P. 129 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12801-12900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	7p3e10 12801	7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
⊢ (7 + 3) = ;10
Theorem	7p4e11 12802	7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (7 + 4) = ;11
Theorem	7p5e12 12803	7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 + 5) = ;12
Theorem	7p6e13 12804	7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 + 6) = ;13
Theorem	7p7e14 12805	7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 + 7) = ;14
Theorem	8p2e10 12806	8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
⊢ (8 + 2) = ;10
Theorem	8p3e11 12807	8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 + 3) = ;11
Theorem	8p4e12 12808	8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 4) = ;12
Theorem	8p5e13 12809	8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 5) = ;13
Theorem	8p6e14 12810	8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 6) = ;14
Theorem	8p7e15 12811	8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 7) = ;15
Theorem	8p8e16 12812	8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 8) = ;16
Theorem	9p2e11 12813	9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (9 + 2) = ;11
Theorem	9p3e12 12814	9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 3) = ;12
Theorem	9p4e13 12815	9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 4) = ;13
Theorem	9p5e14 12816	9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 5) = ;14
Theorem	9p6e15 12817	9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 6) = ;15
Theorem	9p7e16 12818	9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 7) = ;16
Theorem	9p8e17 12819	9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 8) = ;17
Theorem	9p9e18 12820	9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 9) = ;18
Theorem	10p10e20 12821	10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (;10 + ;10) = ;20
Theorem	10m1e9 12822	10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
⊢ (;10 − 1) = 9
Theorem	4t3lem 12823	Lemma for 4t3e12 12824 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 12824	4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (4 · 3) = ;12
Theorem	4t4e16 12825	4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (4 · 4) = ;16
Theorem	5t2e10 12826	5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
⊢ (5 · 2) = ;10
Theorem	5t3e15 12827	5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (5 · 3) = ;15
Theorem	5t4e20 12828	5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (5 · 4) = ;20
Theorem	5t5e25 12829	5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (5 · 5) = ;25
Theorem	6t2e12 12830	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (6 · 2) = ;12
Theorem	6t3e18 12831	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (6 · 3) = ;18
Theorem	6t4e24 12832	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (6 · 4) = ;24
Theorem	6t5e30 12833	6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (6 · 5) = ;30
Theorem	6t6e36 12834	6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (6 · 6) = ;36
Theorem	7t2e14 12835	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 2) = ;14
Theorem	7t3e21 12836	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 3) = ;21
Theorem	7t4e28 12837	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 4) = ;28
Theorem	7t5e35 12838	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 5) = ;35
Theorem	7t6e42 12839	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 6) = ;42
Theorem	7t7e49 12840	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 · 7) = ;49
Theorem	8t2e16 12841	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 2) = ;16
Theorem	8t3e24 12842	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 3) = ;24
Theorem	8t4e32 12843	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 4) = ;32
Theorem	8t5e40 12844	8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 · 5) = ;40
Theorem	8t6e48 12845	8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 · 6) = ;48
Theorem	8t7e56 12846	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 7) = ;56
Theorem	8t8e64 12847	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 8) = ;64
Theorem	9t2e18 12848	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 2) = ;18
Theorem	9t3e27 12849	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 3) = ;27
Theorem	9t4e36 12850	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 4) = ;36
Theorem	9t5e45 12851	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 5) = ;45
Theorem	9t6e54 12852	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 6) = ;54
Theorem	9t7e63 12853	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 7) = ;63
Theorem	9t8e72 12854	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 8) = ;72
Theorem	9t9e81 12855	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 9) = ;81
Theorem	9t11e99 12856	9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (9 · ;11) = ;99
Theorem	9t11e99OLD 12857	Obsolete version of 9t11e99 12856 as of 10-Jun-2026. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (9 · ;11) = ;99
Theorem	9lt10 12858	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 9 < ;10
Theorem	8lt10 12859	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 8 < ;10
Theorem	7lt10 12860	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 7 < ;10
Theorem	6lt10 12861	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 6 < ;10
Theorem	5lt10 12862	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 5 < ;10
Theorem	4lt10 12863	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 4 < ;10
Theorem	3lt10 12864	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 3 < ;10
Theorem	2lt10 12865	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 2 < ;10
Theorem	1lt10 12866	1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 1 < ;10
Theorem	1lt10OLD 12867	Obsolete version of 1lt10 12866 as of 10-Jun-2026. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 1 < ;10
Theorem	decbin0 12868	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ (4 · 𝐴) = (2 · (2 · 𝐴))
Theorem	decbin2 12869	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1))
Theorem	decbin3 12870	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 12871	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 12872	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 12873*	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 12874 for its value, uzssz 12893 for its relationship to ℤ, nnuz 12911 and nn0uz 12910 for its relationships to ℕ and ℕ0, and eluz1 12876 and eluz2 12878 for its membership relations. (Contributed by NM, 5-Sep-2005.)
⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘})
Theorem	uzval 12874*	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 12875	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 12876	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 12877	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 12878	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 12879	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 12880	Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
⊢ 𝑀 ∈ ℤ    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
Theorem	eluzuzle 12881	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 12882	A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
Theorem	eluzelre 12883	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 12884	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 12885	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁)
Theorem	eluz 12886	Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
Theorem	uzid 12887	Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
Theorem	uzidd 12888	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 12889	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅)
Theorem	uztrn 12890	Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁))
Theorem	uztrn2 12891	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 12892	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ (ℤ≥‘-𝑁))
Theorem	uzssz 12893	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 12894	An upper set of integers is a subset of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (ℤ≥‘𝑀) ⊆ ℝ
Theorem	uzss 12895	Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀))
Theorem	uztric 12896	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 12897	The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ↔ 𝑀 = 𝑁))
Theorem	eluzp1m1 12898	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 12899	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 12900	Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1)))