P. 123 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12201-12300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnaddcom 12201	Addition is commutative for natural numbers. Uses fewer axioms than addcom 11332. (Contributed by Steven Nguyen, 9-Dec-2022.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	nnaddcomli 12202	Version of addcomli 11338 for natural numbers. (Contributed by Steven Nguyen, 1-Aug-2023.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    &   ⊢ (𝐴 + 𝐵) = 𝐶    ⇒   ⊢ (𝐵 + 𝐴) = 𝐶
Theorem	nnmtmip 12203	"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 11589, 𝐴 and 𝐵 are complex numbers because of nncn 12182, and (𝐴 · 𝐵) ∈ ℕ because of nnmulcl 12198. This also holds for positive reals, see rpmtmip 12968. 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)
Theorem	nn2ge 12204*	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 (𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥))
Theorem	nnge1 12205	A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴)
Theorem	nngt1ne1 12206	A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
⊢ (𝐴 ∈ ℕ → (1 < 𝐴 ↔ 𝐴 ≠ 1))
Theorem	nnle1eq1 12207	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))
Theorem	nngt0 12208	A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
⊢ (𝐴 ∈ ℕ → 0 < 𝐴)
Theorem	nnnlt1 12209	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)
Theorem	nnnle0 12210	A positive integer is not less than or equal to zero. (Contributed by AV, 13-May-2020.)
⊢ (𝐴 ∈ ℕ → ¬ 𝐴 ≤ 0)
Theorem	nnne0 12211	A positive integer is nonzero. See nnne0ALT 12215 for a shorter proof using ax-pre-mulgt0 11115. This proof avoids 0lt1 11672, and thus ax-pre-mulgt0 11115, by splitting ax-1ne0 11107 into the two separate cases 0 < 1 and 1 < 0. (Contributed by NM, 27-Sep-1999.) Remove dependency on ax-pre-mulgt0 11115. (Revised by Steven Nguyen, 30-Jan-2023.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
Theorem	nnneneg 12212	No positive integer is equal to its negation. (Contributed by AV, 20-Jun-2023.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ -𝐴)
Theorem	0nnn 12213	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) Remove dependency on ax-pre-mulgt0 11115. (Revised by Steven Nguyen, 30-Jan-2023.)
⊢ ¬ 0 ∈ ℕ
Theorem	0nnnALT 12214	Alternate proof of 0nnn 12213, which requires ax-pre-mulgt0 11115 but is not based on nnne0 12211 (and which can therefore be used in nnne0ALT 12215). (Contributed by NM, 25-Aug-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ¬ 0 ∈ ℕ
Theorem	nnne0ALT 12215	Alternate version of nnne0 12211. 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 12216	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 0 < 𝐴
Theorem	nnne0i 12217	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ≠ 0
Theorem	nndivre 12218	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 12219	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)
Theorem	nnrecgt0 12220	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 0 < (1 / 𝐴))
Theorem	nnsub 12221	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 12222	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℕ)
Theorem	nndiv 12223*	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 12224	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 12225	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 1 ≤ 𝐴)
Theorem	nngt0d 12226	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 0 < 𝐴)
Theorem	nnne0d 12227	A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0)
Theorem	nnrecred 12228	The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
Theorem	nnaddcld 12229	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 12230	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 12231	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ)
Theorem	1t1e1ALT 12232	Alternate proof of 1t1e1 12338 using a different set of axioms (add ax-mulrcl 11101, ax-i2m1 11106, ax-1ne0 11107, ax-rrecex 11110 and remove ax-resscn 11095, ax-mulcom 11102, ax-mulass 11104, ax-distr 11105). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (1 · 1) = 1
Theorem	nnadddir 12233	Right-distributivity for natural numbers without ax-mulcom 11102. (Contributed by SN, 5-Feb-2024.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
Theorem	nnmul1com 12234	Multiplication with 1 is commutative for natural numbers, without ax-mulcom 11102. Since (𝐴 · 1) is 𝐴 by ax-1rid 11108, this is equivalent to remullid 42866 for natural numbers, but using fewer axioms (avoiding ax-resscn 11095, ax-addass 11103, ax-mulass 11104, ax-rnegex 11109, ax-pre-lttri 11112, ax-pre-lttrn 11113, ax-pre-ltadd 11114). (Contributed by SN, 5-Feb-2024.)
⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1))
Theorem	nnmulcom 12235	Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
5.4.3  Decimal representation of numbers The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 11045 through df-9 12251), 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 11045 and df-1 11046). With the decimal constructor df-dec 12645, it is possible to easily express larger integers in base 10. See deccl 12659 and the theorems that follow it. See also 4001prm 17115 (4001 is prime) and the proof of bpos 27256. 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 16155. 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 12236	Extend class notation to include the number 2.
class 2
Syntax	c3 12237	Extend class notation to include the number 3.
class 3
Syntax	c4 12238	Extend class notation to include the number 4.
class 4
Syntax	c5 12239	Extend class notation to include the number 5.
class 5
Syntax	c6 12240	Extend class notation to include the number 6.
class 6
Syntax	c7 12241	Extend class notation to include the number 7.
class 7
Syntax	c8 12242	Extend class notation to include the number 8.
class 8
Syntax	c9 12243	Extend class notation to include the number 9.
class 9
Definition	df-2 12244	Define the number 2. (Contributed by NM, 27-May-1999.)
