mmtheorems75 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7401-7500 - Page 75 of 107

Type	Label	Description
Statement
Theorem	sinf 7401	Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.)
⊢ sin:ℂ–→ℂ
Theorem	cosf 7402	Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.)
⊢ cos:ℂ–→ℂ
Theorem	sinnegt 7403	The sine of a negative is the negative of the sine.
⊢ (A ∈ ℂ → (sin ‘-A) = -(sin ‘A))
Theorem	cosnegt 7404	The cosines of a number and its negative are the same.
⊢ (A ∈ ℂ → (cos ‘-A) = (cos ‘A))
Theorem	sin0 7405	Value of the sine function at 0. (Contributed by Steve Rodriguez, 5-Jul-2006.)
⊢ (sin ‘0) = 0
Theorem	sin0ALT 7406	Value of the sine function at 0.
⊢ (sin ‘0) = 0
Theorem	cos0 7407	Value of the cosine function at 0.
⊢ (cos ‘0) = 1
Theorem	efivalt 7408	The exponential function in terms of sine and cosine.
⊢ (A ∈ ℂ → (exp ‘(i · A)) = ((cos ‘A) + (i · (sin ‘A))))
Theorem	efmivalt 7409	The exponential function in terms of sine and cosine.
⊢ (A ∈ ℂ → (exp ‘(-i · A)) = ((cos ‘A) − (i · (sin ‘A))))
Theorem	efeult 7410	Eulerian representation of the complex exponential. (Suggested by Jeffrey Hankins, 3-Jul-2006.)
⊢ (A ∈ ℂ → (exp ‘A) = ((exp ‘(ℜ ‘A)) · ((cos ‘(ℑ ‘A)) + (i · (sin ‘(ℑ ‘A))))))
Theorem	efieq 7411	The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.)
⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → ((exp ‘(i · A)) = (exp ‘(i · B)) ↔ ((cos ‘A) = (cos ‘B) ⋀ (sin ‘A) = (sin ‘B))))
Theorem	sinadd 7412	Sine addition formula for complex arguments. Equation 14 of [Gleason] p. 310.
⊢ A ∈ ℂ    &   ⊢ B ∈ ℂ    ⇒   ⊢ (sin ‘(A + B)) = (((sin ‘A) · (cos ‘B)) + ((cos ‘A) · (sin ‘B)))
Theorem	cosadd 7413	Addition formula for cosine. Equation 15 of [Gleason] p. 310.
⊢ A ∈ ℂ    &   ⊢ B ∈ ℂ    ⇒   ⊢ (cos ‘(A + B)) = (((cos ‘A) · (cos ‘B)) − ((sin ‘A) · (sin ‘B)))
Theorem	sinaddt 7414	Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.)
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (sin ‘(A + B)) = (((sin ‘A) · (cos ‘B)) + ((cos ‘A) · (sin ‘B))))
Theorem	cosaddt 7415	Addition formula for cosine. Equation 15 of [Gleason] p. 310.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (cos ‘(A + B)) = (((cos ‘A) · (cos ‘B)) − ((sin ‘A) · (sin ‘B))))
Theorem	sinsubt 7416	Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (sin ‘(A − B)) = (((sin ‘A) · (cos ‘B)) − ((cos ‘A) · (sin ‘B))))
Theorem	cossubt 7417	Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (cos ‘(A − B)) = (((cos ‘A) · (cos ‘B)) + ((sin ‘A) · (sin ‘B))))
Theorem	addsint 7418	Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → ((sin ‘A) + (sin ‘B)) = (2 · ((sin ‘((A + B) / 2)) · (cos ‘((A − B) / 2)))))
Theorem	subsint 7419	Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → ((sin ‘A) − (sin ‘B)) = (2 · ((cos ‘((A + B) / 2)) · (sin ‘((A − B) / 2)))))
Theorem	addcost 7420	Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → ((cos ‘A) + (cos ‘B)) = (2 · ((cos ‘((A + B) / 2)) · (cos ‘((A − B) / 2)))))
Theorem	subcost 7421	Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → ((cos ‘A) − (cos ‘B)) = (-2 · ((sin ‘((A + B) / 2)) · (sin ‘((A − B) / 2)))))
Theorem	sincossqt 7422	Sine squared plus cosine squared is 1. Equation 17 of [Gleason] p. 311. Note that this holds for non-real arguments, even though individually each term is unbounded.
⊢ (A ∈ ℂ → (((sin ‘A)↑2) + ((cos ‘A)↑2)) = 1)
Theorem	sin2tt 7423	Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)
⊢ (A ∈ ℂ → (sin ‘(2 · A)) = (2 · ((sin ‘A) · (cos ‘A))))
Theorem	cos2tt 7424	Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
⊢ (A ∈ ℂ → (cos ‘(2 · A)) = ((2 · ((cos ‘A)↑2)) − 1))
Theorem	cos2tOLD 7425	Double-angle formula for cosine. (Contributed by Paul Chapman, 25-Nov-2007.)
