P. 77 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lediv12a 7601	Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)
⊢ ((((A ∈ ℝ ∧ B ∈ ℝ) ∧ (0 ≤ A ∧ A ≤ B)) ∧ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (0 < 𝐶 ∧ 𝐶 ≤ 𝐷))) → (A / 𝐷) ≤ (B / 𝐶))
Theorem	lediv2a 7602	Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((((A ∈ ℝ ∧ 0 < A) ∧ (B ∈ ℝ ∧ 0 < B) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ A ≤ B) → (𝐶 / B) ≤ (𝐶 / A))
Theorem	reclt1 7603	The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)
⊢ ((A ∈ ℝ ∧ 0 < A) → (A < 1 ↔ 1 < (1 / A)))
Theorem	recgt1 7604	The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)
⊢ ((A ∈ ℝ ∧ 0 < A) → (1 < A ↔ (1 / A) < 1))
Theorem	recgt1i 7605	The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)
⊢ ((A ∈ ℝ ∧ 1 < A) → (0 < (1 / A) ∧ (1 / A) < 1))
Theorem	recp1lt1 7606	Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
⊢ ((A ∈ ℝ ∧ 0 ≤ A) → (A / (1 + A)) < 1)
Theorem	recreclt 7607	Given a positive number A, construct a new positive number less than both A and 1. (Contributed by NM, 28-Dec-2005.)
⊢ ((A ∈ ℝ ∧ 0 < A) → ((1 / (1 + (1 / A))) < 1 ∧ (1 / (1 + (1 / A))) < A))
Theorem	le2msq 7608	The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ (B ∈ ℝ ∧ 0 ≤ B)) → (A ≤ B ↔ (A · A) ≤ (B · B)))
Theorem	msq11 7609	The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ (B ∈ ℝ ∧ 0 ≤ B)) → ((A · A) = (B · B) ↔ A = B))
Theorem	ledivp1 7610	Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ (B ∈ ℝ ∧ 0 ≤ B)) → ((A / (B + 1)) · B) ≤ A)
Theorem	squeeze0 7611*	If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)
⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → A = 0)
Theorem	ltp1i 7612	A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
⊢ A ∈ ℝ    ⇒   ⊢ A < (A + 1)
Theorem	recgt0i 7613	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
⊢ A ∈ ℝ    ⇒   ⊢ (0 < A → 0 < (1 / A))
Theorem	recgt0ii 7614	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
⊢ A ∈ ℝ    &   ⊢ 0 < A    ⇒   ⊢ 0 < (1 / A)
Theorem	prodgt0i 7615	Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ∧ 0 < (A · B)) → 0 < B)
Theorem	prodge0i 7616	Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 < A ∧ 0 ≤ (A · B)) → 0 ≤ B)
Theorem	divgt0i 7617	The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 < A ∧ 0 < B) → 0 < (A / B))
Theorem	divge0i 7618	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ∧ 0 < B) → 0 ≤ (A / B))
Theorem	ltreci 7619	The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 < A ∧ 0 < B) → (A < B ↔ (1 / B) < (1 / A)))
Theorem	lereci 7620	The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 < A ∧ 0 < B) → (A ≤ B ↔ (1 / B) ≤ (1 / A)))
Theorem	lt2msqi 7621	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ∧ 0 ≤ B) → (A < B ↔ (A · A) < (B · B)))
Theorem	le2msqi 7622	The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ∧ 0 ≤ B) → (A ≤ B ↔ (A · A) ≤ (B · B)))
Theorem	msq11i 7623	The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ∧ 0 ≤ B) → ((A · A) = (B · B) ↔ A = B))
Theorem	divgt0i2i 7624	The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 0 < B    ⇒   ⊢ (0 < A → 0 < (A / B))
Theorem	ltrecii 7625	The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 0 < A    &   ⊢ 0 < B    ⇒   ⊢ (A < B ↔ (1 / B) < (1 / A))
Theorem	divgt0ii 7626	The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 0 < A    &   ⊢ 0 < B    ⇒   ⊢ 0 < (A / B)
Theorem	ltmul1i 7627	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (A < B ↔ (A · 𝐶) < (B · 𝐶)))
Theorem	ltdiv1i 7628	Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (A < B ↔ (A / 𝐶) < (B / 𝐶)))
