P. 124 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12301-12400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnneneg 12301	No positive integer is equal to its negation. (Contributed by AV, 20-Jun-2023.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ -𝐴)
Theorem	0nnn 12302	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) Remove dependency on ax-pre-mulgt0 11232. (Revised by Steven Nguyen, 30-Jan-2023.)
⊢ ¬ 0 ∈ ℕ
Theorem	0nnnALT 12303	Alternate proof of 0nnn 12302, which requires ax-pre-mulgt0 11232 but is not based on nnne0 12300 (and which can therefore be used in nnne0ALT 12304). (Contributed by NM, 25-Aug-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ¬ 0 ∈ ℕ
Theorem	nnne0ALT 12304	Alternate version of nnne0 12300. A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
Theorem	nngt0i 12305	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 0 < 𝐴
Theorem	nnne0i 12306	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ≠ 0
Theorem	nndivre 12307	The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ)
Theorem	nnrecre 12308	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)
Theorem	nnrecgt0 12309	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 0 < (1 / 𝐴))
Theorem	nnsub 12310	Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℕ))
Theorem	nnsubi 12311	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℕ)
Theorem	nndiv 12312*	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))
Theorem	nndivtr 12313	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)
Theorem	nnge1d 12314	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 1 ≤ 𝐴)
Theorem	nngt0d 12315	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 0 < 𝐴)
Theorem	nnne0d 12316	A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0)
Theorem	nnrecred 12317	The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
Theorem	nnaddcld 12318	Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ)
Theorem	nnmulcld 12319	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ)
Theorem	nndivred 12320	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ 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 11162 through df-9 12336), 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 11162 and df-1 11163). With the decimal constructor df-dec 12734, it is possible to easily express larger integers in base 10. See deccl 12748 and the theorems that follow it. See also 4001prm 17182 (4001 is prime) and the proof of bpos 27337. 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 16224. 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.
Syntax	c2 12321	Extend class notation to include the number 2.
class 2
Syntax	c3 12322	Extend class notation to include the number 3.
class 3
Syntax	c4 12323	Extend class notation to include the number 4.
class 4
Syntax	c5 12324	Extend class notation to include the number 5.
class 5
Syntax	c6 12325	Extend class notation to include the number 6.
class 6
Syntax	c7 12326	Extend class notation to include the number 7.
class 7
Syntax	c8 12327	Extend class notation to include the number 8.
class 8
Syntax	c9 12328	Extend class notation to include the number 9.
class 9
Definition	df-2 12329	Define the number 2. (Contributed by NM, 27-May-1999.)
⊢ 2 = (1 + 1)
Definition	df-3 12330	Define the number 3. (Contributed by NM, 27-May-1999.)
⊢ 3 = (2 + 1)
Definition	df-4 12331	Define the number 4. (Contributed by NM, 27-May-1999.)
⊢ 4 = (3 + 1)
Definition	df-5 12332	Define the number 5. (Contributed by NM, 27-May-1999.)
⊢ 5 = (4 + 1)
Definition	df-6 12333	Define the number 6. (Contributed by NM, 27-May-1999.)
⊢ 6 = (5 + 1)
Definition	df-7 12334	Define the number 7. (Contributed by NM, 27-May-1999.)
⊢ 7 = (6 + 1)
Definition	df-8 12335	Define the number 8. (Contributed by NM, 27-May-1999.)
⊢ 8 = (7 + 1)
Definition	df-9 12336	Define the number 9. (Contributed by NM, 27-May-1999.)
⊢ 9 = (8 + 1)
Theorem	0ne1 12337	Zero is different from one (the commuted form is Axiom ax-1ne0 11224). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 1
Theorem	1m1e0 12338	One minus one equals zero. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 − 1) = 0
Theorem	2nn 12339	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
⊢ 2 ∈ ℕ
Theorem	2re 12340	The number 2 is real. (Contributed by NM, 27-May-1999.)
⊢ 2 ∈ ℝ
Theorem	2cn 12341	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
Theorem	2cnALT 12342	Alternate proof of 2cn 12341. Shorter but uses more axioms. Similar proofs are possible for 3cn 12347, ... , 9cn 12366. (Contributed by NM, 30-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 2 ∈ ℂ
Theorem	2ex 12343	The number 2 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 2 ∈ V
Theorem	2cnd 12344	The number 2 is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 2 ∈ ℂ)
Theorem	3nn 12345	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 3 ∈ ℕ
Theorem	3re 12346	The number 3 is real. (Contributed by NM, 27-May-1999.)
