mmtheorems61 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6001-6100 - Page 61 of 108

Type	Label	Description
Statement
Theorem	2nn 6001	2 is a natural number.
⊢ 2 ∈ ℕ
Theorem	3nn 6002	3 is a natural number.
⊢ 3 ∈ ℕ
Some properties of specific numbers
Theorem	2p2e4 6003	Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpegif/mmset.html#trivia.
⊢ (2 + 2) = 4
Theorem	4nn 6004	4 is a natural number.
⊢ 4 ∈ ℕ
Theorem	2times 6005	Two times a number.
⊢ A ∈ ℂ    ⇒   ⊢ (2 · A) = (A + A)
Theorem	2timest 6006	Two times a number.
⊢ (A ∈ ℂ → (2 · A) = (A + A))
Theorem	times2t 6007	A number times 2.
⊢ (A ∈ ℂ → (A · 2) = (A + A))
Theorem	times2 6008	A number times 2.
⊢ A ∈ ℂ    ⇒   ⊢ (A · 2) = (A + A)
Theorem	3p2e5 6009	3 + 2 = 5.
⊢ (3 + 2) = 5
Theorem	3p3e6 6010	3 + 3 = 6.
⊢ (3 + 3) = 6
Theorem	4p2e6 6011	4 + 2 = 6.
⊢ (4 + 2) = 6
Theorem	4p3e7 6012	4 + 3 = 7.
⊢ (4 + 3) = 7
Theorem	4p4e8 6013	4 + 4 = 8.
⊢ (4 + 4) = 8
Theorem	5p2e7 6014	5 + 2 = 7.
⊢ (5 + 2) = 7
Theorem	5p3e8 6015	5 + 3 = 8.
⊢ (5 + 3) = 8
Theorem	5p4e9 6016	5 + 4 = 9.
⊢ (5 + 4) = 9
Theorem	5p5e10 6017	5 + 5 = 10.
⊢ (5 + 5) = 10
Theorem	6p2e8 6018	6 + 2 = 8.
⊢ (6 + 2) = 8
Theorem	6p3e9 6019	6 + 3 = 9.
⊢ (6 + 3) = 9
Theorem	6p4e10 6020	6 + 4 = 10.
⊢ (6 + 4) = 10
Theorem	7p2e9 6021	7 + 2 = 9.
⊢ (7 + 2) = 9
Theorem	7p3e10 6022	7 + 3 = 10.
⊢ (7 + 3) = 10
Theorem	8p2e10 6023	8 + 2 = 10.
⊢ (8 + 2) = 10
Theorem	2t2e4 6024	2 times 2 equals 4.
⊢ (2 · 2) = 4
Theorem	3t2e6 6025	3 times 2 equals 6.
⊢ (3 · 2) = 6
Theorem	3t3e9 6026	3 times 3 equals 9.
⊢ (3 · 3) = 9
Theorem	4t2e8 6027	4 times 2 equals 8.
⊢ (4 · 2) = 8
Theorem	5t2e10 6028	5 times 2 equals 10.
⊢ (5 · 2) = 10
Theorem	4d2e2 6029	One half of four is two.
⊢ (4 / 2) = 2
Theorem	1lt2 6030	1 is less than 2.
⊢ 1 < 2
Theorem	halfgt0 6031	One-half is greater than zero.
⊢ 0 < (1 / 2)
Theorem	halflt1 6032	One-half is less than one.
⊢ (1 / 2) < 1
Theorem	8th4div3 6033	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
⊢ ((1 / 8) · (4 / 3)) = (1 / 6)
Theorem	halfpm6th 6034	One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
⊢ (((1 / 2) − (1 / 6)) = (1 / 3) ⋀ ((1 / 2) + (1 / 6)) = (2 / 3))
Theorem	halfclt 6035	Closure of half of a number (frequently used special case).
⊢ (A ∈ ℂ → (A / 2) ∈ ℂ)
Theorem	rehalfclt 6036	Real closure of half.
⊢ (A ∈ ℝ → (A / 2) ∈ ℝ)
Theorem	half0t 6037	Half of a number is zero iff the number is zero.