Theorem	ltmuldivi 7629	'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → ((A · 𝐶) < B ↔ A < (B / 𝐶)))
Theorem	ltmul2i 7630	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (A < B ↔ (𝐶 · A) < (𝐶 · B)))
Theorem	lemul1i 7631	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (A ≤ B ↔ (A · 𝐶) ≤ (B · 𝐶)))
Theorem	lemul2i 7632	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (A ≤ B ↔ (𝐶 · A) ≤ (𝐶 · B)))
Theorem	ltdiv23i 7633	Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((0 < B ∧ 0 < 𝐶) → ((A / B) < 𝐶 ↔ (A / 𝐶) < B))
Theorem	ltdiv23ii 7634	Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 0 < B    &   ⊢ 0 < 𝐶    ⇒   ⊢ ((A / B) < 𝐶 ↔ (A / 𝐶) < B)
Theorem	ltmul1ii 7635	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 0 < 𝐶    ⇒   ⊢ (A < B ↔ (A · 𝐶) < (B · 𝐶))
Theorem	ltdiv1ii 7636	Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 0 < 𝐶    ⇒   ⊢ (A < B ↔ (A / 𝐶) < (B / 𝐶))
Theorem	ltp1d 7637	A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    ⇒   ⊢ (φ → A < (A + 1))
Theorem	lep1d 7638	A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    ⇒   ⊢ (φ → A ≤ (A + 1))
Theorem	ltm1d 7639	A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    ⇒   ⊢ (φ → (A − 1) < A)
Theorem	lem1d 7640	A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    ⇒   ⊢ (φ → (A − 1) ≤ A)
Theorem	recgt0d 7641	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → 0 < A)    ⇒   ⊢ (φ → 0 < (1 / A))
Theorem	divgt0d 7642	The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 0 < A)    &   ⊢ (φ → 0 < B)    ⇒   ⊢ (φ → 0 < (A / B))
Theorem	mulgt1d 7643	The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 1 < A)    &   ⊢ (φ → 1 < B)    ⇒   ⊢ (φ → 1 < (A · B))
Theorem	lemulge11d 7644	Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 0 ≤ A)    &   ⊢ (φ → 1 ≤ B)    ⇒   ⊢ (φ → A ≤ (A · B))
Theorem	lemulge12d 7645	Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 0 ≤ A)    &   ⊢ (φ → 1 ≤ B)    ⇒   ⊢ (φ → A ≤ (B · A))
Theorem	lemul1ad 7646	Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 𝐶 ∈ ℝ)    &   ⊢ (φ → 0 ≤ 𝐶)    &   ⊢ (φ → A ≤ B)    ⇒   ⊢ (φ → (A · 𝐶) ≤ (B · 𝐶))
Theorem	lemul2ad 7647	Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 𝐶 ∈ ℝ)    &   ⊢ (φ → 0 ≤ 𝐶)    &   ⊢ (φ → A ≤ B)    ⇒   ⊢ (φ → (𝐶 · A) ≤ (𝐶 · B))
Theorem	ltmul12ad 7648	Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 𝐶 ∈ ℝ)    &   ⊢ (φ → 𝐷 ∈ ℝ)    &   ⊢ (φ → 0 ≤ A)    &   ⊢ (φ → A < B)    &   ⊢ (φ → 0 ≤ 𝐶)    &   ⊢ (φ → 𝐶 < 𝐷)    ⇒   ⊢ (φ → (A · 𝐶) < (B · 𝐷))
Theorem	lemul12ad 7649	Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 𝐶 ∈ ℝ)    &   ⊢ (φ → 𝐷 ∈ ℝ)    &   ⊢ (φ → 0 ≤ A)    &   ⊢ (φ → 0 ≤ 𝐶)    &   ⊢ (φ → A ≤ B)    &   ⊢ (φ → 𝐶 ≤ 𝐷)    ⇒   ⊢ (φ → (A · 𝐶) ≤ (B · 𝐷))
Theorem	lemul12bd 7650	Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 𝐶 ∈ ℝ)    &   ⊢ (φ → 𝐷 ∈ ℝ)    &   ⊢ (φ → 0 ≤ A)    &   ⊢ (φ → 0 ≤ 𝐷)    &   ⊢ (φ → A ≤ B)    &   ⊢ (φ → 𝐶 ≤ 𝐷)    ⇒   ⊢ (φ → (A · 𝐶) ≤ (B · 𝐷))
3.3.10  Imaginary and complex number properties
Theorem	crap0 7651	The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → ((A # 0 ∨ B # 0) ↔ (A + (i · B)) # 0))
Theorem	creur 7652*	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)))
Theorem	creui 7653*	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)))
Theorem	cju 7654*	The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
⊢ (A ∈ ℂ → ∃!x ∈ ℂ ((A + x) ∈ ℝ ∧ (i · (A − x)) ∈ ℝ))
3.4  Integer sets
3.4.1  Positive integers (as a subset of complex numbers)
Syntax	cn 7655	Extend class notation to include the class of positive integers.