⊢ 2 = (1 + 1)
Definition	df-3 12245	Define the number 3. (Contributed by NM, 27-May-1999.)
⊢ 3 = (2 + 1)
Definition	df-4 12246	Define the number 4. (Contributed by NM, 27-May-1999.)
⊢ 4 = (3 + 1)
Definition	df-5 12247	Define the number 5. (Contributed by NM, 27-May-1999.)
⊢ 5 = (4 + 1)
Definition	df-6 12248	Define the number 6. (Contributed by NM, 27-May-1999.)
⊢ 6 = (5 + 1)
Definition	df-7 12249	Define the number 7. (Contributed by NM, 27-May-1999.)
⊢ 7 = (6 + 1)
Definition	df-8 12250	Define the number 8. (Contributed by NM, 27-May-1999.)
⊢ 8 = (7 + 1)
Definition	df-9 12251	Define the number 9. (Contributed by NM, 27-May-1999.)
⊢ 9 = (8 + 1)
Theorem	0ne1 12252	Zero is different from one (the commuted form is Axiom ax-1ne0 11107). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 1
Theorem	1m1e0 12253	One minus one equals zero. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 − 1) = 0
Theorem	2nn 12254	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
⊢ 2 ∈ ℕ
Theorem	2re 12255	The number 2 is real. (Contributed by NM, 27-May-1999.)
⊢ 2 ∈ ℝ
Theorem	2cn 12256	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 12257	Alternate proof of 2cn 12256. Shorter but uses more axioms. Similar proofs are possible for 3cn 12262, ... , 9cn 12281. (Contributed by NM, 30-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 2 ∈ ℂ
Theorem	2ex 12258	The number 2 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 2 ∈ V
Theorem	2cnd 12259	The number 2 is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 2 ∈ ℂ)
Theorem	3nn 12260	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 3 ∈ ℕ
Theorem	3re 12261	The number 3 is real. (Contributed by NM, 27-May-1999.)
⊢ 3 ∈ ℝ
Theorem	3cn 12262	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 12263	The number 3 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 3 ∈ V
Theorem	4nn 12264	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 4 ∈ ℕ
Theorem	4re 12265	The number 4 is real. (Contributed by NM, 27-May-1999.)
⊢ 4 ∈ ℝ
Theorem	4cn 12266	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 12267	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 ∈ ℕ
Theorem	5re 12268	The number 5 is real. (Contributed by NM, 27-May-1999.)
⊢ 5 ∈ ℝ
Theorem	5cn 12269	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 12270	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 ∈ ℕ
Theorem	6re 12271	The number 6 is real. (Contributed by NM, 27-May-1999.)
⊢ 6 ∈ ℝ
Theorem	6cn 12272	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 12273	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 ∈ ℕ
Theorem	7re 12274	The number 7 is real. (Contributed by NM, 27-May-1999.)
⊢ 7 ∈ ℝ
Theorem	7cn 12275	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 12276	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 8 ∈ ℕ
Theorem	8re 12277	The number 8 is real. (Contributed by NM, 27-May-1999.)
⊢ 8 ∈ ℝ
Theorem	8cn 12278	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 12279	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
⊢ 9 ∈ ℕ
Theorem	9re 12280	The number 9 is real. (Contributed by NM, 27-May-1999.)
⊢ 9 ∈ ℝ
Theorem	9cn 12281	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 12282	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ≤ 0
Theorem	0le2 12283	The number 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ 0 ≤ 2
Theorem	2pos 12284	The number 2 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 2
Theorem	2ne0 12285	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
⊢ 2 ≠ 0
Theorem	3pos 12286	The number 3 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 3
Theorem	3ne0 12287	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ 3 ≠ 0
Theorem	4pos 12288	The number 4 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 4
Theorem	4ne0 12289	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ 4 ≠ 0
Theorem	5pos 12290	The number 5 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 5
Theorem	6pos 12291	The number 6 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 6
Theorem	7pos 12292	The number 7 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 7
Theorem	8pos 12293	The number 8 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 8
Theorem	9pos 12294	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	1pneg1e0 12295	1 + -1 is 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + -1) = 0
Theorem	0m0e0 12296	0 minus 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (0 − 0) = 0
Theorem	1m0e1 12297	1 - 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 − 0) = 1
Theorem	0p1e1 12298	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (0 + 1) = 1
Theorem	fv0p1e1 12299	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 12300	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + 0) = 1