⊢ (A ∈ ℂ → ((A / 2) = 0 ↔ A = 0))
Theorem	halfpost 6038	A positive number is greater than its half.
⊢ (A ∈ ℝ → (0 < A ↔ (A / 2) < A))
Theorem	halfpos2t 6039	A number is positive iff its half is positive.
⊢ (A ∈ ℝ → (0 < A ↔ 0 < (A / 2)))
Theorem	halfnneg2t 6040	A number is nonnegative iff its half is nonnegative.
⊢ (A ∈ ℝ → (0 ≤ A ↔ 0 ≤ (A / 2)))
Theorem	2halvest 6041	Two halves make a whole.
⊢ (A ∈ ℂ → ((A / 2) + (A / 2)) = A)
Theorem	halfaddsubcl 6042	Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (((A + B) / 2) ∈ ℂ ⋀ ((A − B) / 2) ∈ ℂ))
Theorem	halfaddsubt 6043	Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → ((((A + B) / 2) + ((A − B) / 2)) = A ⋀ (((A + B) / 2) − ((A − B) / 2)) = B))
Theorem	lt2halvest 6044	A sum is less than the whole if each term is less than half.
⊢ ((A ∈ ℝ ⋀ B ∈ ℝ ⋀ C ∈ ℝ) → ((A < (C / 2) ⋀ B < (C / 2)) → (A + B) < C))
Theorem	nominpos 6045	There is no smallest positive real number.
⊢ ¬ ∃x ∈ ℝ (0 < x ⋀ ¬ ∃y ∈ ℝ (0 < y ⋀ y < x))
Theorem	avglet 6046	The average of two numbers is less than or equal to at least one of them.
⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → (((A + B) / 2) ≤ A ⋁ ((A + B) / 2) ≤ B))
Completeness Axiom and Suprema
Theorem	lbreu 6047	If a set of reals contains a lower bound, it contains a unique lower bound.
⊢ ((S ⊆ ℝ ⋀ ∃x ∈ S ∀y ∈ S x ≤ y) → ∃!x ∈ S ∀y ∈ S x ≤ y)
Theorem	lbcl 6048	If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set.
⊢ ((S ⊆ ℝ ⋀ ∃x ∈ S ∀y ∈ S x ≤ y) → ∪{x ∈ S∣∀y ∈ S x ≤ y} ∈ S)
Theorem	lble 6049	If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set.
⊢ ((S ⊆ ℝ ⋀ ∃x ∈ S ∀y ∈ S x ≤ y ⋀ A ∈ S) → ∪{x ∈ S∣∀y ∈ S x ≤ y} ≤ A)
Theorem	lbinfm 6050	If a set of reals contains a lower bound, the lower bound is its infimum.
⊢ ((S ⊆ ℝ ⋀ ∃x ∈ S ∀y ∈ S x ≤ y) → sup(S, ℝ, ◡ < ) = ∪{x ∈ S∣∀y ∈ S x ≤ y})
Theorem	lbinfmcl 6051	If a set of reals contains a lower bound, it contains its infimum.
⊢ ((S ⊆ ℝ ⋀ ∃x ∈ S ∀y ∈ S x ≤ y) → sup(S, ℝ, ◡ < ) ∈ S)
Theorem	lbinfmle 6052	If a set of reals contains a lower bound, its infmimum is less than or equal to all members of the set.
⊢ ((S ⊆ ℝ ⋀ ∃x ∈ S ∀y ∈ S x ≤ y ⋀ A ∈ S) → sup(S, ℝ, ◡ < ) ≤ A)
Theorem	sup2 6053	A non-empty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent).
⊢ ((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A (y < x ⋁ y = x)) → ∃x ∈ ℝ (∀y ∈ A ¬ x < y ⋀ ∀y ∈ ℝ (y < x → ∃z ∈ A y < z)))
Theorem	sup3 6054	A version of the completeness axiom for reals.
