P. 95 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9401-9500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	9t4e36 9401	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 4) = ;36
Theorem	9t5e45 9402	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 5) = ;45
Theorem	9t6e54 9403	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 6) = ;54
Theorem	9t7e63 9404	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 7) = ;63
Theorem	9t8e72 9405	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 8) = ;72
Theorem	9t9e81 9406	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 9) = ;81
Theorem	9t11e99 9407	9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
⊢ (9 · ;11) = ;99
Theorem	9lt10 9408	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 9 < ;10
Theorem	8lt10 9409	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 8 < ;10
Theorem	7lt10 9410	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 7 < ;10
Theorem	6lt10 9411	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 6 < ;10
Theorem	5lt10 9412	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 5 < ;10
Theorem	4lt10 9413	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 4 < ;10
Theorem	3lt10 9414	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 3 < ;10
Theorem	2lt10 9415	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 2 < ;10
Theorem	1lt10 9416	1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 1 < ;10
Theorem	decbin0 9417	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ (4 · 𝐴) = (2 · (2 · 𝐴))
Theorem	decbin2 9418	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1))
Theorem	decbin3 9419	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ ((4 · 𝐴) + 3) = ((2 · ((2 · 𝐴) + 1)) + 1)
Theorem	halfthird 9420	Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
⊢ ((1 / 2) − (1 / 3)) = (1 / 6)
Theorem	5recm6rec 9421	One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
⊢ ((1 / 5) − (1 / 6)) = (1 / ;30)
4.4.11  Upper sets of integers
Syntax	cuz 9422	Extend class notation with the upper integer function. Read "ℤ≥‘𝑀 " as "the set of integers greater than or equal to 𝑀."
class ℤ≥
Definition	df-uz 9423*	Define a function whose value at 𝑗 is the semi-infinite set of contiguous integers starting at 𝑗, which we will also call the upper integers starting at 𝑗. Read "ℤ≥‘𝑀 " as "the set of integers greater than or equal to 𝑀." See uzval 9424 for its value, uzssz 9441 for its relationship to ℤ, nnuz 9457 and nn0uz 9456 for its relationships to ℕ and ℕ0, and eluz1 9426 and eluz2 9428 for its membership relations. (Contributed by NM, 5-Sep-2005.)
⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘})
Theorem	uzval 9424*	The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ (𝑁 ∈ ℤ → (ℤ≥‘𝑁) = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘})
Theorem	uzf 9425	The domain and range of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ℤ≥:ℤ⟶𝒫 ℤ
Theorem	eluz1 9426	Membership in the upper set of integers starting at 𝑀. (Contributed by NM, 5-Sep-2005.)
⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)))
Theorem	eluzel2 9427	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
Theorem	eluz2 9428	Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show 𝑀 ∈ ℤ. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
Theorem	eluz1i 9429	Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
⊢ 𝑀 ∈ ℤ    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
Theorem	eluzuzle 9430	An integer in an upper set of integers is an element of an upper set of integers with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴) → (𝐶 ∈ (ℤ≥‘𝐴) → 𝐶 ∈ (ℤ≥‘𝐵)))
Theorem	eluzelz 9431	A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
Theorem	eluzelre 9432	A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ)
Theorem	eluzelcn 9433	A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℂ)
Theorem	eluzle 9434	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁)
Theorem	eluz 9435	Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
Theorem	uzid 9436	Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
Theorem	uzn0 9437	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅)
Theorem	uztrn 9438	Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁))
Theorem	uztrn2 9439	Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
⊢ 𝑍 = (ℤ≥‘𝐾)    ⇒   ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ 𝑍)
Theorem	uzneg 9440	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ (ℤ≥‘-𝑁))
Theorem	uzssz 9441	An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ (ℤ≥‘𝑀) ⊆ ℤ
Theorem	uzss 9442	Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀))
