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Theorem List for Metamath Proof Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoreminfrecl 11601* 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 11602* 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 11603* 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 11604* 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 11605 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 11606 The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
¬ i ∈ ℝ

Theoremrimul 11607 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 11608 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 11609 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 11610* 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 11611* 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 11612* 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 11613 Function analogue of subeq0 10890. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → ((𝐹f𝐺) = (𝐴 × {0}) ↔ 𝐹 = 𝐺))

Theoremofnegsub 11614 Function analogue of negsub 10912. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → (𝐹f + ((𝐴 × {-1}) ∘f · 𝐺)) = (𝐹f𝐺))

Theoremofsubge0 11615 Function analogue of subge0 11131. (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 11616 Extend class notation to include the class of positive integers.
class

Definitiondf-nn 11617 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 11621), in contrast to the more elementary ordinal natural numbers ω, df-om 7559). See nnind 11634 for the principle of mathematical induction. See df-n0 11877 for the set of nonnegative integers 0. See dfn2 11889 for defined in terms of 0.

This is a technical definition that helps us avoid the Axiom of Infinity ax-inf2 9082 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 11630 (or its slight variant dfnn2 11629). (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 3-May-2014.)

ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)

TheoremnnexALT 11618 Alternate proof of nnex 11622, more direct, that makes use of ax-rep 5166. (Contributed by Mario Carneiro, 3-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
ℕ ∈ V

Theorempeano5nni 11619* 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 11620 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 11621 The positive integers are a subset of the complex numbers. Remark: this could also be proven from nnssre 11620 and ax-resscn 10572 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

Theoremnnex 11622 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℕ ∈ V

Theoremnnre 11623 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℝ)

Theoremnncn 11624 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℂ)

Theoremnnrei 11625 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
𝐴 ∈ ℕ       𝐴 ∈ ℝ

Theoremnncni 11626 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

Theorem1nn 11627 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
1 ∈ ℕ

Theorempeano2nn 11628 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 11629* Alternate definition of the set of positive integers. This was our original definition, before the current df-nn 11617 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 11630* 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 11631 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℝ)

Theoremnncnd 11632 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℂ)

Theorempeano2nnd 11633 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 11634* 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 11639 for an example of its use. See nn0ind 12056 for induction on nonnegative integers and uzind 12053, uzind4 12285 for induction on an arbitrary upper set of integers. See indstr 12295 for strong induction. See also nnindALT 11635. 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 11635* 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 11634 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> MINIMIZE_WITH 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)

Theoremnnindd 11636* Principle of Mathematical Induction (inference schema) on integers, a deduction version. (Contributed by Thierry Arnoux, 19-Jul-2020.)
(𝑥 = 1 → (𝜓𝜒))    &   (𝑥 = 𝑦 → (𝜓𝜃))    &   (𝑥 = (𝑦 + 1) → (𝜓𝜏))    &   (𝑥 = 𝐴 → (𝜓𝜂))    &   (𝜑𝜒)    &   (((𝜑𝑦 ∈ ℕ) ∧ 𝜃) → 𝜏)       ((𝜑𝐴 ∈ ℕ) → 𝜂)

Theoremnn1m1nn 11637 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
(𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))

Theoremnn1suc 11638* 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 11639 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 11640 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) Remove dependency on ax-mulcom 10579 and ax-mulass 10581. (Revised by Steven Nguyen, 24-Sep-2022.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)

Theoremnnmulcli 11641 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ       (𝐴 · 𝐵) ∈ ℕ

Theoremnnmtmip 11642 "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 11057, 𝐴 and 𝐵 are complex numbers because of nncn 11624, and (𝐴 · 𝐵) ∈ ℕ because of nnmulcl 11640. This also holds for positive reals, see rpmtmip 12392. 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 11643* 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 11644 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
(𝐴 ∈ ℕ → 1 ≤ 𝐴)

Theoremnngt1ne1 11645 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 11646 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 11647 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
(𝐴 ∈ ℕ → 0 < 𝐴)

Theoremnnnlt1 11648 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 11649 A positive integer is not less than or equal to zero . (Contributed by AV, 13-May-2020.)
(𝐴 ∈ ℕ → ¬ 𝐴 ≤ 0)

Theoremnnne0 11650 A positive integer is nonzero. See nnne0ALT 11654 for a shorter proof using ax-pre-mulgt0 10592. This proof avoids 0lt1 11140, and thus ax-pre-mulgt0 10592, by splitting ax-1ne0 10584 into the two separate cases 0 < 1 and 1 < 0. (Contributed by NM, 27-Sep-1999.) Remove dependency on ax-pre-mulgt0 10592. (Revised by Steven Nguyen, 30-Jan-2023.)
(𝐴 ∈ ℕ → 𝐴 ≠ 0)

Theoremnnneneg 11651 No positive integer is equal to its negation. (Contributed by AV, 20-Jun-2023.)
(𝐴 ∈ ℕ → 𝐴 ≠ -𝐴)

Theorem0nnn 11652 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) Remove dependency on ax-pre-mulgt0 10592. (Revised by Steven Nguyen, 30-Jan-2023.)
¬ 0 ∈ ℕ