⊢ ((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x) → ∃x ∈ ℝ (∀y ∈ A ¬ x < y ⋀ ∀y ∈ ℝ (y < x → ∃z ∈ A y < z)))
Theorem	infm3lem 6055	Lemma for infm3 6056.
Theorem	infm3 6056	The completeness axiom for reals in terms of infimum: a non-empty, bounded-below set of reals has a infimum. (This theorem is the dual of sup3 6054.)
⊢ ((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A x ≤ y) → ∃x ∈ ℝ (∀y ∈ A ¬ y < x ⋀ ∀y ∈ ℝ (x < y → ∃z ∈ A z < y)))
Theorem	suprcl 6057	Closure of supremum of a non-empty bounded set of reals.
⊢ ((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x) → sup(A, ℝ, < ) ∈ ℝ)
Theorem	suprub 6058	A member of a non-empty bounded set of reals is less than or equal to the set's upper bound.
⊢ (((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x) ⋀ B ∈ A) → B ≤ sup(A, ℝ, < ))
Theorem	suprlub 6059	The supremum of a non-empty bounded set of reals is the least upper bound.
⊢ (((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x) ⋀ (B ∈ ℝ ⋀ B < sup(A, ℝ, < ))) → ∃z ∈ A B < z)
Theorem	suprnub 6060	An upper bound is not less than the supremum of a non-empty bounded set of reals.
⊢ (((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x) ⋀ (B ∈ ℝ ⋀ ∀z ∈ A ¬ B < z)) → ¬ B < sup(A, ℝ, < ))
Theorem	suprleub 6061	The supremum of a non-empty bounded set of reals is less than or equal to an upper bound.
⊢ (((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x) ⋀ (B ∈ ℝ ⋀ ∀z ∈ A z ≤ B)) → sup(A, ℝ, < ) ≤ B)
Theorem	sup3i 6062	A version of the completeness axiom for reals.
⊢ (A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x)    ⇒   ⊢ ∃x ∈ ℝ (∀y ∈ A ¬ x < y ⋀ ∀y ∈ ℝ (y < x → ∃z ∈ A y < z))
Theorem	suprcli 6063	Closure of supremum of a non-empty bounded set of reals.
⊢ (A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x)    ⇒   ⊢ sup(A, ℝ, < ) ∈ ℝ
Theorem	suprubi 6064	A member of a non-empty bounded set of reals is less than or equal to the set's upper bound.
⊢ (A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x)    ⇒   ⊢ (B ∈ A → B ≤ sup(A, ℝ, < ))
Theorem	suprlubi 6065	The supremum of a non-empty bounded set of reals is the least upper bound.
⊢ (A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x)    ⇒   ⊢ ((B ∈ ℝ ⋀ B < sup(A, ℝ, < )) → ∃z ∈ A B < z)
Theorem	suprnubi 6066	An upper bound is not less than the supremum of a non-empty bounded set of reals.
⊢ (A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x)    ⇒   ⊢ ((B ∈ ℝ ⋀ ∀z ∈ A ¬ B < z) → ¬ B < sup(A, ℝ, < ))
Theorem	suprleubi 6067	The supremum of a non-empty bounded set of reals is less than or equal to an upper bound.
⊢ (A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x)    ⇒   ⊢ ((B ∈ ℝ ⋀ ∀z ∈ A z ≤ B) → sup(A, ℝ, < ) ≤ B)
Theorem	reuunineg 6068	The negative of the unique real such that φ.
⊢ (x = -y → (φ ↔ ψ))    ⇒   ⊢ (∃!x ∈ ℝ φ → ∪{x ∈ ℝ∣φ} = -∪{y ∈ ℝ∣ψ})
Theorem	dfinfmr 6069	The infimum (expressed as supremum with converse 'less-than') of a set of reals A.
⊢ (A ⊆ ℝ → sup(A, ℝ, ◡ < ) = ∪{x ∈ ℝ∣(∀y ∈ A x ≤ y ⋀ ∀y ∈ ℝ (x < y → ∃z ∈ A z < y))})
Theorem	infmsup 6070	The infimum (expressed as supremum with converse 'less-than') of a set of reals A is the negative of the supremum of the negatives of its elements. The antecedent ensures that A is nonempty and has a lower bound.
