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Theorem List for Metamath Proof Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrecreclt 10801 Given a positive number 𝐴, construct a new positive number less than both 𝐴 and 1. (Contributed by NM, 28-Dec-2005.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((1 / (1 + (1 / 𝐴))) < 1 ∧ (1 / (1 + (1 / 𝐴))) < 𝐴))

Theoremle2msq 10802 The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵)))

Theoremmsq11 10803 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 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 · 𝐴) = (𝐵 · 𝐵) ↔ 𝐴 = 𝐵))

Theoremledivp1 10804 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 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 / (𝐵 + 1)) · 𝐵) ≤ 𝐴)

Theoremsqueeze0 10805* If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → 𝐴 = 0)

Theoremltp1i 10806 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
𝐴 ∈ ℝ       𝐴 < (𝐴 + 1)

Theoremrecgt0i 10807 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
𝐴 ∈ ℝ       (0 < 𝐴 → 0 < (1 / 𝐴))

Theoremrecgt0ii 10808 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
𝐴 ∈ ℝ    &   0 < 𝐴       0 < (1 / 𝐴)

Theoremprodgt0i 10809 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 < (𝐴 · 𝐵)) → 0 < 𝐵)

Theoremprodge0i 10810 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 < 𝐴 ∧ 0 ≤ (𝐴 · 𝐵)) → 0 ≤ 𝐵)

Theoremdivgt0i 10811 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 / 𝐵))

Theoremdivge0i 10812 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 < 𝐵) → 0 ≤ (𝐴 / 𝐵))

Theoremltreci 10813 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 < 𝐴 ∧ 0 < 𝐵) → (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴)))

Theoremlereci 10814 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 < 𝐴 ∧ 0 < 𝐵) → (𝐴𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴)))

Theoremlt2msqi 10815 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐴) < (𝐵 · 𝐵)))

Theoremle2msqi 10816 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵)))

Theoremmsq11i 10817 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((𝐴 · 𝐴) = (𝐵 · 𝐵) ↔ 𝐴 = 𝐵))

Theoremdivgt0i2i 10818 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   0 < 𝐵       (0 < 𝐴 → 0 < (𝐴 / 𝐵))

Theoremltrecii 10819 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   0 < 𝐴    &   0 < 𝐵       (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴))

Theoremdivgt0ii 10820 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   0 < 𝐴    &   0 < 𝐵       0 < (𝐴 / 𝐵)

Theoremltmul1i 10821 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 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))

Theoremltdiv1i 10822 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶)))

Theoremltmuldivi 10823 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → ((𝐴 · 𝐶) < 𝐵𝐴 < (𝐵 / 𝐶)))

Theoremltmul2i 10824 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 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵)))

Theoremlemul1i 10825 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → (𝐴𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)))

Theoremlemul2i 10826 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → (𝐴𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)))

Theoremltdiv23i 10827 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((0 < 𝐵 ∧ 0 < 𝐶) → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵))

Theoremledivp1i 10828 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐶𝐴 ≤ (𝐵 / (𝐶 + 1))) → (𝐴 · 𝐶) ≤ 𝐵)

Theoremltdivp1i 10829 Less-than and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐶𝐴 < (𝐵 / (𝐶 + 1))) → (𝐴 · 𝐶) < 𝐵)

Theoremltdiv23ii 10830 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   0 < 𝐵    &   0 < 𝐶       ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵)

Theoremltmul1ii 10831 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 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   0 < 𝐶       (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶))

Theoremltdiv1ii 10832 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   0 < 𝐶       (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))

Theoremltp1d 10833 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 < (𝐴 + 1))

Theoremlep1d 10834 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≤ (𝐴 + 1))

Theoremltm1d 10835 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 − 1) < 𝐴)

Theoremlem1d 10836 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 10837 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 10838 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 10839 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 10840 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 10841 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 10842 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 10843 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 10844 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 10845 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 10846 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))

5.3.8  Completeness Axiom and Suprema

Theoremfimaxre 10847* A finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑦𝑥)