Theorem0nnnALT 11653 Alternate proof of 0nnn 11652, which requires ax-pre-mulgt0 10592 but is not based on nnne0 11650 (and which can therefore be used in nnne0ALT 11654). (Contributed by NM, 25-Aug-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
¬ 0 ∈ ℕ

Theoremnnne0ALT 11654 Alternate version of nnne0 11650. A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 ∈ ℕ → 𝐴 ≠ 0)

Theoremnngt0i 11655 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
𝐴 ∈ ℕ       0 < 𝐴

Theoremnnne0i 11656 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℕ       𝐴 ≠ 0

Theoremnndivre 11657 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 11658 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
(𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)

Theoremnnrecgt0 11659 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
(𝐴 ∈ ℕ → 0 < (1 / 𝐴))

Theoremnnsub 11660 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 11661 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ       (𝐴 < 𝐵 ↔ (𝐵𝐴) ∈ ℕ)

Theoremnndiv 11662* 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 11663 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 11664 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 1 ≤ 𝐴)

Theoremnngt0d 11665 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 0 < 𝐴)

Theoremnnne0d 11666 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ≠ 0)

Theoremnnrecred 11667 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → (1 / 𝐴) ∈ ℝ)

Theoremnnaddcld 11668 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 + 𝐵) ∈ ℕ)

Theoremnnmulcld 11669 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 · 𝐵) ∈ ℕ)

Theoremnndivred 11670 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 / 𝐵) ∈ ℝ)

5.4.3  Decimal representation of numbers

The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 10522 through df-9 11686), which are explicitly defined in the following. Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 10522 and df-1 10523).

With the decimal constructor df-dec 12078, it is possible to easily express larger integers in base 10. See deccl 12092 and the theorems that follow it. See also 4001prm 16457 (4001 is prime) and the proof of bpos 25856. Note that the decimal constructor builds on the definitions in this section.

Note: The number 10 will be represented by its digits using the decimal constructor only, i.e., by 10. Therefore, only decimal digits are needed (as symbols) for the decimal representation of a number.

Integers can also be exhibited as sums of powers of 10 (e.g. the number 103 can be expressed as ((10↑2) + 3)) or as some other expression built from operations on the numbers 0 through 9. For example, the prime number 823541 can be expressed as (7↑7) − 2. Decimals can be expressed as ratios of integers, as in cos2bnd 15521.

Most abstract math rarely requires numbers larger than 4. Even in Wiles' proof of Fermat's Last Theorem, the largest number used appears to be 12.

Syntaxc2 11671 Extend class notation to include the number 2.
class 2

Syntaxc3 11672 Extend class notation to include the number 3.
class 3

Syntaxc4 11673 Extend class notation to include the number 4.
class 4

Syntaxc5 11674 Extend class notation to include the number 5.
class 5

Syntaxc6 11675 Extend class notation to include the number 6.
class 6

Syntaxc7 11676 Extend class notation to include the number 7.
class 7

Syntaxc8 11677 Extend class notation to include the number 8.
class 8

Syntaxc9 11678 Extend class notation to include the number 9.
class 9

Definitiondf-2 11679 Define the number 2. (Contributed by NM, 27-May-1999.)
2 = (1 + 1)

Definitiondf-3 11680 Define the number 3. (Contributed by NM, 27-May-1999.)
3 = (2 + 1)

Definitiondf-4 11681 Define the number 4. (Contributed by NM, 27-May-1999.)
4 = (3 + 1)

Definitiondf-5 11682 Define the number 5. (Contributed by NM, 27-May-1999.)
5 = (4 + 1)

Definitiondf-6 11683 Define the number 6. (Contributed by NM, 27-May-1999.)
6 = (5 + 1)

Definitiondf-7 11684 Define the number 7. (Contributed by NM, 27-May-1999.)
7 = (6 + 1)

Definitiondf-8 11685 Define the number 8. (Contributed by NM, 27-May-1999.)
8 = (7 + 1)

Definitiondf-9 11686 Define the number 9. (Contributed by NM, 27-May-1999.)
9 = (8 + 1)

Theorem0ne1 11687 Zero is different from one (the commuted form is the axiom ax-1ne0 10584). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 1

Theorem1m1e0 11688 One minus one equals zero. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 − 1) = 0

Theorem2nn 11689 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
2 ∈ ℕ

Theorem2re 11690 The number 2 is real. (Contributed by NM, 27-May-1999.)
2 ∈ ℝ

Theorem2cn 11691 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
2 ∈ ℂ

Theorem2cnALT 11692 Alternate proof of 2cn 11691. Shorter but uses more axioms. Similar proofs are possible for 3cn 11697, ... , 9cn 11716. (Contributed by NM, 30-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
2 ∈ ℂ

Theorem2ex 11693 The number 2 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
2 ∈ V

Theorem2cnd 11694 The number 2 is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 2 ∈ ℂ)

Theorem3nn 11695 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
3 ∈ ℕ

Theorem3re 11696 The number 3 is real. (Contributed by NM, 27-May-1999.)
3 ∈ ℝ

Theorem3cn 11697 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
3 ∈ ℂ

Theorem3ex 11698 The number 3 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
3 ∈ V

Theorem4nn 11699 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
4 ∈ ℕ

Theorem4re 11700 The number 4 is real. (Contributed by NM, 27-May-1999.)
4 ∈ ℝ

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