P. 78 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnex 7701	The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℕ ∈ V
Theorem	nnre 7702	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
⊢ (A ∈ ℕ → A ∈ ℝ)
Theorem	nncn 7703	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
⊢ (A ∈ ℕ → A ∈ ℂ)
Theorem	nnrei 7704	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
⊢ A ∈ ℕ    ⇒   ⊢ A ∈ ℝ
Theorem	nncni 7705	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
⊢ A ∈ ℕ    ⇒   ⊢ A ∈ ℂ
Theorem	1nn 7706	Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
⊢ 1 ∈ ℕ
Theorem	peano2nn 7707	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)
Theorem	nnred 7708	A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → A ∈ ℝ)
Theorem	nncnd 7709	A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → A ∈ ℂ)
Theorem	peano2nnd 7710	Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → (A + 1) ∈ ℕ)
3.4.2  Principle of mathematical induction
Theorem	nnind 7711*	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 7715 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
⊢ (x = 1 → (φ ↔ ψ))    &   ⊢ (x = y → (φ ↔ χ))    &   ⊢ (x = (y + 1) → (φ ↔ θ))    &   ⊢ (x = A → (φ ↔ τ))    &   ⊢ ψ    &   ⊢ (y ∈ ℕ → (χ → θ))    ⇒   ⊢ (A ∈ ℕ → τ)
Theorem	nnindALT 7712*	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 7711 has a different hypothesis order. It may be easier to use with the metamath program's 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 nnind /allow". (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (y ∈ ℕ → (χ → θ))    &   ⊢ ψ    &   ⊢ (x = 1 → (φ ↔ ψ))    &   ⊢ (x = y → (φ ↔ χ))    &   ⊢ (x = (y + 1) → (φ ↔ θ))    &   ⊢ (x = A → (φ ↔ τ))    ⇒   ⊢ (A ∈ ℕ → τ)
Theorem	nn1m1nn 7713	Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
⊢ (A ∈ ℕ → (A = 1 ∨ (A − 1) ∈ ℕ))
Theorem	nn1suc 7714*	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 → (φ ↔ ψ))    &   ⊢ (x = (y + 1) → (φ ↔ χ))    &   ⊢ (x = A → (φ ↔ θ))    &   ⊢ ψ    &   ⊢ (y ∈ ℕ → χ)    ⇒   ⊢ (A ∈ ℕ → θ)
Theorem	nnaddcl 7715	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)
Theorem	nnmulcl 7716	Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → (A · B) ∈ ℕ)
Theorem	nnmulcli 7717	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ A ∈ ℕ    &   ⊢ B ∈ ℕ    ⇒   ⊢ (A · B) ∈ ℕ
Theorem	nnge1 7718	A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
⊢ (A ∈ ℕ → 1 ≤ A)
Theorem	nnle1eq1 7719	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 7720	A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
⊢ (A ∈ ℕ → 0 < A)
Theorem	nnnlt1 7721	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	0nnn 7722	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
⊢ ¬ 0 ∈ ℕ
Theorem	nnne0 7723	A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
⊢ (A ∈ ℕ → A ≠ 0)
Theorem	nnap0 7724	A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
⊢ (A ∈ ℕ → A # 0)
Theorem	nngt0i 7725	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
⊢ A ∈ ℕ    ⇒   ⊢ 0 < A
Theorem	nnne0i 7726	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
⊢ A ∈ ℕ    ⇒   ⊢ A ≠ 0
Theorem	nn2ge 7727*	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	nn1gt1 7728	A positive integer is either one or greater than one. This is for ℕ; 0elnn 4283 is a similar theorem for 𝜔 (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)
⊢ (A ∈ ℕ → (A = 1 ∨ 1 < A))
Theorem	nngt1ne1 7729	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	nndivre 7730	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 7731	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)
Theorem	nnrecgt0 7732	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
⊢ (A ∈ ℕ → 0 < (1 / A))
Theorem	nnsub 7733	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 7734	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
⊢ A ∈ ℕ    &   ⊢ B ∈ ℕ    ⇒   ⊢ (A < B ↔ (B − A) ∈ ℕ)
Theorem	nndiv 7735*	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 7736	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 7737	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → 1 ≤ A)
Theorem	nngt0d 7738	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → 0 < A)
Theorem	nnne0d 7739	A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → A ≠ 0)
Theorem	nnrecred 7740	The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → (1 / A) ∈ ℝ)
Theorem	nnaddcld 7741	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 7742	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 7743	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℕ)    ⇒   ⊢ (φ → (A / B) ∈ ℝ)
3.4.3  Decimal representation of numbers Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 6718 and df-1 6719). Only the digits 0 through 9 (df-0 6718 through df-9 7760) and the number 10 (df-10 7761) are explicitly defined. 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 7744	Extend class notation to include the number 2.
class 2
Syntax	c3 7745	Extend class notation to include the number 3.
class 3
Syntax	c4 7746	Extend class notation to include the number 4.
class 4
Syntax	c5 7747	Extend class notation to include the number 5.
class 5
Syntax	c6 7748	Extend class notation to include the number 6.
class 6
Syntax	c7 7749	Extend class notation to include the number 7.
class 7
Syntax	c8 7750	Extend class notation to include the number 8.
class 8
Syntax	c9 7751	Extend class notation to include the number 9.
class 9
Syntax	c10 7752	Extend class notation to include the number 10.
class 10
Definition	df-2 7753	Define the number 2. (Contributed by NM, 27-May-1999.)