Theorem	uztric 9443	Trichotomy of the ordering relation on integers, stated in terms of upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jun-2013.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ∨ 𝑀 ∈ (ℤ≥‘𝑁)))
Theorem	uz11 9444	The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ↔ 𝑀 = 𝑁))
Theorem	eluzp1m1 9445	Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑀))
Theorem	eluzp1l 9446	Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑁)
Theorem	eluzp1p1 9447	Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
Theorem	eluzaddi 9448	Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
⊢ 𝑀 ∈ ℤ    &   ⊢ 𝐾 ∈ ℤ    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
Theorem	eluzsubi 9449	Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
⊢ 𝑀 ∈ ℤ    &   ⊢ 𝐾 ∈ ℤ    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀))
Theorem	eluzadd 9450	Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
Theorem	eluzsub 9451	Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀))
Theorem	uzm1 9452	Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈ (ℤ≥‘𝑀)))
Theorem	uznn0sub 9453	The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0)
Theorem	uzin 9454	Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((ℤ≥‘𝑀) ∩ (ℤ≥‘𝑁)) = (ℤ≥‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
Theorem	uzp1 9455	Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))))
Theorem	nn0uz 9456	Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ ℕ0 = (ℤ≥‘0)
Theorem	nnuz 9457	Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ ℕ = (ℤ≥‘1)
Theorem	elnnuz 9458	A positive integer expressed as a member of an upper set of integers. (Contributed by NM, 6-Jun-2006.)
⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1))
Theorem	elnn0uz 9459	A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0))
Theorem	eluz2nn 9460	An integer is greater than or equal to 2 is a positive integer. (Contributed by AV, 3-Nov-2018.)
⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ)
Theorem	eluz4eluz2 9461	An integer greater than or equal to 4 is an integer greater than or equal to 2. (Contributed by AV, 30-May-2023.)
⊢ (𝑋 ∈ (ℤ≥‘4) → 𝑋 ∈ (ℤ≥‘2))
Theorem	eluz4nn 9462	An integer greater than or equal to 4 is a positive integer. (Contributed by AV, 30-May-2023.)
⊢ (𝑋 ∈ (ℤ≥‘4) → 𝑋 ∈ ℕ)
Theorem	eluzge2nn0 9463	If an integer is greater than or equal to 2, then it is a nonnegative integer. (Contributed by AV, 27-Aug-2018.) (Proof shortened by AV, 3-Nov-2018.)
⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ0)
Theorem	eluz2n0 9464	An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.)
⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ≠ 0)
Theorem	uzuzle23 9465	An integer in the upper set of integers starting at 3 is element of the upper set of integers starting at 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
⊢ (𝐴 ∈ (ℤ≥‘3) → 𝐴 ∈ (ℤ≥‘2))
Theorem	eluzge3nn 9466	If an integer is greater than 3, then it is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
Theorem	uz3m2nn 9467	An integer greater than or equal to 3 decreased by 2 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ)
Theorem	1eluzge0 9468	1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
⊢ 1 ∈ (ℤ≥‘0)
Theorem	2eluzge0 9469	2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ 2 ∈ (ℤ≥‘0)
Theorem	2eluzge1 9470	2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
⊢ 2 ∈ (ℤ≥‘1)
Theorem	uznnssnn 9471	The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)
⊢ (𝑁 ∈ ℕ → (ℤ≥‘𝑁) ⊆ ℕ)
Theorem	raluz 9472*	Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
⊢ (𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑)))
Theorem	raluz2 9473*	Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑)))
Theorem	rexuz 9474*	Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑)))
Theorem	rexuz2 9475*	Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
⊢ (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑)))
Theorem	2rexuz 9476*	Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 ≤ 𝑛 ∧ 𝜑))
Theorem	peano2uz 9477	Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀))
Theorem	peano2uzs 9478	Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑁 ∈ 𝑍 → (𝑁 + 1) ∈ 𝑍)
Theorem	peano2uzr 9479	Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀))
Theorem	uzaddcl 9480	Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℕ0) → (𝑁 + 𝐾) ∈ (ℤ≥‘𝑀))
Theorem	nn0pzuz 9481	The sum of a nonnegative integer and an integer is an integer greater than or equal to that integer. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ (ℤ≥‘𝑍))
Theorem	uzind4 9482*	Induction on the upper set of integers that starts at an integer 𝑀. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 7-Sep-2005.)
