𝐶 / A)))
 
Theoremltdiv23 7619 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)
((A (B 0 < B) (𝐶 0 < 𝐶)) → ((A / B) < 𝐶 ↔ (A / 𝐶) < B))
 
Theoremlediv23 7620 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
((A (B 0 < B) (𝐶 0 < 𝐶)) → ((A / B) ≤ 𝐶 ↔ (A / 𝐶) ≤ B))
 
Theoremlediv12a 7621 Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)
((((A B ℝ) (0 ≤ A AB)) ((𝐶 𝐷 ℝ) (0 < 𝐶 𝐶𝐷))) → (A / 𝐷) ≤ (B / 𝐶))
 
Theoremlediv2a 7622 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
((((A 0 < A) (B 0 < B) (𝐶 0 ≤ 𝐶)) AB) → (𝐶 / B) ≤ (𝐶 / A))
 
Theoremreclt1 7623 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)
((A 0 < A) → (A < 1 ↔ 1 < (1 / A)))
 
Theoremrecgt1 7624 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)
((A 0 < A) → (1 < A ↔ (1 / A) < 1))
 
Theoremrecgt1i 7625 The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)
((A 1 < A) → (0 < (1 / A) (1 / A) < 1))
 
Theoremrecp1lt1 7626 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
((A 0 ≤ A) → (A / (1 + A)) < 1)
 
Theoremrecreclt 7627 Given a positive number A, construct a new positive number less than both A and 1. (Contributed by NM, 28-Dec-2005.)
((A 0 < A) → ((1 / (1 + (1 / A))) < 1 (1 / (1 + (1 / A))) < A))
 
Theoremle2msq 7628 The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((A 0 ≤ A) (B 0 ≤ B)) → (AB ↔ (A · A) ≤ (B · B)))
 
Theoremmsq11 7629 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((A 0 ≤ A) (B 0 ≤ B)) → ((A · A) = (B · B) ↔ A = B))
 
Theoremledivp1 7630 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.)
(((A 0 ≤ A) (B 0 ≤ B)) → ((A / (B + 1)) · B) ≤ A)
 
Theoremsqueeze0 7631* If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)
((A 0 ≤ A x ℝ (0 < xA < x)) → A = 0)
 
Theoremltp1i 7632 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
A        A < (A + 1)
 
Theoremrecgt0i 7633 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
A        (0 < A → 0 < (1 / A))
 
Theoremrecgt0ii 7634 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
A     &   0 < A       0 < (1 / A)
 
Theoremprodgt0i 7635 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)
A     &   B        ((0 ≤ A 0 < (A · B)) → 0 < B)
 
Theoremprodge0i 7636 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)
A     &   B        ((0 < A 0 ≤ (A · B)) → 0 ≤ B)
 
Theoremdivgt0i 7637 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
A     &   B        ((0 < A 0 < B) → 0 < (A / B))
 
Theoremdivge0i 7638 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)
A     &   B        ((0 ≤ A 0 < B) → 0 ≤ (A / B))
 
Theoremltreci 7639 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
A     &   B        ((0 < A 0 < B) → (A < B ↔ (1 / B) < (1 / A)))
 
Theoremlereci 7640 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)
A     &   B        ((0 < A 0 < B) → (AB ↔ (1 / B) ≤ (1 / A)))
 
Theoremlt2msqi 7641 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)
A     &   B        ((0 ≤ A 0 ≤ B) → (A < B ↔ (A · A) < (B · B)))
 
Theoremle2msqi 7642 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
A     &   B        ((0 ≤ A 0 ≤ B) → (AB ↔ (A · A) ≤ (B · B)))
 
Theoremmsq11i 7643 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)
A     &   B        ((0 ≤ A 0 ≤ B) → ((A · A) = (B · B) ↔ A = B))
 
Theoremdivgt0i2i 7644 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
A     &   B     &   0 < B       (0 < A → 0 < (A / B))
 
Theoremltrecii 7645 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
A     &   B     &   0 < A    &   0 < B       (A < B ↔ (1 / B) < (1 / A))
 
Theoremdivgt0ii 7646 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
A     &   B     &   0 < A    &   0 < B       0 < (A / B)
 
Theoremltmul1i 7647 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
A     &   B     &   𝐶        (0 < 𝐶 → (A < B ↔ (A · 𝐶) < (B · 𝐶)))
 
Theoremltdiv1i 7648 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
A     &   B     &   𝐶        (0 < 𝐶 → (A < B ↔ (A / 𝐶) < (B / 𝐶)))
 
Theoremltmuldivi 7649 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
A     &   B     &   𝐶        (0 < 𝐶 → ((A · 𝐶) < BA < (B / 𝐶)))
 
Theoremltmul2i 7650 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
A     &   B     &   𝐶        (0 < 𝐶 → (A < B ↔ (𝐶 · A) < (𝐶 · B)))
 
Theoremlemul1i 7651 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)
A     &   B     &   𝐶        (0 < 𝐶 → (AB ↔ (A · 𝐶) ≤ (B · 𝐶)))
 
Theoremlemul2i 7652 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)
A     &   B     &   𝐶        (0 < 𝐶 → (AB ↔ (𝐶 · A) ≤ (𝐶 · B)))
 
Theoremltdiv23i 7653 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
A     &   B     &   𝐶        ((0 < B 0 < 𝐶) → ((A / B) < 𝐶 ↔ (A / 𝐶) < B))
 
Theoremltdiv23ii 7654 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
A     &   B     &   𝐶     &   0 < B    &   0 < 𝐶       ((A / B) < 𝐶 ↔ (A / 𝐶) < B)
 
Theoremltmul1ii 7655 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.)
A     &   B     &   𝐶     &   0 < 𝐶       (A < B ↔ (A · 𝐶) < (B · 𝐶))
 