⊢ 2 = (1 + 1)
Definition	df-3 7754	Define the number 3. (Contributed by NM, 27-May-1999.)
⊢ 3 = (2 + 1)
Definition	df-4 7755	Define the number 4. (Contributed by NM, 27-May-1999.)
⊢ 4 = (3 + 1)
Definition	df-5 7756	Define the number 5. (Contributed by NM, 27-May-1999.)
⊢ 5 = (4 + 1)
Definition	df-6 7757	Define the number 6. (Contributed by NM, 27-May-1999.)
⊢ 6 = (5 + 1)
Definition	df-7 7758	Define the number 7. (Contributed by NM, 27-May-1999.)
⊢ 7 = (6 + 1)
Definition	df-8 7759	Define the number 8. (Contributed by NM, 27-May-1999.)
⊢ 8 = (7 + 1)
Definition	df-9 7760	Define the number 9. (Contributed by NM, 27-May-1999.)
⊢ 9 = (8 + 1)
Definition	df-10 7761	Define the number 10. See remarks under df-2 7753. (Contributed by NM, 5-Feb-2007.)
⊢ 10 = (9 + 1)
Theorem	0ne1 7762	0 ≠ 1 (common case). See aso 1ap0 7374. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 1
Theorem	1ne0 7763	1 ≠ 0. See aso 1ap0 7374. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 1 ≠ 0
Theorem	1m1e0 7764	(1 − 1) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 − 1) = 0
Theorem	2re 7765	The number 2 is real. (Contributed by NM, 27-May-1999.)
⊢ 2 ∈ ℝ
Theorem	2cn 7766	The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
⊢ 2 ∈ ℂ
Theorem	2ex 7767	2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 2 ∈ V
Theorem	2cnd 7768	2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (φ → 2 ∈ ℂ)
Theorem	3re 7769	The number 3 is real. (Contributed by NM, 27-May-1999.)
⊢ 3 ∈ ℝ
Theorem	3cn 7770	The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
⊢ 3 ∈ ℂ
Theorem	3ex 7771	3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 3 ∈ V
Theorem	4re 7772	The number 4 is real. (Contributed by NM, 27-May-1999.)
⊢ 4 ∈ ℝ
Theorem	4cn 7773	The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 4 ∈ ℂ
Theorem	5re 7774	The number 5 is real. (Contributed by NM, 27-May-1999.)
⊢ 5 ∈ ℝ
Theorem	5cn 7775	The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 5 ∈ ℂ
Theorem	6re 7776	The number 6 is real. (Contributed by NM, 27-May-1999.)
⊢ 6 ∈ ℝ
Theorem	6cn 7777	The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 6 ∈ ℂ
Theorem	7re 7778	The number 7 is real. (Contributed by NM, 27-May-1999.)
⊢ 7 ∈ ℝ
Theorem	7cn 7779	The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 7 ∈ ℂ
Theorem	8re 7780	The number 8 is real. (Contributed by NM, 27-May-1999.)
⊢ 8 ∈ ℝ
Theorem	8cn 7781	The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 8 ∈ ℂ
Theorem	9re 7782	The number 9 is real. (Contributed by NM, 27-May-1999.)
⊢ 9 ∈ ℝ
Theorem	9cn 7783	The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 9 ∈ ℂ
Theorem	10re 7784	The number 10 is real. (Contributed by NM, 5-Feb-2007.)
⊢ 10 ∈ ℝ
Theorem	0le0 7785	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ≤ 0
Theorem	0le2 7786	0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ 0 ≤ 2
Theorem	2pos 7787	The number 2 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 2
Theorem	2ne0 7788	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
⊢ 2 ≠ 0
Theorem	2ap0 7789	The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 2 # 0
Theorem	3pos 7790	The number 3 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 3
Theorem	3ne0 7791	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ 3 ≠ 0
Theorem	4pos 7792	The number 4 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 4
Theorem	4ne0 7793	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ 4 ≠ 0
Theorem	5pos 7794	The number 5 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 5
Theorem	6pos 7795	The number 6 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 6
Theorem	7pos 7796	The number 7 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 7
Theorem	8pos 7797	The number 8 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 8
Theorem	9pos 7798	The number 9 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 9
Theorem	10pos 7799	The number 10 is positive. (Contributed by NM, 5-Feb-2007.)
⊢ 0 < 10
3.4.4  Some properties of specific numbers This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10.
Theorem	neg1cn 7800	-1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ -1 ∈ ℂ