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Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprodge0 7601 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(((A B ℝ) (0 < A 0 ≤ (A · B))) → 0 ≤ B)
 
Theoremprodge02 7602 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)
(((A B ℝ) (0 < B 0 ≤ (A · B))) → 0 ≤ A)
 
Theoremltmul2 7603 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)
((A B (𝐶 0 < 𝐶)) → (A < B ↔ (𝐶 · A) < (𝐶 · B)))
 
Theoremlemul2 7604 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)
((A B (𝐶 0 < 𝐶)) → (AB ↔ (𝐶 · A) ≤ (𝐶 · B)))
 
Theoremlemul1a 7605 Multiplication of both sides of 'less than or equal to' by a nonnegative number. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 21-Feb-2005.)
(((A B (𝐶 0 ≤ 𝐶)) AB) → (A · 𝐶) ≤ (B · 𝐶))
 
Theoremlemul2a 7606 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
(((A B (𝐶 0 ≤ 𝐶)) AB) → (𝐶 · A) ≤ (𝐶 · B))
 
Theoremltmul12a 7607 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
((((A B ℝ) (0 ≤ A A < B)) ((𝐶 𝐷 ℝ) (0 ≤ 𝐶 𝐶 < 𝐷))) → (A · 𝐶) < (B · 𝐷))
 
Theoremlemul12b 7608 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
((((A 0 ≤ A) B ℝ) (𝐶 (𝐷 0 ≤ 𝐷))) → ((AB 𝐶𝐷) → (A · 𝐶) ≤ (B · 𝐷)))
 
Theoremlemul12a 7609 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
((((A 0 ≤ A) B ℝ) ((𝐶 0 ≤ 𝐶) 𝐷 ℝ)) → ((AB 𝐶𝐷) → (A · 𝐶) ≤ (B · 𝐷)))
 
Theoremmulgt1 7610 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)
(((A B ℝ) (1 < A 1 < B)) → 1 < (A · B))
 
Theoremltmulgt11 7611 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
((A B 0 < A) → (1 < BA < (A · B)))
 
Theoremltmulgt12 7612 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
((A B 0 < A) → (1 < BA < (B · A)))
 
Theoremlemulge11 7613 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)
(((A B ℝ) (0 ≤ A 1 ≤ B)) → A ≤ (A · B))
 
Theoremlemulge12 7614 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)
(((A B ℝ) (0 ≤ A 1 ≤ B)) → A ≤ (B · A))
 
Theoremltdiv1 7615 Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
((A B (𝐶 0 < 𝐶)) → (A < B ↔ (A / 𝐶) < (B / 𝐶)))
 
Theoremlediv1 7616 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)
((A B (𝐶 0 < 𝐶)) → (AB ↔ (A / 𝐶) ≤ (B / 𝐶)))
 
Theoremgt0div 7617 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)
((A B 0 < B) → (0 < A ↔ 0 < (A / B)))
 
Theoremge0div 7618 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)
((A B 0 < B) → (0 ≤ A ↔ 0 ≤ (A / B)))
 
Theoremdivgt0 7619 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)
(((A 0 < A) (B 0 < B)) → 0 < (A / B))
 
Theoremdivge0 7620 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)
(((A 0 ≤ A) (B 0 < B)) → 0 ≤ (A / B))
 
Theoremltmuldiv 7621 '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 7622 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
((A B (𝐶 0 < 𝐶)) → ((𝐶 · A) < BA < (B / 𝐶)))
 
Theoremltdivmul 7623 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
((A B (𝐶 0 < 𝐶)) → ((A / 𝐶) < BA < (𝐶 · B)))
 
Theoremledivmul 7624 '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 7625 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)
((A B (𝐶 0 < 𝐶)) → ((A / 𝐶) < BA < (B · 𝐶)))
 
Theoremlt2mul2div 7626 '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 7627 '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 7628 '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 7629 '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 7630 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 7631 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 7632 Lemma for lt2msq 7633. (Contributed by Mario Carneiro, 27-May-2016.)
(((A 0 ≤ A) B A < B) → (A · A) < (B · B))
 
Theoremlt2msq 7633 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 7634 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 7635 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 7636 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 7637 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 7638 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 7639 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 7640 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 7641 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 7642 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 7643 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 7644 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 7645 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 7646 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
((A 0 ≤ A) → (A / (1 + A)) < 1)
 
Theoremrecreclt 7647 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 7648 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 7649 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 7650 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 7651* 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 7652 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
A        A < (A + 1)
 
Theoremrecgt0i 7653 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 7654 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 7655 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 7656 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 7657 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
A     &   B        ((0 < A 0 < B) → 0 < (A / B))
 
Theoremdivge0i 7658 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 7659 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 7660 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 7661 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 7662 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 7663 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 7664 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
A     &   B     &   0 < B       (0 < A → 0 < (A / B))
 
Theoremltrecii 7665 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 7666 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
A     &   B     &   0 < A    &   0 < B       0 < (A / B)
 
Theoremltmul1i 7667 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 7668 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 7669 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
A     &   B     &   𝐶        (0 < 𝐶 → ((A · 𝐶) < BA < (B / 𝐶)))
 
Theoremltmul2i 7670 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 7671 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 7672 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 7673 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 7674 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 7675 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 7676 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 7677 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φA < (A + 1))
 
Theoremlep1d 7678 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φA ≤ (A + 1))
 
Theoremltm1d 7679 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φ → (A − 1) < A)
 
Theoremlem1d 7680 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 7681 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 7682 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 7683 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 7684 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 7685 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 7686 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 7687 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 7688 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 7689 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 7690 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 7691 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 7692* 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 7693* 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 7694* 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 7695 Extend class notation to include the class of positive integers.
class
 
Definitiondf-inn 7696* Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 7697 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 7697* Definition of the set of positive integers. Another name for df-inn 7696. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
ℕ = {x ∣ (1 x y x (y + 1) x)}
 
Theorempeano5nni 7698* 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 7699 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 7700 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℕ ⊆ ℂ
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