Theoremltdiv1ii 7656 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
A     &   B     &   𝐶     &   0 < 𝐶       (A < B ↔ (A / 𝐶) < (B / 𝐶))
 
Theoremltp1d 7657 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φA < (A + 1))
 
Theoremlep1d 7658 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φA ≤ (A + 1))
 
Theoremltm1d 7659 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φ → (A − 1) < A)
 
Theoremlem1d 7660 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φ → (A − 1) ≤ A)
 
Theoremrecgt0d 7661 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φ → 0 < A)       (φ → 0 < (1 / A))
 
Theoremdivgt0d 7662 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 < A)    &   (φ → 0 < B)       (φ → 0 < (A / B))
 
Theoremmulgt1d 7663 The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 1 < A)    &   (φ → 1 < B)       (φ → 1 < (A · B))
 
Theoremlemulge11d 7664 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 ≤ A)    &   (φ → 1 ≤ B)       (φA ≤ (A · B))
 
Theoremlemulge12d 7665 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 ≤ A)    &   (φ → 1 ≤ B)       (φA ≤ (B · A))
 
Theoremlemul1ad 7666 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ → 0 ≤ 𝐶)    &   (φAB)       (φ → (A · 𝐶) ≤ (B · 𝐶))
 
Theoremlemul2ad 7667 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ → 0 ≤ 𝐶)    &   (φAB)       (φ → (𝐶 · A) ≤ (𝐶 · B))
 
Theoremltmul12ad 7668 Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φ → 0 ≤ A)    &   (φA < B)    &   (φ → 0 ≤ 𝐶)    &   (φ𝐶 < 𝐷)       (φ → (A · 𝐶) < (B · 𝐷))
 
Theoremlemul12ad 7669 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φ → 0 ≤ A)    &   (φ → 0 ≤ 𝐶)    &   (φAB)    &   (φ𝐶𝐷)       (φ → (A · 𝐶) ≤ (B · 𝐷))
 
Theoremlemul12bd 7670 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φ → 0 ≤ A)    &   (φ → 0 ≤ 𝐷)    &   (φAB)    &   (φ𝐶𝐷)       (φ → (A · 𝐶) ≤ (B · 𝐷))
 
3.3.10  Imaginary and complex number properties
 
Theoremcrap0 7671 The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
((A B ℝ) → ((A # 0 B # 0) ↔ (A + (i · B)) # 0))
 
Theoremcreur 7672* The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(A ℂ → ∃!x y A = (x + (i · y)))
 
Theoremcreui 7673* The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(A ℂ → ∃!y x A = (x + (i · y)))
 
Theoremcju 7674* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
(A ℂ → ∃!x ℂ ((A + x) (i · (Ax)) ℝ))
 
3.4  Integer sets
 
3.4.1  Positive integers (as a subset of complex numbers)
 
Syntaxcn 7675 Extend class notation to include the class of positive integers.
class
 
Definitiondf-inn 7676* Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 7677 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)
ℕ = {x ∣ (1 x y x (y + 1) x)}
 
Theoremdfnn2 7677* Definition of the set of positive integers. Another name for df-inn 7676. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
ℕ = {x ∣ (1 x y x (y + 1) x)}
 
Theorempeano5nni 7678* Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
((1 A x A (x + 1) A) → ℕ ⊆ A)
 
Theoremnnssre 7679 The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
ℕ ⊆ ℝ
 
Theoremnnsscn 7680 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℕ ⊆ ℂ
 
Theoremnnex 7681 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
V
 
Theoremnnre 7682 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
(A ℕ → A ℝ)
 
Theoremnncn 7683 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
(A ℕ → A ℂ)
 
Theoremnnrei 7684 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
A        A
 
Theoremnncni 7685 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
A        A
 
Theorem1nn 7686 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
1
 
Theorempeano2nn 7687 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
(A ℕ → (A + 1) ℕ)
 
Theoremnnred 7688 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φA ℝ)
 
Theoremnncnd 7689 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φA ℂ)
 
Theorempeano2nnd 7690 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φ → (A + 1) ℕ)
 
3.4.2  Principle of mathematical induction
 
Theoremnnind 7691* Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 7695 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
(x = 1 → (φψ))    &   (x = y → (φχ))    &   (x = (y + 1) → (φθ))    &   (x = A → (φτ))    &   ψ    &   (y ℕ → (χθ))       (A ℕ → τ)
 
TheoremnnindALT 7692* Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction step and the basis.

This ALT version of nnind 7691 has a different hypothesis order. It may be easier to use with the metamath program's Proof Assistant, because "MM-PA> assign last" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> minimize nnind /allow". (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

(y ℕ → (χθ))    &   ψ    &   (x = 1 → (φψ))    &   (x = y → (φχ))    &   (x = (y + 1) → (φθ))    &   (x = A → (φτ))       (A ℕ → τ)
 
Theoremnn1m1nn 7693 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
(A ℕ → (A = 1 (A − 1) ℕ))
 
Theoremnn1suc 7694* If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
(x = 1 → (φψ))    &   (x = (y + 1) → (φχ))    &   (x = A → (φθ))    &   ψ    &   (y ℕ → χ)       (A ℕ → θ)
 
Theoremnnaddcl 7695 Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
((A B ℕ) → (A + B) ℕ)
 
Theoremnnmulcl 7696 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
((A B ℕ) → (A · B) ℕ)
 
Theoremnnmulcli 7697 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
A     &   B        (A · B)
 
Theoremnnge1 7698 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
(A ℕ → 1 ≤ A)
 
Theoremnnle1eq1 7699 A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)
(A ℕ → (A ≤ 1 ↔ A = 1))
 
Theoremnngt0 7700 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
(A ℕ → 0 < A)
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