⊢ A ∈ ℂ    ⇒   ⊢ (cos ‘(2 · A)) = ((2 · ((cos ‘A)↑2)) − 1)
Theorem	sinbndt 7426	The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311.
⊢ (A ∈ ℝ → (-1 ≤ (sin ‘A) ⋀ (sin ‘A) ≤ 1))
Theorem	cosbndt 7427	The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311.
⊢ (A ∈ ℝ → (-1 ≤ (cos ‘A) ⋀ (cos ‘A) ≤ 1))
Theorem	sin01bndlem1 7428	Lemma for sin01bnd 7433 and cos01bnd 7434.
Theorem	sin01bndlem2 7429	Lemma for sin01bnd 7433.
Theorem	sin01bndlem3 7430	Lemma for sin01bnd 7433.
Theorem	cos01bndlem2 7431	Lemma for cos01bnd 7434.
Theorem	cos01bndlem3 7432	Lemma for cos01bnd 7434.
Theorem	sin01bnd 7433	Bounds on the sine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (A ∈ (0(,]1) → ((A − ((A↑3) / 3)) < (sin ‘A) ⋀ (sin ‘A) < A))
Theorem	cos01bnd 7434	Bounds on the cosine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (A ∈ (0(,]1) → ((1 − (2 · ((A↑2) / 3))) < (cos ‘A) ⋀ (cos ‘A) < (1 − ((A↑2) / 3))))
Theorem	cos1bnd 7435	Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ ((1 / 3) < (cos ‘1) ⋀ (cos ‘1) < (2 / 3))
Theorem	cos2bnd 7436	Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (-(7 / 9) < (cos ‘2) ⋀ (cos ‘2) < -(1 / 9))
Theorem	sin01gt0 7437	The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (A ∈ (0(,]1) → 0 < (sin ‘A))
Theorem	cos01gt0 7438	The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (A ∈ (0(,]1) → 0 < (cos ‘A))
Theorem	sin02gt0 7439	The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (A ∈ (0(,]2) → 0 < (sin ‘A))
Theorem	sincos1sgn 7440	The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (0 < (sin ‘1) ⋀ 0 < (cos ‘1))
Theorem	sincos2sgn 7441	The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (0 < (sin ‘2) ⋀ (cos ‘2) < 0)
Theorem	sin4lt0 7442	The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (sin ‘4) < 0
Theorem	absefit 7443	The absolute value of the exponential function of an imaginary number is one. Equation 48 of [Rudin] p. 167. (Contributed by Jason Orendorff, 9-Feb-2007.)
⊢ (A ∈ ℝ → (abs ‘(exp ‘(i · A))) = 1)
Theorem	abseft 7444	The absolute value of the exponential function is the exponential function of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)
⊢ (A ∈ ℂ → (abs ‘(exp ‘A)) = (exp ‘(ℜ ‘A)))
Theorem	demoivre 7445	De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. (Contributed by Steve Rodriguez, 10-Nov-2006.) Warning: The HTML proof page is 0.6 megabyte in size.
⊢ ((A ∈ ℂ ⋀ N ∈ ℕ0) → (((cos ‘A) + (i · (sin ‘A)))↑N) = ((cos ‘(N · A)) + (i · (sin ‘(N · A)))))
Theorem	demoivreALT 7446	Shorter proof of demoivre 7445 using the exponential function.
⊢ ((A ∈ ℂ ⋀ N ∈ ℕ0) → (((cos ‘A) + (i · (sin ‘A)))↑N) = ((cos ‘(N · A)) + (i · (sin ‘(N · A)))))
Axiom of dependent choice
Theorem	acdc3lem 7447	Lemma for acdc3 7448. Build a sequence G starting at value c, as follows. Given a previous value x of G, we construct, for the next value of G, the v such that ∀u ∈ (F ‘x)¬ urv, which is unique when r is a well-ordering on A.
Theorem	acdc3 7448	Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value C. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence.
⊢ A ∈ V    ⇒   ⊢ ((F:A–→(℘A ∖ {∅}) ⋀ C ∈ A) → ∃g(g:ℕ–→A ⋀ (g ‘1) = C ⋀ ∀k ∈ ℕ (g ‘(k + 1)) ∈ (F ‘(g ‘k))))
Theorem	acdc2lem1 7449	Lemma for acdc2 7451.
Theorem	acdc2lem2 7450	Lemma for acdc2 7451. Build a sequence G starting at value c, as follows. Given a previous value x of G, we construct, for the next value of G, the v such that ∀u ∈ (yFx)¬ urv, which is unique when r is a well-ordering on A.
Theorem	acdc2 7451	A more general version of acdc 7456 that allows the function F to vary with k.
