mmtheorems67 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6601-6700 - Page 67 of 107

Type	Label	Description
Statement
Theorem	sq01t 6601	If a complex number equals its square, it must be 0 or 1.
⊢ (A ∈ ℂ → ((A↑2) = A ↔ (A = 0 ⋁ A = 1)))
Theorem	bernneq 6602	Bernoulli's inequality, due to Johan Bernoulli (1667-1748).
⊢ ((A ∈ ℝ ⋀ N ∈ ℕ0 ⋀ -1 ≤ A) → (1 + (A · N)) ≤ ((1 + A)↑N))
Theorem	bernneq2 6603	Variation of Bernoulli's inequality bernneq 6602.
⊢ ((A ∈ ℝ ⋀ N ∈ ℕ0 ⋀ 0 ≤ A) → (((A − 1) · N) + 1) ≤ (A↑N))
Theorem	expnbndt 6604	Exponentiation with a mantissa greater than 1 has no upper bound.
⊢ ((A ∈ ℝ ⋀ B ∈ ℝ ⋀ 1 < B) → ∃k ∈ ℕ A < (B↑k))
Discriminant
Theorem	discrlem1 6605	Lemma for discriminant theorem.
Theorem	discrlem2 6606	Lemma for discriminant theorem.
Theorem	discrlem3 6607	Lemma for discriminant theorem.
Theorem	discrlem 6608	If a quadratic polynomial with real coefficients is nonnegative for all values, then its discriminant is non-positive. The antecedent 0 ≤ A is redundant but simplifies the proof.
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ C ∈ ℝ    &   ⊢ ∀x ∈ ℝ 0 ≤ (((A · (x↑2)) + (B · x)) + C)    ⇒   ⊢ (0 ≤ A → ((B↑2) − (4 · (A · C))) ≤ 0)
More natural number properties
Theorem	nnsqcl 6609	The square of a natural number is a natural number.
⊢ N ∈ ℕ    ⇒   ⊢ (N↑2) ∈ ℕ
Theorem	nnlesq 6610	A natural number is less than or equal to its square.
⊢ N ∈ ℕ    ⇒   ⊢ N ≤ (N↑2)
Theorem	nnesq 6611	A natural number is even iff its square is even.
⊢ N ∈ ℕ    ⇒   ⊢ ((N / 2) ∈ ℕ ↔ ((N↑2) / 2) ∈ ℕ)
Ordered pair theorem for nonnegative integers
Theorem	nn0le2msqt 6612	The square function on nonnegative integers is monotonic. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ A ∈ ℕ0    &   ⊢ B ∈ ℕ0    ⇒   ⊢ (A ≤ B ↔ (A · A) ≤ (B · B))
Theorem	nn0opthlem1 6613	A rather pretty lemma for nn0opth 6615. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ A ∈ ℕ0    &   ⊢ C ∈ ℕ0    ⇒   ⊢ (A < C ↔ ((A · A) + (2 · A)) < (C · C))
Theorem	nn0opthlem2 6614	Lemma for nn0opth 6615.
Theorem	nn0opth 6615	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers A and B by (((A + B) · (A + B)) + B). If two such ordered pairs are equal, their first elements are equal and their second elements are equal. Contrast this ordered pair representation with the standard one df-op 2412 that works for any set. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ A ∈ ℕ0    &   ⊢ B ∈ ℕ0    &   ⊢ C ∈ ℕ0    &   ⊢ D ∈ ℕ0    ⇒   ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) ↔ (A = C ⋀ B = D))
Theorem	nn0opth2 6616	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opth 6615.
⊢ A ∈ ℕ0    &   ⊢ B ∈ ℕ0    &   ⊢ C ∈ ℕ0    &   ⊢ D ∈ ℕ0    ⇒   ⊢ ((((A + B)↑2) + B) = (((C + D)↑2) + D) ↔ (A = C ⋀ B = D))
Theorem	nn0opth2t 6617	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opth 6615.
⊢ (((A ∈ ℕ0 ⋀ B ∈ ℕ0) ⋀ (C ∈ ℕ0 ⋀ D ∈ ℕ0)) → ((((A + B)↑2) + B) = (((C + D)↑2) + D) ↔ (A = C ⋀ B = D)))
Square root
Syntax	csqr 6618	Extend class notation to include positive square root of a positive real number.
class √
Definition	df-sqr 6619	Define a function whose value is the square root of a nonnegative real number. The square root of x is the supremum of all reals whose square is less than x. See sqrcl 6649 for its closure, sqrval 6620 for its value, sqrsq 6669 and sqsqr 6670 for its relationship to squares, and sqr11 6652 for uniqueness.
⊢ √ = {⟨x, y⟩∣((x ∈ ℝ ⋀ 0 ≤ x) ⋀ y = sup({z ∈ ℝ∣(0 ≤ z ⋀ (z · z) ≤ x)}, ℝ, < ))}
Theorem	sqrval 6620	Value of square root function.
⊢ ((A ∈ ℝ ⋀ 0 ≤ A) → (√ ‘A) = sup({x ∈ ℝ∣(0 ≤ x ⋀ (x · x) ≤ A)}, ℝ, < ))
Theorem	sqr0 6621	Square root of zero.