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑀 ∈ ℤ → 𝜓)    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏)
Theorem	uzind4ALT 9483*	Induction on the upper set of integers that starts at an integer 𝑀. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 9482 or uzind4ALT 9483 may be used; see comment for nnind 8832. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝑀 ∈ ℤ → 𝜓)    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃))    &   ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏)
Theorem	uzind4s 9484*	Induction on the upper set of integers that starts at an integer 𝑀, using explicit substitution. The hypotheses are the basis and the induction step. (Contributed by NM, 4-Nov-2005.)
⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑘]𝜑)    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜑 → [(𝑘 + 1) / 𝑘]𝜑))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑘]𝜑)
Theorem	uzind4s2 9485*	Induction on the upper set of integers that starts at an integer 𝑀, using explicit substitution. The hypotheses are the basis and the induction step. Use this instead of uzind4s 9484 when 𝑗 and 𝑘 must be distinct in [(𝑘 + 1) / 𝑗]𝜑. (Contributed by NM, 16-Nov-2005.)
⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑)    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ([𝑘 / 𝑗]𝜑 → [(𝑘 + 1) / 𝑗]𝜑))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑗]𝜑)
Theorem	uzind4i 9486*	Induction on the upper integers that start at 𝑀. The first four give us the substitution instances we need, and the last two are the basis and the induction step. This is a stronger version of uzind4 9482 assuming that 𝜓 holds unconditionally. Notice that 𝑁 ∈ (ℤ≥‘𝑀) implies that the lower bound 𝑀 is an integer (𝑀 ∈ ℤ, see eluzel2 9427). (Contributed by NM, 4-Sep-2005.) (Revised by AV, 13-Jul-2022.)
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏)
Theorem	indstr 9487*	Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑))    ⇒   ⊢ (𝑥 ∈ ℕ → 𝜑)
Theorem	infrenegsupex 9488*	The infimum of a set of reals 𝐴 is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 14-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    ⇒   ⊢ (𝜑 → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))
Theorem	supinfneg 9489*	If a set of real numbers has a least upper bound, the set of the negation of those numbers has a greatest lower bound. For a theorem which is similar but only for the boundedness part, see ublbneg 9504. (Contributed by Jim Kingdon, 15-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦)))
Theorem	infsupneg 9490*	If a set of real numbers has a greatest lower bound, the set of the negation of those numbers has a least upper bound. To go in the other direction see supinfneg 9489. (Contributed by Jim Kingdon, 15-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑦 < 𝑧)))
Theorem	supminfex 9491*	A supremum is the negation of the infimum of that set's image under negation. (Contributed by Jim Kingdon, 14-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
Theorem	eluznn0 9492	Membership in a nonnegative upper set of integers implies membership in ℕ0. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ0)
Theorem	eluznn 9493	Membership in a positive upper set of integers implies membership in ℕ. (Contributed by JJ, 1-Oct-2018.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ)
Theorem	eluz2b1 9494	Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁))
Theorem	eluz2gt1 9495	An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
⊢ (𝑁 ∈ (ℤ≥‘2) → 1 < 𝑁)
Theorem	eluz2b2 9496	Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))
Theorem	eluz2b3 9497	Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
Theorem	uz2m1nn 9498	One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ ℕ)
Theorem	1nuz2 9499	1 is not in (ℤ≥‘2). (Contributed by Paul Chapman, 21-Nov-2012.)
⊢ ¬ 1 ∈ (ℤ≥‘2)
Theorem	elnn1uz2 9500	A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)))