class ℕ
Definition	df-inn 7656*	Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 7657 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)
⊢ ℕ = ∩ {x ∣ (1 ∈ x ∧ ∀y ∈ x (y + 1) ∈ x)}
Theorem	dfnn2 7657*	Definition of the set of positive integers. Another name for df-inn 7656. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
⊢ ℕ = ∩ {x ∣ (1 ∈ x ∧ ∀y ∈ x (y + 1) ∈ x)}
Theorem	peano5nni 7658*	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 ∧ ∀x ∈ A (x + 1) ∈ A) → ℕ ⊆ A)
Theorem	nnssre 7659	The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
⊢ ℕ ⊆ ℝ
Theorem	nnsscn 7660	The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
⊢ ℕ ⊆ ℂ
Theorem	nnex 7661	The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℕ ∈ V
Theorem	nnre 7662	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
⊢ (A ∈ ℕ → A ∈ ℝ)
Theorem	nncn 7663	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
⊢ (A ∈ ℕ → A ∈ ℂ)
Theorem	nnrei 7664	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
⊢ A ∈ ℕ    ⇒   ⊢ A ∈ ℝ
Theorem	nncni 7665	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
⊢ A ∈ ℕ    ⇒   ⊢ A ∈ ℂ
Theorem	1nn 7666	Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
⊢ 1 ∈ ℕ
Theorem	peano2nn 7667	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 7668	A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → A ∈ ℝ)
Theorem	nncnd 7669	A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → A ∈ ℂ)
Theorem	peano2nnd 7670	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 7671*	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 7675 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 7672*	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 7671 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 7673	Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
⊢ (A ∈ ℕ → (A = 1 ∨ (A − 1) ∈ ℕ))
Theorem	nn1suc 7674*	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 7675	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 7676	Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → (A · B) ∈ ℕ)
Theorem	nnmulcli 7677	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ A ∈ ℕ    &   ⊢ B ∈ ℕ    ⇒   ⊢ (A · B) ∈ ℕ
Theorem	nnge1 7678	A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
⊢ (A ∈ ℕ → 1 ≤ A)
Theorem	nnle1eq1 7679	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 7680	A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
⊢ (A ∈ ℕ → 0 < A)
Theorem	nnnlt1 7681	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 7682	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
⊢ ¬ 0 ∈ ℕ
Theorem	nnne0 7683	A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
⊢ (A ∈ ℕ → A ≠ 0)
Theorem	nnap0 7684	A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
⊢ (A ∈ ℕ → A # 0)
Theorem	nngt0i 7685	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
⊢ A ∈ ℕ    ⇒   ⊢ 0 < A
Theorem	nnne0i 7686	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
⊢ A ∈ ℕ    ⇒   ⊢ A ≠ 0
Theorem	nn2ge 7687*	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 7688	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 7689	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 7690	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 7691	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)
Theorem	nnrecgt0 7692	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
⊢ (A ∈ ℕ → 0 < (1 / A))
Theorem	nnsub 7693	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 7694	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
⊢ A ∈ ℕ    &   ⊢ B ∈ ℕ    ⇒   ⊢ (A < B ↔ (B − A) ∈ ℕ)
Theorem	nndiv 7695*	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 7696	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 7697	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → 1 ≤ A)
Theorem	nngt0d 7698	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → 0 < A)
Theorem	nnne0d 7699	A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → A ≠ 0)
Theorem	nnrecred 7700	The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → (1 / A) ∈ ℝ)