⊢ ((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A x ≤ y) → sup(A, ℝ, ◡ < ) = -sup({z ∈ ℝ∣-z ∈ A}, ℝ, < ))
Theorem	infmrcl 6071	Closure of infimum of a non-empty bounded set of reals.
⊢ ((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A x ≤ y) → sup(A, ℝ, ◡ < ) ∈ ℝ)
Theorem	nnunb 6072	The set of natural numbers is unbounded above. Theorem I.28 of [Apostol] p. 26.
⊢ ¬ ∃x ∈ ℝ ∀y ∈ ℕ (y < x ⋁ y = x)
Theorem	arch 6073	Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26.
⊢ (A ∈ ℝ → ∃n ∈ ℕ A < n)
Theorem	nnreclt 6074	There exists a natural number whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28.
⊢ ((A ∈ ℝ ⋀ 0 < A) → ∃n ∈ ℕ (1 / n) < A)
Theorem	bndndx 6075	A bounded real sequence A(k) is less than or equal to at least one of its indices.
⊢ (∃x ∈ ℝ ∀k ∈ ℕ (A ∈ ℝ ⋀ A ≤ x) → ∃k ∈ ℕ A ≤ k)
Supremum on the extended reals
Theorem	xrsupexmnf 6076	Adding minus infinity to a set does not affect the existence of its supremum.
⊢ (∃x ∈ ℝ* (∀y ∈ A ¬ x < y ⋀ ∀y ∈ ℝ* (y < x → ∃z ∈ A y < z)) → ∃x ∈ ℝ* (∀y ∈ (A ∪ { -∞}) ¬ x < y ⋀ ∀y ∈ ℝ* (y < x → ∃z ∈ (A ∪ { -∞})y < z)))
Theorem	xrinfmexpnf 6077	Adding plus infinity to a set does not affect the existence of its infimum.
⊢ (∃x ∈ ℝ* (∀y ∈ A ¬ y < x ⋀ ∀y ∈ ℝ* (x < y → ∃z ∈ A z < y)) → ∃x ∈ ℝ* (∀y ∈ (A ∪ { +∞}) ¬ y < x ⋀ ∀y ∈ ℝ* (x < y → ∃z ∈ (A ∪ { +∞})z < y)))
Theorem	xrsupsslem 6078	Lemma for xrsupss 6080.
Theorem	xrinfmsslem 6079	Lemma for xrinfmss 6081.
Theorem	xrsupss 6080	Any subset of extended reals has a supremum.
⊢ (A ⊆ ℝ* → ∃x ∈ ℝ* (∀y ∈ A ¬ x < y ⋀ ∀y ∈ ℝ* (y < x → ∃z ∈ A y < z)))
Theorem	xrinfmss 6081	Any subset of extended reals has an infimum.
⊢ (A ⊆ ℝ* → ∃x ∈ ℝ* (∀y ∈ A ¬ y < x ⋀ ∀y ∈ ℝ* (x < y → ∃z ∈ A z < y)))
Theorem	xrub 6082	By quantifying only over reals, we can specify any extended real upper bound for any set of extended reals.
⊢ ((A ⊆ ℝ* ⋀ B ∈ ℝ*) → (∀x ∈ ℝ (x < B → ∃y ∈ A x < y) ↔ ∀x ∈ ℝ* (x < B → ∃y ∈ A x < y)))
Theorem	supxr 6083	The supremum of a set of extended reals.
⊢ (((A ⊆ ℝ* ⋀ B ∈ ℝ*) ⋀ (∀x ∈ A ¬ B < x ⋀ ∀x ∈ ℝ (x < B → ∃y ∈ A x < y))) → sup(A, ℝ*, < ) = B)
Theorem	supxr2 6084	The supremum of a set of extended reals.
⊢ (((A ⊆ ℝ* ⋀ B ∈ ℝ*) ⋀ (∀x ∈ A x ≤ B ⋀ ∀x ∈ ℝ (x < B → ∃y ∈ A x < y))) → sup(A, ℝ*, < ) = B)
Theorem	supxrre 6085	The real and extended real suprema match when the real supremum exists.
