P. 128 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12701-12800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	9p5e14 12701	9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 5) = ;14
Theorem	9p6e15 12702	9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 6) = ;15
Theorem	9p7e16 12703	9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 7) = ;16
Theorem	9p8e17 12704	9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 8) = ;17
Theorem	9p9e18 12705	9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 9) = ;18
Theorem	10p10e20 12706	10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (;10 + ;10) = ;20
Theorem	10m1e9 12707	10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
⊢ (;10 − 1) = 9
Theorem	4t3lem 12708	Lemma for 4t3e12 12709 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 12709	4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (4 · 3) = ;12
Theorem	4t4e16 12710	4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (4 · 4) = ;16
Theorem	5t2e10 12711	5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
⊢ (5 · 2) = ;10
Theorem	5t3e15 12712	5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (5 · 3) = ;15
Theorem	5t4e20 12713	5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (5 · 4) = ;20
Theorem	5t5e25 12714	5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (5 · 5) = ;25
Theorem	6t2e12 12715	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (6 · 2) = ;12
Theorem	6t3e18 12716	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (6 · 3) = ;18
Theorem	6t4e24 12717	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (6 · 4) = ;24
Theorem	6t5e30 12718	6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (6 · 5) = ;30
Theorem	6t6e36 12719	6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (6 · 6) = ;36
Theorem	7t2e14 12720	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 2) = ;14
Theorem	7t3e21 12721	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 3) = ;21
Theorem	7t4e28 12722	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 4) = ;28
Theorem	7t5e35 12723	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 5) = ;35
Theorem	7t6e42 12724	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 6) = ;42
Theorem	7t7e49 12725	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 7) = ;49
Theorem	8t2e16 12726	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 2) = ;16
Theorem	8t3e24 12727	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 3) = ;24
Theorem	8t4e32 12728	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 4) = ;32
Theorem	8t5e40 12729	8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 · 5) = ;40
Theorem	8t6e48 12730	8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 · 6) = ;48
Theorem	8t7e56 12731	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 7) = ;56
Theorem	8t8e64 12732	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 8) = ;64
Theorem	9t2e18 12733	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 2) = ;18
Theorem	9t3e27 12734	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 3) = ;27
Theorem	9t4e36 12735	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 4) = ;36
Theorem	9t5e45 12736	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 5) = ;45
Theorem	9t6e54 12737	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 6) = ;54
Theorem	9t7e63 12738	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 7) = ;63
Theorem	9t8e72 12739	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 8) = ;72
Theorem	9t9e81 12740	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 9) = ;81
Theorem	9t11e99 12741	9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
⊢ (9 · ;11) = ;99
Theorem	9lt10 12742	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 9 < ;10
Theorem	8lt10 12743	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 8 < ;10
Theorem	7lt10 12744	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 7 < ;10
Theorem	6lt10 12745	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 6 < ;10
Theorem	5lt10 12746	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 5 < ;10
Theorem	4lt10 12747	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 4 < ;10
Theorem	3lt10 12748	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 3 < ;10
Theorem	2lt10 12749	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 2 < ;10
Theorem	1lt10 12750	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 12751	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ (4 · 𝐴) = (2 · (2 · 𝐴))
Theorem	decbin2 12752	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1))
Theorem	decbin3 12753	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ ((4 · 𝐴) + 3) = ((2 · ((2 · 𝐴) + 1)) + 1)
Theorem	5recm6rec 12754	One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
⊢ ((1 / 5) − (1 / 6)) = (1 / ;30)
5.4.11  Upper sets of integers
Syntax	cuz 12755	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 12756*	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 12757 for its value, uzssz 12776 for its relationship to ℤ, nnuz 12794 and nn0uz 12793 for its relationships to ℕ and ℕ0, and eluz1 12759 and eluz2 12761 for its membership relations. (Contributed by NM, 5-Sep-2005.)
⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘})
Theorem	uzval 12757*	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 12758	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 12759	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 12760	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 12761	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	eluzmn 12762	Membership in an earlier upper set of integers. (Contributed by Thierry Arnoux, 8-Oct-2018.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → 𝑀 ∈ (ℤ≥‘(𝑀 − 𝑁)))
Theorem	eluz1i 12763	Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
⊢ 𝑀 ∈ ℤ    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
Theorem	eluzuzle 12764	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 12765	A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
Theorem	eluzelre 12766	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 12767	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 12768	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁)
Theorem	eluz 12769	Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
Theorem	uzid 12770	Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
Theorem	uzidd 12771	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 12772	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅)
Theorem	uztrn 12773	Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁))
Theorem	uztrn2 12774	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 12775	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ (ℤ≥‘-𝑁))
Theorem	uzssz 12776	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	uzssre 12777	An upper set of integers is a subset of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (ℤ≥‘𝑀) ⊆ ℝ
Theorem	uzss 12778	Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀))
Theorem	uztric 12779	Totality 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 12780	The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ↔ 𝑀 = 𝑁))
Theorem	eluzp1m1 12781	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 12782	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 12783	Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
Theorem	eluzadd 12784	Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by SN, 7-Feb-2025.)
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
Theorem	eluzsub 12785	Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by SN, 7-Feb-2025.)
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀))
Theorem	eluzaddi 12786	Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.) Shorten and remove 𝑀 ∈ ℤ hypothesis. (Revised by SN, 7-Feb-2025.)
⊢ 𝐾 ∈ ℤ    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
Theorem	eluzsubi 12787	Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.) (Proof shortened by SN, 7-Feb-2025.)
⊢ 𝑀 ∈ ℤ    &   ⊢ 𝐾 ∈ ℤ    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀))
Theorem	subeluzsub 12788	Membership of a difference in an earlier upper set of integers. (Contributed by AV, 10-May-2022.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → (𝑀 − 𝐾) ∈ (ℤ≥‘(𝑀 − 𝑁)))
Theorem	uzm1 12789	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 12790	The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0)
Theorem	uzin 12791	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 12792	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 12793	Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ ℕ0 = (ℤ≥‘0)
Theorem	nnuz 12794	Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ ℕ = (ℤ≥‘1)
Theorem	elnnuz 12795	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 12796	A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0))
Theorem	1eluzge0 12797	1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
⊢ 1 ∈ (ℤ≥‘0)
Theorem	2eluzge0 12798	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 12799	2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
⊢ 2 ∈ (ℤ≥‘1)
Theorem	5eluz3 12800	5 is an integer greater than or equal to 3. (Contributed by AV, 7-Sep-2025.)
⊢ 5 ∈ (ℤ≥‘3)