⊢ A ∈ V    ⇒   ⊢ ((A ≠ ∅ ⋀ F:(ℕ × A)–→(℘A ∖ {∅})) → ∃g(g:ℕ–→A ⋀ ∀k ∈ ℕ (g ‘(k + 1)) ∈ ((k + 1)F(g ‘k))))
Theorem	acdc5lem1 7452	Lemma for acdc5 7454.
Theorem	acdc5lem2 7453	Lemma for acdc5 7454. Build a sequence G starting at value c, as follows. Given a previous value x of G, we construct, for the next value of G, the v such that ∀u ∈ (yFx)¬ urv, which is unique when r is a well-ordering on A.
Theorem	acdc5 7454	A more general version of acdc 7456 that has an initial value and where the function F depends on k.
⊢ A ∈ V    ⇒   ⊢ ((F:(ℕ × A)–→(℘A ∖ {∅}) ⋀ C ∈ A) → ∃g(g:ℕ–→A ⋀ (g ‘1) = C ⋀ ∀k ∈ ℕ (g ‘(k + 1)) ∈ ((k + 1)F(g ‘k))))
Theorem	acdclem 7455	Lemma for acdc 7456. Build a sequence G starting at value c, as follows. Given a previous value x of G, we construct, for the next value of G, the v such that ∀u ∈ (F ‘x)¬ urv, which is unique when r is a well-ordering on A.
Theorem	acdc 7456	Dependent Choice. Axiom DC1 of [Schechter] p. 149. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence.
⊢ A ∈ V    ⇒   ⊢ ((A ≠ ∅ ⋀ F:A–→(℘A ∖ {∅})) → ∃g(g:ℕ–→A ⋀ ∀k ∈ ℕ (g ‘(k + 1)) ∈ (F ‘(g ‘k))))
Theorem	acdcALT 7457	Dependent Choice. Axiom DC1 of [Schechter] p. 149. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence.
⊢ A ∈ V    ⇒   ⊢ ((A ≠ ∅ ⋀ F:A–→(℘A ∖ {∅})) → ∃g(g:ℕ–→A ⋀ ∀k ∈ ℕ (g ‘(k + 1)) ∈ (F ‘(g ‘k))))
Cardinality and cardinal arithmetic (cont.)
Countability of integers and rationals
Theorem	nn0ennn 7458	The nonnegative integers are equinumerous to the natural numbers.
⊢ ℕ0 ≈ ℕ
Theorem	nnenom 7459	The set of natural numbers (as a subset of complex numbers) is equinumerous to omega (the set of finite ordinal numbers).
⊢ ℕ ≈ ω
Theorem	xpnnen 7460	The cross product of the set of natural numbers with itself is equinumerous to the set of natural numbers. The key idea is to use nn0opth2t 6617 to show that the mapping from natural numbers z and w to ((z + w)↑2) + w is one-to-one.
⊢ (ℕ × ℕ) ≈ ℕ
Theorem	xpomen 7461	The cross product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133 (which proves this with a direct, but longer, proof; ours uses instead the Schroeder-Bernstein Theorem sbth 4453 in xpnnen 7460).
⊢ (ω × ω) ≈ ω
Theorem	znnenlem 7462	Lemma for znnen 7464.
Theorem	znnenlemOLD 7463	Lemma for znnen 7464.
Theorem	znnen 7464	The set of integers and the set of natural numbers are equinumerous. Exercise 1 of [Gleason] p. 140.
⊢ ℤ ≈ ℕ
Theorem	qnnen 7465	The rational numbers are countable. (This unusual proof uses the Axiom of Choice via fodom 4789 to make it much shorter, but this theorem can also be proved without it. See, for example, Exercise 2 of [Enderton] p. 133.)
⊢ ℚ ≈ ℕ
Infinite primes theorem
Theorem	unbenlem 7466	Lemma for unben 7467.
Theorem	unben 7467	An unbounded set of natural numbers is infinite.
⊢ ((A ⊆ ℕ ⋀ ∀m ∈ ℕ ∃n ∈ A m < n) → A ≈ ℕ)
Theorem	infpnlem1 7468	Lemma for infpn 7470. The smallest divisor (greater than 1) M of N! + 1 is a prime greater than N.
Theorem	infpnlem2 7469	Lemma for infpn 7470. For any natural number N, there exists a prime number j greater than N.
Theorem	infpn 7470	There exist infinitely many prime numbers: for any natural number N, there exists a prime number j greater than N. (See infpn2 7471 for the equinumerosity version.)
⊢ (N ∈ ℕ → ∃j ∈ ℕ (N < j ⋀ ∀k ∈ ℕ ((j / k) ∈ ℕ → (k = 1 ⋁ k = j))))
Theorem	infpn2 7471	There exist infinitely many prime numbers: the set of all primes S is unbounded by infpn 7470, so by unben 7467 it is infinite.
