P. 97 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9601-9700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	9p6e15 9601	9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 6) = ;15
Theorem	9p7e16 9602	9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 7) = ;16
Theorem	9p8e17 9603	9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 8) = ;17
Theorem	9p9e18 9604	9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 9) = ;18
Theorem	10p10e20 9605	10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (;10 + ;10) = ;20
Theorem	10m1e9 9606	10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
⊢ (;10 − 1) = 9
Theorem	4t3lem 9607	Lemma for 4t3e12 9608 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 = (𝐵 + 1)    &   ⊢ (𝐴 · 𝐵) = 𝐷    &   ⊢ (𝐷 + 𝐴) = 𝐸    ⇒   ⊢ (𝐴 · 𝐶) = 𝐸
Theorem	4t3e12 9608	4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (4 · 3) = ;12
Theorem	4t4e16 9609	4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (4 · 4) = ;16
Theorem	5t2e10 9610	5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
⊢ (5 · 2) = ;10
Theorem	5t3e15 9611	5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (5 · 3) = ;15
Theorem	5t4e20 9612	5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (5 · 4) = ;20
Theorem	5t5e25 9613	5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (5 · 5) = ;25
Theorem	6t2e12 9614	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (6 · 2) = ;12
Theorem	6t3e18 9615	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (6 · 3) = ;18
Theorem	6t4e24 9616	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (6 · 4) = ;24
Theorem	6t5e30 9617	6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (6 · 5) = ;30
Theorem	6t6e36 9618	6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (6 · 6) = ;36
Theorem	7t2e14 9619	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 2) = ;14
Theorem	7t3e21 9620	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 3) = ;21
Theorem	7t4e28 9621	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 4) = ;28
Theorem	7t5e35 9622	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 5) = ;35
Theorem	7t6e42 9623	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 6) = ;42
Theorem	7t7e49 9624	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 7) = ;49
Theorem	8t2e16 9625	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 2) = ;16
Theorem	8t3e24 9626	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 3) = ;24
Theorem	8t4e32 9627	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 4) = ;32
Theorem	8t5e40 9628	8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 · 5) = ;40
Theorem	8t6e48 9629	8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 · 6) = ;48
Theorem	8t7e56 9630	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 7) = ;56
Theorem	8t8e64 9631	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 8) = ;64
Theorem	9t2e18 9632	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 2) = ;18
Theorem	9t3e27 9633	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 3) = ;27
Theorem	9t4e36 9634	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 4) = ;36
Theorem	9t5e45 9635	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 5) = ;45
Theorem	9t6e54 9636	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 6) = ;54
Theorem	9t7e63 9637	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 7) = ;63
Theorem	9t8e72 9638	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 8) = ;72
Theorem	9t9e81 9639	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 9) = ;81
Theorem	9t11e99 9640	9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
⊢ (9 · ;11) = ;99
Theorem	9lt10 9641	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 9 < ;10
Theorem	8lt10 9642	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 8 < ;10
Theorem	7lt10 9643	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 7 < ;10
Theorem	6lt10 9644	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 6 < ;10
Theorem	5lt10 9645	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 5 < ;10
Theorem	4lt10 9646	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 4 < ;10
Theorem	3lt10 9647	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 3 < ;10
Theorem	2lt10 9648	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 2 < ;10
Theorem	1lt10 9649	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 9650	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ (4 · 𝐴) = (2 · (2 · 𝐴))
Theorem	decbin2 9651	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1))
Theorem	decbin3 9652	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 9653	Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
⊢ ((1 / 2) − (1 / 3)) = (1 / 6)
Theorem	5recm6rec 9654	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 9655	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 9656*	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 9657 for its value, uzssz 9675 for its relationship to ℤ, nnuz 9691 and nn0uz 9690 for its relationships to ℕ and ℕ0, and eluz1 9659 and eluz2 9661 for its membership relations. (Contributed by NM, 5-Sep-2005.)
⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘})
Theorem	uzval 9657*	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 9658	The domain and codomain 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 9659	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 9660	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 9661	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 9662	Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
⊢ 𝑀 ∈ ℤ    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
Theorem	eluzuzle 9663	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 9664	A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
Theorem	eluzelre 9665	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 9666	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 9667	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁)
Theorem	eluz 9668	Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
Theorem	uzid 9669	Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
Theorem	uzidd 9670	Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
Theorem	uzn0 9671	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅)
Theorem	uztrn 9672	Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁))
Theorem	uztrn2 9673	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 9674	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ (ℤ≥‘-𝑁))
Theorem	uzssz 9675	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 9676	Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀))
Theorem	uztric 9677	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 9678	The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ↔ 𝑀 = 𝑁))
Theorem	eluzp1m1 9679	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 9680	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 9681	Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
Theorem	eluzaddi 9682	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 9683	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 9684	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 9685	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 9686	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 9687	The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0)
Theorem	uzin 9688	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 9689	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 9690	Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ ℕ0 = (ℤ≥‘0)
Theorem	nnuz 9691	Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ ℕ = (ℤ≥‘1)
Theorem	elnnuz 9692	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 9693	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 9694	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 9695	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 9696	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 9697	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 9698	An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.)
⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ≠ 0)
Theorem	uzuzle23 9699	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 9700	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)