⊢ 3 ∈ ℝ
Theorem	3cn 12347	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
Theorem	3ex 12348	The number 3 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 3 ∈ V
Theorem	4nn 12349	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 4 ∈ ℕ
Theorem	4re 12350	The number 4 is real. (Contributed by NM, 27-May-1999.)
⊢ 4 ∈ ℝ
Theorem	4cn 12351	The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 4 ∈ ℂ
Theorem	5nn 12352	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 ∈ ℕ
Theorem	5re 12353	The number 5 is real. (Contributed by NM, 27-May-1999.)
⊢ 5 ∈ ℝ
Theorem	5cn 12354	The number 5 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 5 ∈ ℂ
Theorem	6nn 12355	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 ∈ ℕ
Theorem	6re 12356	The number 6 is real. (Contributed by NM, 27-May-1999.)
⊢ 6 ∈ ℝ
Theorem	6cn 12357	The number 6 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 6 ∈ ℂ
Theorem	7nn 12358	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 ∈ ℕ
Theorem	7re 12359	The number 7 is real. (Contributed by NM, 27-May-1999.)
⊢ 7 ∈ ℝ
Theorem	7cn 12360	The number 7 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 7 ∈ ℂ
Theorem	8nn 12361	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 8 ∈ ℕ
Theorem	8re 12362	The number 8 is real. (Contributed by NM, 27-May-1999.)
⊢ 8 ∈ ℝ
Theorem	8cn 12363	The number 8 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 8 ∈ ℂ
Theorem	9nn 12364	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
⊢ 9 ∈ ℕ
Theorem	9re 12365	The number 9 is real. (Contributed by NM, 27-May-1999.)
⊢ 9 ∈ ℝ
Theorem	9cn 12366	The number 9 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 9 ∈ ℂ
Theorem	0le0 12367	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ≤ 0
Theorem	0le2 12368	The number 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ 0 ≤ 2
Theorem	2pos 12369	The number 2 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 2
Theorem	2ne0 12370	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
⊢ 2 ≠ 0
Theorem	3pos 12371	The number 3 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 3
Theorem	3ne0 12372	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ 3 ≠ 0
Theorem	4pos 12373	The number 4 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 4
Theorem	4ne0 12374	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ 4 ≠ 0
Theorem	5pos 12375	The number 5 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 5
Theorem	6pos 12376	The number 6 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 6
Theorem	7pos 12377	The number 7 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 7
Theorem	8pos 12378	The number 8 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 8
Theorem	9pos 12379	The number 9 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 9
5.4.4  Some properties of specific numbers This section includes specific theorems about one-digit natural numbers (membership, addition, subtraction, multiplication, division, ordering).
Theorem	neg1cn 12380	-1 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ -1 ∈ ℂ
Theorem	neg1rr 12381	-1 is a real number. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ -1 ∈ ℝ
Theorem	neg1ne0 12382	-1 is nonzero. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -1 ≠ 0
Theorem	neg1lt0 12383	-1 is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -1 < 0
Theorem	negneg1e1 12384	--1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ --1 = 1
Theorem	1pneg1e0 12385	1 + -1 is 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + -1) = 0
Theorem	0m0e0 12386	0 minus 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (0 − 0) = 0
Theorem	1m0e1 12387	1 - 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 − 0) = 1
Theorem	0p1e1 12388	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (0 + 1) = 1
Theorem	fv0p1e1 12389	Function value at 𝑁 + 1 with 𝑁 replaced by 0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
⊢ (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1))
Theorem	1p0e1 12390	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + 0) = 1
Theorem	1p1e2 12391	1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
⊢ (1 + 1) = 2
Theorem	2m1e1 12392	2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 12421. (Contributed by David A. Wheeler, 4-Jan-2017.)
⊢ (2 − 1) = 1
Theorem	1e2m1 12393	1 = 2 - 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 = (2 − 1)
Theorem	3m1e2 12394	3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.) (Proof shortened by AV, 6-Sep-2021.)
⊢ (3 − 1) = 2
Theorem	4m1e3 12395	4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
⊢ (4 − 1) = 3
Theorem	5m1e4 12396	5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
⊢ (5 − 1) = 4
Theorem	6m1e5 12397	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
⊢ (6 − 1) = 5
Theorem	7m1e6 12398	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
⊢ (7 − 1) = 6
Theorem	8m1e7 12399	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
⊢ (8 − 1) = 7
Theorem	9m1e8 12400	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
⊢ (9 − 1) = 8