Theoremfimaxre2 10848* A nonempty finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.)
((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)

Theoremfimaxre3 10849* A nonempty finite set of real numbers has a maximum (image set version). (Contributed by Mario Carneiro, 13-Feb-2014.)
((𝐴 ∈ Fin ∧ ∀𝑦𝐴 𝐵 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝐵𝑥)

Theoremnegfi 10850* The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.)
((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → {𝑛 ∈ ℝ ∣ -𝑛𝐴} ∈ Fin)

Theoremfiminre 10851* A nonempty finite set of real numbers has a minimum. Analogous to fimaxre 10847. (Contributed by AV, 9-Aug-2020.)
((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)

Theoremlbreu 10852* If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → ∃!𝑥𝑆𝑦𝑆 𝑥𝑦)

Theoremlbcl 10853* If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → (𝑥𝑆𝑦𝑆 𝑥𝑦) ∈ 𝑆)

Theoremlble 10854* If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦𝐴𝑆) → (𝑥𝑆𝑦𝑆 𝑥𝑦) ≤ 𝐴)

Theoremlbinf 10855* If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → inf(𝑆, ℝ, < ) = (𝑥𝑆𝑦𝑆 𝑥𝑦))

Theoremlbinfcl 10856* If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → inf(𝑆, ℝ, < ) ∈ 𝑆)

Theoremlbinfle 10857* If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦𝐴𝑆) → inf(𝑆, ℝ, < ) ≤ 𝐴)

Theoremsup2 10858* A nonempty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent). (Contributed by NM, 19-Jan-1997.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 (𝑦 < 𝑥𝑦 = 𝑥)) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

Theoremsup3 10859* A version of the completeness axiom for reals. (Contributed by NM, 12-Oct-2004.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

Theoreminfm3lem 10860* Lemma for infm3 10861. (Contributed by NM, 14-Jun-2005.)
(𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ 𝑥 = -𝑦)

Theoreminfm3 10861* The completeness axiom for reals in terms of infimum: a nonempty, bounded-below set of reals has an infimum. (This theorem is the dual of sup3 10859.) (Contributed by NM, 14-Jun-2005.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))

Theoremsuprcl 10862* Closure of supremum of a nonempty bounded set of reals. (Contributed by NM, 12-Oct-2004.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ)

Theoremsuprub 10863* A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Oct-2004.)
(((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) ∧ 𝐵𝐴) → 𝐵 ≤ sup(𝐴, ℝ, < ))

Theoremsuprlub 10864* The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)
(((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) ∧ 𝐵 ∈ ℝ) → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧𝐴 𝐵 < 𝑧))

Theoremsuprnub 10865* An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)
(((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) ∧ 𝐵 ∈ ℝ) → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧))

Theoremsuprleub 10866* The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
(((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) ∧ 𝐵 ∈ ℝ) → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧𝐴 𝑧𝐵))

Theoremsupaddc 10867* The supremum function distributes over addition in a sense similar to that in supmul1 10869. (Contributed by Brendan Leahy, 25-Sep-2017.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)    &   (𝜑𝐵 ∈ ℝ)    &   𝐶 = {𝑧 ∣ ∃𝑣𝐴 𝑧 = (𝑣 + 𝐵)}       (𝜑 → (sup(𝐴, ℝ, < ) + 𝐵) = sup(𝐶, ℝ, < ))

Theoremsupadd 10868* The supremum function distributes over addition in a sense similar to that in supmul 10872. (Contributed by Brendan Leahy, 26-Sep-2017.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)    &   (𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐵 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐵 𝑦𝑥)    &   𝐶 = {𝑧 ∣ ∃𝑣𝐴𝑏𝐵 𝑧 = (𝑣 + 𝑏)}       (𝜑 → (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) = sup(𝐶, ℝ, < ))