⊢ S = {n ∈ ℕ∣(1 < n ⋀ ∀m ∈ ℕ ((n / m) ∈ ℕ → (m = 1 ⋁ m = n)))}    ⇒   ⊢ S ≈ ℕ
The reals are uncountable
Theorem	ruclem1 7472	Lemma for ruc 7511 (the reals are uncountable). This is an arithmetic fact that will be used to compute ordering relations.
Theorem	ruclem2 7473	Lemma for ruc 7511. Arithmetic fact that will be used to compute ordering relations.
Theorem	ruclem3 7474	Lemma for ruc 7511. Arithmetic fact that will be used to compute ordering relations.
Theorem	ruclem4 7475	Lemma for ruc 7511. Helper lemma showing a tedious equality used several times.
Theorem	ruclem5 7476	Lemma for ruc 7511. Helper lemma showing the input function used for our recursive sequence builder (defined in ruclem13 7484) is a set.
Theorem	ruclem6 7477	Lemma for ruc 7511. Helper lemma showing the input function used for our recursive sequence builder (defined in ruclem13 7484) matches our input mapping F for successor values.
Theorem	ruclem7 7478	Lemma for ruc 7511. Helper lemma showing the initial value of the input function for our recursive sequence builder (defined in ruclem13 7484).
Theorem	ruclem8 7479	Lemma for ruc 7511. Helper lemma showing the successor value of the input function for our recursive sequence builder (defined in ruclem13 7484).
Theorem	ruclem9 7480	Lemma for ruc 7511. Helper lemma showing the operation used for our recursive sequence builder (defined in ruclem13 7484) is a set.
Theorem	ruclem10 7481	Lemma for ruc 7511. The values of our recursive sequence builder are pairs of real numbers. The values of our constructed function G are the first of these pairs.
Theorem	ruclem11 7482	Lemma for ruc 7511. The values of our recursive sequence builder are pairs of real numbers. The values of our constructed function H are the second of these pairs.
Theorem	ruclem12 7483	Lemma for ruc 7511. A helper lemma that changes bound variables.
Theorem	ruclem13 7484	Lemma for ruc 7511. A helper lemma showing the recursive sequence builder used for our construction maps natural numbers to pairs of reals.
Theorem	ruclem14 7485	Lemma for ruc 7511. A helper lemma showing the initial value of the recursive sequence builder used for our construction.
Theorem	ruclem15 7486	Lemma for ruc 7511. A helper lemma showing the successor value of the recursive sequence builder used for our construction.
Theorem	ruclem16 7487	Lemma for ruc 7511. A helper lemma showing the initial value of our constructed G.
Theorem	ruclem17 7488	Lemma for ruc 7511. A helper lemma showing our constructed function G maps ℕ to real numbers.
Theorem	ruclem18 7489	Lemma for ruc 7511. The value of our constructed function G when the value of the input function F lies between the previous values of G and H. This assignment to G defines a new interval between G and H (see also ruclem19 7490) that avoids the value of F.
Theorem	ruclem19 7490	Lemma for ruc 7511. The value of our constructed function H when the value of the input function F lies between the previous values of G and H. This assignment to H defines a new interval between G and H (see also ruclem18 7489) that avoids the value of F.
Theorem	ruclem20 7491	Lemma for ruc 7511. The value of our constructed function G when the value of the input function F does not lie between the previous values of G and H. This assignment to G just shrinks the interval between G and H by some arbitrary amount.
Theorem	ruclem21 7492	Lemma for ruc 7511. The value of our constructed function H when the value of the input function F does not lie between the previous values of G and H. This assignment to H just shrinks the interval between G and H by some arbitrary amount.
Theorem	ruclem22 7493	Lemma for ruc 7511. Each value of our constructed function G is a real number.
Theorem	ruclem23 7494	Lemma for ruc 7511. Each value of our constructed function H is a real number.
Theorem	ruclem24 7495	Lemma for ruc 7511. A helper lemma for the induction hypothesis in ruclem25 7496.
Theorem	ruclem25 7496	Lemma for ruc 7511. At any index A, the value of G is less than the value of H.
Theorem	ruclem26 7497	Lemma for ruc 7511. Our constructed function G has an ever-increasing set of values.
Theorem	ruclem27 7498	Lemma for ruc 7511. Our constructed function H has an ever-decreasing set of values.
Theorem	ruclem28 7499	Lemma for ruc 7511. A helper lemma for ruclem29 7500.
Theorem	ruclem29 7500	Lemma for ruc 7511. At any index A, the interval between our constructed functions G and H does not include the corresponding value of input function F. In other words, our constructed functions define, by ruclem26 7497 and ruclem27 7498, an ever-shrinking interval that eventually squeezes out all values of F.