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Theorem List for Metamath Proof Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnegfi 11301* 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 11302* A nonempty finite set of real numbers has a minimum. Analogous to fimaxre 11298. (Contributed by AV, 9-Aug-2020.)
((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)

Theoremlbreu 11303* 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 11304* 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 11305* 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 11306* 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 11307* 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 11308* 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 11309* 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 11310* 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 11311* Lemma for infm3 11312. (Contributed by NM, 14-Jun-2005.)
(𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ 𝑥 = -𝑦)

Theoreminfm3 11312* 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 11310.) (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 11313* 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 11314* 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, < ))

Theoremsuprubd 11315* Natural deduction form of suprubd 11315. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)    &   (𝜑𝐵𝐴)       (𝜑𝐵 ≤ sup(𝐴, ℝ, < ))

Theoremsuprcld 11316* Natural deduction form of suprcl 11313. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)       (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)

Theoremsuprlub 11317* 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 11318* 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 11319* 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 11320* The supremum function distributes over addition in a sense similar to that in supmul1 11322. (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 11321* The supremum function distributes over addition in a sense similar to that in supmul 11325. (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 11322* 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 11325 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 11323* Lemma for supmul 11325. (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 11324* Lemma for supmul 11325. (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 11325* 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 10121). (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 11326* 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 11327* 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 11328* 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 11329* 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 11330* 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 11331* 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 11332* 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 11333 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 11334* 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 11335* 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 11336* 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 11337* 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 11338* 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 11339 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 11340 The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
¬ i ∈ ℝ

Theoremrimul 11341 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 11342 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 11343 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 11344* 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 11345* 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 11346* 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 11347 Function analogue of subeq0 10628. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → ((𝐹𝑓𝐺) = (𝐴 × {0}) ↔ 𝐹 = 𝐺))

Theoremofnegsub 11348 Function analogue of negsub 10650. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → (𝐹𝑓 + ((𝐴 × {-1}) ∘𝑓 · 𝐺)) = (𝐹𝑓𝐺))

Theoremofsubge0 11349 Function analogue of subge0 10865. (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 11350 Extend class notation to include the class of positive integers.
class

Definitiondf-nn 11351 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 11355), in contrast to the more elementary ordinal natural numbers ω, df-om 7327). See nnind 11370 for the principle of mathematical induction. See df-n0 11619 for the set of nonnegative integers 0. See dfn2 11633 for defined in terms of 0.

This is a technical definition that helps us avoid the Axiom of Infinity ax-inf2 8815 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 11366 (or its slight variant dfnn2 11365). (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 3-May-2014.)

ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)

TheoremnnexALT 11352 Alternate proof of nnex 11357, more direct, that makes use of ax-rep 4994. (Contributed by Mario Carneiro, 3-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
ℕ ∈ V

Theorempeano5nni 11353* 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)

Theoremnnssre 11354 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 11355 The positive integers are a subset of the complex numbers. Remark: this could also be proven from nnssre 11354 and ax-resscn 10309 at the cost of using more axioms. (Contributed by NM, 2-Aug-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
ℕ ⊆ ℂ

TheoremnnsscnOLD 11356 Obsolete version of nnsscn 11355 as of 4-Oct-2022. The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
ℕ ⊆ ℂ

Theoremnnex 11357 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℕ ∈ V

Theoremnnre 11358 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℝ)

Theoremnncn 11359 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℂ)

Theoremnnrei 11360 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
𝐴 ∈ ℕ       𝐴 ∈ ℝ

Theoremnncni 11361 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
𝐴 ∈ ℕ       𝐴 ∈ ℂ

TheoremnncniOLD 11362 Obsolete version of nncni 11361 as of 4-Oct-2022. A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ ℕ       𝐴 ∈ ℂ

Theorem1nn 11363 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
1 ∈ ℕ

Theorempeano2nn 11364 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)

Theoremdfnn2 11365* Alternate definition of the set of positive integers. This was our original definition, before the current df-nn 11351 replaced it. This definition requires the axiom of infinity to ensure it has the properties we expect. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}

Theoremdfnn3 11366* Alternate definition of the set of positive integers. Definition of positive integers in [Apostol] p. 22. (Contributed by NM, 3-Jul-2005.)
ℕ = {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}

Theoremnnred 11367 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℝ)

Theoremnncnd 11368 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℂ)

Theorempeano2nnd 11369 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)

5.4.2  Principle of mathematical induction

Theoremnnind 11370* 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 11374 for an example of its use. See nn0ind 11800 for induction on nonnegative integers and uzind 11797, uzind4 12028 for induction on an arbitrary upper set of integers. See indstr 12039 for strong induction. See also nnindALT 11371. 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 11371* 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 11370 has a different hypothesis order. It may be easier to use with the Metamath program 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> MINIMIZEWITH nnind / MAYGROW";. (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 11372 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
(𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))