Theoremsupmul1 10869* The supremum function distributes over multiplication, in the sense that 𝐴 · (sup𝐵) = sup(𝐴 · 𝐵), where 𝐴 · 𝐵 is shorthand for {𝐴 · 𝑏𝑏𝐵} and is defined as 𝐶 below. This is the simple version, with only one set argument; see supmul 10872 for the more general case with two set arguments. (Contributed by Mario Carneiro, 5-Jul-2013.)
𝐶 = {𝑧 ∣ ∃𝑣𝐵 𝑧 = (𝐴 · 𝑣)}    &   (𝜑 ↔ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥𝐵 0 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐵 𝑦𝑥)))       (𝜑 → (𝐴 · sup(𝐵, ℝ, < )) = sup(𝐶, ℝ, < ))

Theoremsupmullem1 10870* Lemma for supmul 10872. (Contributed by Mario Carneiro, 5-Jul-2013.)
𝐶 = {𝑧 ∣ ∃𝑣𝐴𝑏𝐵 𝑧 = (𝑣 · 𝑏)}    &   (𝜑 ↔ ((∀𝑥𝐴 0 ≤ 𝑥 ∧ ∀𝑥𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐵 𝑦𝑥)))       (𝜑 → ∀𝑤𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, < )))

Theoremsupmullem2 10871* Lemma for supmul 10872. (Contributed by Mario Carneiro, 5-Jul-2013.)
𝐶 = {𝑧 ∣ ∃𝑣𝐴𝑏𝐵 𝑧 = (𝑣 · 𝑏)}    &   (𝜑 ↔ ((∀𝑥𝐴 0 ≤ 𝑥 ∧ ∀𝑥𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐵 𝑦𝑥)))       (𝜑 → (𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤𝐶 𝑤𝑥))

Theoremsupmul 10872* The supremum function distributes over multiplication, in the sense that (sup𝐴) · (sup𝐵) = sup(𝐴 · 𝐵), where 𝐴 · 𝐵 is shorthand for {𝑎 · 𝑏𝑎𝐴, 𝑏𝐵} and is defined as 𝐶 below. We made use of this in our definition of multiplication in the Dedekind cut construction of the reals (see df-mp 9685). (Contributed by Mario Carneiro, 5-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2014.)
𝐶 = {𝑧 ∣ ∃𝑣𝐴𝑏𝐵 𝑧 = (𝑣 · 𝑏)}    &   (𝜑 ↔ ((∀𝑥𝐴 0 ≤ 𝑥 ∧ ∀𝑥𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐵 𝑦𝑥)))       (𝜑 → (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, < )) = sup(𝐶, ℝ, < ))

Theoremsup3ii 10873* A version of the completeness axiom for reals. (Contributed by NM, 23-Aug-1999.)
(𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)       𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))

Theoremsuprclii 10874* Closure of supremum of a nonempty bounded set of reals. (Contributed by NM, 12-Sep-1999.)
(𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)       sup(𝐴, ℝ, < ) ∈ ℝ

Theoremsuprubii 10875* A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Sep-1999.)
(𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)       (𝐵𝐴𝐵 ≤ sup(𝐴, ℝ, < ))

Theoremsuprlubii 10876* The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by NM, 15-Oct-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)
(𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)       (𝐵 ∈ ℝ → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧𝐴 𝐵 < 𝑧))

Theoremsuprnubii 10877* An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by NM, 15-Oct-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)
(𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)       (𝐵 ∈ ℝ → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧))

Theoremsuprleubii 10878* The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
(𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)       (𝐵 ∈ ℝ → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧𝐴 𝑧𝐵))

Theoremriotaneg 10879* The negative of the unique real such that 𝜑. (Contributed by NM, 13-Jun-2005.)
(𝑥 = -𝑦 → (𝜑𝜓))       (∃!𝑥 ∈ ℝ 𝜑 → (𝑥 ∈ ℝ 𝜑) = -(𝑦 ∈ ℝ 𝜓))

Theoremnegiso 10880 Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝐹 = (𝑥 ∈ ℝ ↦ -𝑥)       (𝐹 Isom < , < (ℝ, ℝ) ∧ 𝐹 = 𝐹)