⊢ ((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y ≤ x) → sup(A, ℝ*, < ) = sup(A, ℝ, < ))
Theorem	supxrcl 6086	The supremum of an arbitrary set of extended reals is an extended real.
⊢ (A ⊆ ℝ* → sup(A, ℝ*, < ) ∈ ℝ*)
Theorem	supxrun 6087	The supremum of the union of two sets of extended reals equals the largest of their suprema.
⊢ ((A ⊆ ℝ* ⋀ B ⊆ ℝ* ⋀ sup(A, ℝ*, < ) ≤ sup(B, ℝ*, < )) → sup((A ∪ B), ℝ*, < ) = sup(B, ℝ*, < ))
Theorem	infmxrcl 6088	The infimum of an arbitrary set of extended reals is an extended real.
⊢ (A ⊆ ℝ* → sup(A, ℝ*, ◡ < ) ∈ ℝ*)
Theorem	supxrmnf 6089	Adding minus infinity to a set does not affect its supremum.
⊢ (A ⊆ ℝ* → sup((A ∪ { -∞}), ℝ*, < ) = sup(A, ℝ*, < ))
Theorem	supxrpnf 6090	The supremum of a set of extended reals containing plus infnity is plus infinity.
⊢ ((A ⊆ ℝ* ⋀ +∞ ∈ A) → sup(A, ℝ*, < ) = +∞)
Theorem	supxrunb1 6091	The supremum of an unbounded-above set of extended reals is plus infinity.
⊢ (A ⊆ ℝ* → (∀x ∈ ℝ ∃y ∈ A x ≤ y ↔ sup(A, ℝ*, < ) = +∞))
Theorem	supxrunb2 6092	The supremum of an unbounded-above set of extended reals is plus infinity.
⊢ (A ⊆ ℝ* → (∀x ∈ ℝ ∃y ∈ A x < y ↔ sup(A, ℝ*, < ) = +∞))
Theorem	supxrbnd 6093	The supremum of a bounded-above nonempty set of reals is real.
⊢ ((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ sup(A, ℝ*, < ) < +∞) → sup(A, ℝ*, < ) ∈ ℝ)
Theorem	supxrgtmnf 6094	The supremum of a nonempty set of reals is greater than minus infinity.
⊢ ((A ⊆ ℝ ⋀ A ≠ ∅) → -∞ < sup(A, ℝ*, < ))
Theorem	supxrre1 6095	The supremum of a nonempty set of reals is real iff it is less than plus infinity.
⊢ ((A ⊆ ℝ ⋀ A ≠ ∅) → (sup(A, ℝ*, < ) ∈ ℝ ↔ sup(A, ℝ*, < ) < +∞))
Theorem	supxrre2 6096	The supremum of a nonempty set of reals is real iff it is not plus infinity.
⊢ ((A ⊆ ℝ ⋀ A ≠ ∅) → (sup(A, ℝ*, < ) ∈ ℝ ↔ sup(A, ℝ*, < ) ≠ +∞))
Theorem	supxrbnd1 6097	The supremum of a bounded-above set of extended reals is less than infinity.
⊢ (A ⊆ ℝ* → (∃x ∈ ℝ ∀y ∈ A y < x ↔ sup(A, ℝ*, < ) < +∞))
Theorem	supxrbnd2 6098	The supremum of a bounded-above set of extended reals is less than infinity.
⊢ (A ⊆ ℝ* → (∃x ∈ ℝ ∀y ∈ A y ≤ x ↔ sup(A, ℝ*, < ) < +∞))
Theorem	xrsup0 6099	The supremum of an empty set under the extended reals is minus infinity.
⊢ sup(∅, ℝ*, < ) = -∞
Theorem	supxrub 6100	A member of a set of extended reals is less than or equal to the set's supremum.
⊢ ((A ⊆ ℝ* ⋀ B ∈ A) → B ≤ sup(A, ℝ*, < ))