Theoremnn1suc 11373* 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 11374 Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)

Theoremnnmulcl 11375 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) Remove dependency on ax-mulcom 10316 and ax-mulass 10318. (Revised by Steven Nguyen, 24-Sep-2022.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)

TheoremnnmulclOLD 11376 Obsolete version of nnmulcl 11375 as of 24-Sep-2022. Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)

Theoremnnmulcli 11377 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ       (𝐴 · 𝐵) ∈ ℕ

Theoremnnmtmip 11378 "Minus times minus is plus, The reason for this we need not discuss." (W. H. Auden, as quoted in M. Guillen "Bridges to Infinity", p. 64, see also Metamath Book, section 1.1.1, p. 5) This statement, formalized to "The product of two negative integers is a positive integer", is proved by the following theorem, therefore it actually need not be discussed anymore. "The reason for this" is that (-𝐴 · -𝐵) = (𝐴 · 𝐵) for all complex numbers 𝐴 and 𝐵 because of mul2neg 10793, 𝐴 and 𝐵 are complex numbers because of nncn 11359, and (𝐴 · 𝐵) ∈ ℕ because of nnmulcl 11375. This also holds for positive reals, see rpmtmip 12138. Note that the opposites -𝐴 and -𝐵 of the positive integers 𝐴 and 𝐵 are negative integers. (Contributed by AV, 23-Dec-2022.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (-𝐴 · -𝐵) ∈ ℕ)

Theoremnn2ge 11379* There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ (𝐴𝑥𝐵𝑥))

Theoremnnge1 11380 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
(𝐴 ∈ ℕ → 1 ≤ 𝐴)

Theoremnngt1ne1 11381 A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
(𝐴 ∈ ℕ → (1 < 𝐴𝐴 ≠ 1))

Theoremnnle1eq1 11382 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 11383 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
(𝐴 ∈ ℕ → 0 < 𝐴)

Theoremnnnlt1 11384 A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℕ → ¬ 𝐴 < 1)

Theoremnnnle0 11385 A positive integer is not less than or equal to zero . (Contributed by AV, 13-May-2020.)
(𝐴 ∈ ℕ → ¬ 𝐴 ≤ 0)

Theoremnnne0 11386 A positive integer is nonzero. See nnne0ALT 11389 for a shorter proof using ax-pre-mulgt0 10329. This proof avoids 0lt1 10874, and thus ax-pre-mulgt0 10329, by splitting ax-1ne0 10321 into the two separate cases 0 < 1 and 1 < 0. (Contributed by NM, 27-Sep-1999.) Remove dependency on ax-pre-mulgt0 10329. (Revised by Steven Nguyen, 30-Jan-2023.)
(𝐴 ∈ ℕ → 𝐴 ≠ 0)

Theorem0nnn 11387 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) Remove dependency on ax-pre-mulgt0 10329. (Revised by Steven Nguyen, 30-Jan-2023.)
¬ 0 ∈ ℕ

Theorem0nnnALT 11388 Alternate proof of 0nnn 11387, which requires ax-pre-mulgt0 10329 but is not based on nnne0 11386 (and which can therefore be used in nnne0ALT 11389). (Contributed by NM, 25-Aug-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
¬ 0 ∈ ℕ

Theoremnnne0ALT 11389 Alternate version of nnne0 11386. A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 ∈ ℕ → 𝐴 ≠ 0)

Theoremnngt0i 11390 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
𝐴 ∈ ℕ       0 < 𝐴

Theoremnnne0i 11391 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℕ       𝐴 ≠ 0

Theoremnndivre 11392 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ)

Theoremnnrecre 11393 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
(𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)

Theoremnnrecgt0 11394 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
(𝐴 ∈ ℕ → 0 < (1 / 𝐴))

Theoremnnsub 11395 Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐵𝐴) ∈ ℕ))

Theoremnnsubi 11396 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ       (𝐴 < 𝐵 ↔ (𝐵𝐴) ∈ ℕ)

Theoremnndiv 11397* Two ways to express "𝐴 divides 𝐵 " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑥 ∈ ℕ (𝐴 · 𝑥) = 𝐵 ↔ (𝐵 / 𝐴) ∈ ℕ))

Theoremnndivtr 11398 Transitive property of divisibility: if 𝐴 divides 𝐵 and 𝐵 divides 𝐶, then 𝐴 divides 𝐶. Typically, 𝐶 would be an integer, although the theorem holds for complex 𝐶. (Contributed by NM, 3-May-2005.)
(((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) ∧ ((𝐵 / 𝐴) ∈ ℕ ∧ (𝐶 / 𝐵) ∈ ℕ)) → (𝐶 / 𝐴) ∈ ℕ)

Theoremnnge1d 11399 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 1 ≤ 𝐴)

Theoremnngt0d 11400 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 0 < 𝐴)

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