Theoremdfinfre 10881* The infimum of a set of reals 𝐴. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
(𝐴 ⊆ ℝ → inf(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 𝑥𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))})

Theoreminfrecl 10882* Closure of infimum of a nonempty bounded set of reals. (Contributed by NM, 8-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) → inf(𝐴, ℝ, < ) ∈ ℝ)

Theoreminfrenegsup 10883* The infimum of a set of reals 𝐴 is the negative of the supremum of the negatives of its elements. The antecedent ensures that 𝐴 is nonempty and has a lower bound. (Contributed by NM, 14-Jun-2005.) (Revised by AV, 4-Sep-2020.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧𝐴}, ℝ, < ))

Theoreminfregelb 10884* Any lower bound of a nonempty set of real numbers is less than or equal to its infimum. (Contributed by Jeff Hankins, 1-Sep-2013.) (Revised by AV, 4-Sep-2020.) (Proof modification is discouraged.)
(((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧𝐴 𝐵𝑧))

Theoreminfrelb 10885* If a nonempty set of real numbers has a lower bound, its infimum is less than or equal to any of its elements. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by AV, 4-Sep-2020.)
((𝐵 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐵 𝑥𝑦𝐴𝐵) → inf(𝐵, ℝ, < ) ≤ 𝐴)

Theoremsupfirege 10886 The supremum of a finite set of real numbers is greater than or equal to all the real numbers of the set. (Contributed by AV, 1-Oct-2019.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐶𝐵)    &   (𝜑𝑆 = sup(𝐵, ℝ, < ))       (𝜑𝐶𝑆)

5.3.9  Imaginary and complex number properties

Theoreminelr 10887 The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
¬ i ∈ ℝ

Theoremrimul 10888 A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) → 𝐴 = 0)

Theoremcru 10889 The representation of complex numbers in terms of real and imaginary parts 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 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) = (𝐶 + (i · 𝐷)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremcrne0 10890 The real representation of complex numbers is nonzero iff one of its terms is nonzero. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ (𝐴 + (i · 𝐵)) ≠ 0))

Theoremcreur 10891* 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 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))

Theoremcreui 10892* 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 ∈ ℝ ∃𝑥 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))

Theoremcju 10893* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ))

5.3.10  Function operation analogue theorems

Theoremofsubeq0 10894 Function analogue of subeq0 10186. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → ((𝐹𝑓𝐺) = (𝐴 × {0}) ↔ 𝐹 = 𝐺))

Theoremofnegsub 10895 Function analogue of negsub 10208. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → (𝐹𝑓 + ((𝐴 × {-1}) ∘𝑓 · 𝐺)) = (𝐹𝑓𝐺))

Theoremofsubge0 10896 Function analogue of subge0 10420. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴𝑉𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ) → ((𝐴 × {0}) ∘𝑟 ≤ (𝐹𝑓𝐺) ↔ 𝐺𝑟𝐹))

5.4  Integer sets

5.4.1  Positive integers (as a subset of complex numbers)

Syntaxcn 10897 Extend class notation to include the class of positive integers.
class

Definitiondf-nn 10898 Define the set of positive integers. Some authors, especially in analysis books, call these the natural numbers, whereas other authors choose to include 0 in their definition of natural numbers. Note that is a subset of complex numbers (nnsscn 10902), in contrast to the more elementary ordinal natural numbers ω, df-om 6958). See nnind 10915 for the principle of mathematical induction. See df-n0 11170 for the set of nonnegative integers 0. See dfn2 11182 for defined in terms of 0.

This is a technical definition that helps us avoid the Axiom of Infinity ax-inf2 8421 in certain proofs. For a more conventional and intuitive definition ("the smallest set of reals containing 1 as well as the successor of every member") see dfnn3 10911 (or its slight variant dfnn2 10910). (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 3-May-2014.)

ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)

TheoremnnexALT 10899 Alternate proof of nnex 10903, more direct, that makes use of ax-rep 4699. (Contributed by Mario Carneiro, 3-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
ℕ ∈ V

Theorempeano5nni 10900* 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 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
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