P. 80 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1nn 7901	Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
⊢ 1 ∈ ℕ
Theorem	peano2nn 7902	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 7903	A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	nncnd 7904	A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	peano2nnd 7905	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 7906*	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 7910 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 7907*	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 7906 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 7908	Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))
Theorem	nn1suc 7909*	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 7910	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 7911	Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)
Theorem	nnmulcli 7912	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℕ
Theorem	nnge1 7913	A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴)
Theorem	nnle1eq1 7914	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 7915	A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
⊢ (𝐴 ∈ ℕ → 0 < 𝐴)
Theorem	nnnlt1 7916	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 7917	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
⊢ ¬ 0 ∈ ℕ
Theorem	nnne0 7918	A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
Theorem	nnap0 7919	A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
⊢ (𝐴 ∈ ℕ → 𝐴 # 0)
Theorem	nngt0i 7920	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 0 < 𝐴
Theorem	nnne0i 7921	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ≠ 0
Theorem	nn2ge 7922*	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 7923	A positive integer is either one or greater than one. This is for ℕ; 0elnn 4327 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 7924	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 7925	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 7926	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)
Theorem	nnrecgt0 7927	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 0 < (1 / 𝐴))
Theorem	nnsub 7928	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 7929	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℕ)
Theorem	nndiv 7930*	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 7931	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 7932	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 1 ≤ 𝐴)
Theorem	nngt0d 7933	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 0 < 𝐴)
Theorem	nnne0d 7934	A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0)
Theorem	nnap0d 7935	A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 # 0)
Theorem	nnrecred 7936	The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
Theorem	nnaddcld 7937	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 7938	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 7939	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 6877 and df-1 6878). Only the digits 0 through 9 (df-0 6877 through df-9 7956) and the number 10 (df-10 7957) 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 7940	Extend class notation to include the number 2.
class 2
Syntax	c3 7941	Extend class notation to include the number 3.
class 3
Syntax	c4 7942	Extend class notation to include the number 4.
class 4
Syntax	c5 7943	Extend class notation to include the number 5.
class 5
Syntax	c6 7944	Extend class notation to include the number 6.
class 6
Syntax	c7 7945	Extend class notation to include the number 7.
class 7
Syntax	c8 7946	Extend class notation to include the number 8.
class 8
Syntax	c9 7947	Extend class notation to include the number 9.
class 9
Syntax	c10 7948	Extend class notation to include the number 10.
class 10
Definition	df-2 7949	Define the number 2. (Contributed by NM, 27-May-1999.)
⊢ 2 = (1 + 1)
Definition	df-3 7950	Define the number 3. (Contributed by NM, 27-May-1999.)
⊢ 3 = (2 + 1)
Definition	df-4 7951	Define the number 4. (Contributed by NM, 27-May-1999.)
⊢ 4 = (3 + 1)
Definition	df-5 7952	Define the number 5. (Contributed by NM, 27-May-1999.)
⊢ 5 = (4 + 1)
Definition	df-6 7953	Define the number 6. (Contributed by NM, 27-May-1999.)
⊢ 6 = (5 + 1)
Definition	df-7 7954	Define the number 7. (Contributed by NM, 27-May-1999.)
⊢ 7 = (6 + 1)
Definition	df-8 7955	Define the number 8. (Contributed by NM, 27-May-1999.)
⊢ 8 = (7 + 1)
Definition	df-9 7956	Define the number 9. (Contributed by NM, 27-May-1999.)
⊢ 9 = (8 + 1)
Definition	df-10 7957	Define the number 10. See remarks under df-2 7949. (Contributed by NM, 5-Feb-2007.)
⊢ 10 = (9 + 1)
Theorem	0ne1 7958	0 ≠ 1 (common case). See aso 1ap0 7557. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 1
Theorem	1ne0 7959	1 ≠ 0. See aso 1ap0 7557. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 1 ≠ 0
Theorem	1m1e0 7960	(1 − 1) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 − 1) = 0
Theorem	2re 7961	The number 2 is real. (Contributed by NM, 27-May-1999.)
⊢ 2 ∈ ℝ
Theorem	2cn 7962	The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
⊢ 2 ∈ ℂ
Theorem	2ex 7963	2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 2 ∈ V
Theorem	2cnd 7964	2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 2 ∈ ℂ)
Theorem	3re 7965	The number 3 is real. (Contributed by NM, 27-May-1999.)
⊢ 3 ∈ ℝ
Theorem	3cn 7966	The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
⊢ 3 ∈ ℂ
Theorem	3ex 7967	3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 3 ∈ V
Theorem	4re 7968	The number 4 is real. (Contributed by NM, 27-May-1999.)
⊢ 4 ∈ ℝ
Theorem	4cn 7969	The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 4 ∈ ℂ
Theorem	5re 7970	The number 5 is real. (Contributed by NM, 27-May-1999.)
⊢ 5 ∈ ℝ
Theorem	5cn 7971	The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 5 ∈ ℂ
Theorem	6re 7972	The number 6 is real. (Contributed by NM, 27-May-1999.)
⊢ 6 ∈ ℝ
Theorem	6cn 7973	The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 6 ∈ ℂ
Theorem	7re 7974	The number 7 is real. (Contributed by NM, 27-May-1999.)
⊢ 7 ∈ ℝ
Theorem	7cn 7975	The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 7 ∈ ℂ
Theorem	8re 7976	The number 8 is real. (Contributed by NM, 27-May-1999.)
⊢ 8 ∈ ℝ
Theorem	8cn 7977	The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 8 ∈ ℂ
Theorem	9re 7978	The number 9 is real. (Contributed by NM, 27-May-1999.)
⊢ 9 ∈ ℝ
Theorem	9cn 7979	The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 9 ∈ ℂ
Theorem	10re 7980	The number 10 is real. (Contributed by NM, 5-Feb-2007.)
⊢ 10 ∈ ℝ
Theorem	0le0 7981	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ≤ 0
Theorem	0le2 7982	0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ 0 ≤ 2
Theorem	2pos 7983	The number 2 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 2
Theorem	2ne0 7984	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
⊢ 2 ≠ 0
Theorem	2ap0 7985	The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 2 # 0
Theorem	3pos 7986	The number 3 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 3
Theorem	3ne0 7987	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ 3 ≠ 0
Theorem	4pos 7988	The number 4 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 4
Theorem	4ne0 7989	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ 4 ≠ 0
Theorem	5pos 7990	The number 5 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 5
Theorem	6pos 7991	The number 6 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 6
Theorem	7pos 7992	The number 7 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 7
Theorem	8pos 7993	The number 8 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 8
Theorem	9pos 7994	The number 9 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 9
Theorem	10pos 7995	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 7996	-1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ -1 ∈ ℂ
Theorem	neg1rr 7997	-1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ -1 ∈ ℝ
Theorem	neg1ne0 7998	-1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -1 ≠ 0
Theorem	neg1lt0 7999	-1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -1 < 0
Theorem	neg1ap0 8000	-1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ -1 # 0