⊢ (√ ‘0) = 0
Theorem	sqrlem1 6622	Lemma for square root theorem.
Theorem	sqrlem2 6623	Lemma for square root theorem.
Theorem	sqrlem3 6624	Lemma for square root theorem.
Theorem	sqrlem4 6625	Lemma for square root theorem.
Theorem	sqrlem5 6626	Lemma for square root theorem.
Theorem	sqrlem6 6627	Lemma for square root theorem.
Theorem	sqrlem7 6628	Lemma for square root theorem.
Theorem	sqrlem8 6629	Lemma for square root theorem.
Theorem	sqrlem9 6630	Lemma for square root theorem.
Theorem	sqrlem10 6631	Lemma for square root theorem.
Theorem	sqrlem11 6632	Lemma for square root theorem.
Theorem	sqrlem12 6633	Lemma for square root theorem.
Theorem	sqrlem13 6634	Lemma for square root theorem.
Theorem	sqrlem14 6635	Lemma for square root theorem.
Theorem	sqrlem15 6636	Lemma for square root theorem.
Theorem	sqrlem16 6637	Lemma for square root theorem.
Theorem	sqrlem17 6638	Lemma for square root theorem.
Theorem	sqrlem18 6639	Lemma for square root theorem.
Theorem	sqrlem19 6640	Lemma for square root theorem.
Theorem	sqrlem20 6641	Lemma for square root theorem.
Theorem	sqrlem21 6642	Lemma for square root theorem.
Theorem	sqrlem22 6643	Lemma for square root theorem.
Theorem	sqrlem23 6644	Lemma for square root theorem.
Theorem	sqrlem24 6645	Lemma for square root closure.
Theorem	sqrgt0i 6646	The square root of a positive real is positive.
⊢ A ∈ ℝ    &   ⊢ 0 < A    ⇒   ⊢ 0 < (√ ‘A)
Theorem	sqrlem26 6647	Lemma for square root theorem.
Theorem	sqrth 6648	Square root theorem. Theorem I.35 of [Apostol] p. 29. (A bit of trivia: This theorem was added to the database before the number 2 was defined and before exponents were defined. Thus you will see (1 + 1) and (x · x) throughout its lemmas.)
⊢ A ∈ ℝ    ⇒   ⊢ (0 ≤ A → ((√ ‘A) · (√ ‘A)) = A)
Theorem	sqrcl 6649	The square root of a nonnegative real is a real.
⊢ A ∈ ℝ    ⇒   ⊢ (0 ≤ A → (√ ‘A) ∈ ℝ)
Theorem	sqrgt0 6650	The square root of a positive real is positive.
⊢ A ∈ ℝ    ⇒   ⊢ (0 < A → 0 < (√ ‘A))
Theorem	sqrge0 6651	The square root of a nonnegative real is nonnegative.
⊢ A ∈ ℝ    ⇒   ⊢ (0 ≤ A → 0 ≤ (√ ‘A))
Theorem	sqr11 6652	The square root function is one-to-one.
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ⋀ 0 ≤ B) → ((√ ‘A) = (√ ‘B) ↔ A = B))
Theorem	sqrmuli 6653	Square root distributes over multiplication.
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 0 ≤ A    &   ⊢ 0 ≤ B    ⇒   ⊢ (√ ‘(A · B)) = ((√ ‘A) · (√ ‘B))
Theorem	sqrmul 6654	Square root distributes over multiplication.
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ⋀ 0 ≤ B) → (√ ‘(A · B)) = ((√ ‘A) · (√ ‘B)))
Theorem	sqrmsq2 6655	Relationship between square root and squares.
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ⋀ 0 ≤ B) → ((√ ‘A) = B ↔ A = (B · B)))
Theorem	sqrle 6656	Square root is monotonic.
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ⋀ 0 ≤ B) → (A ≤ B ↔ (√ ‘A) ≤ (√ ‘B)))
Theorem	sqrlt 6657	Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ⋀ 0 ≤ B) → (A < B ↔ (√ ‘A) < (√ ‘B)))
Theorem	sqrmsq 6658	Square root of square.
⊢ A ∈ ℝ    ⇒   ⊢ (0 ≤ A → (√ ‘(A · A)) = A)
Theorem	sqrclt 6659	The square root of a nonnegative real is a real.
⊢ ((A ∈ ℝ ⋀ 0 ≤ A) → (√ ‘A) ∈ ℝ)
Theorem	sqrgt0t 6660	The square root of a positive real is positive.
⊢ ((A ∈ ℝ ⋀ 0 < A) → 0 < (√ ‘A))
Theorem	sqrge0t 6661	The square root of a nonnegative real is nonnegative.
⊢ ((A ∈ ℝ ⋀ 0 ≤ A) → 0 ≤ (√ ‘A))
Theorem	sqrlet 6662	Square root is monotonic.
⊢ (((A ∈ ℝ ⋀ B ∈ ℝ) ⋀ (0 ≤ A ⋀ 0 ≤ B)) → (A ≤ B ↔ (√ ‘A) ≤ (√ ‘B)))
Theorem	sqr00t 6663	A square root is zero iff its argument is 0.
⊢ ((A ∈ ℝ ⋀ 0 ≤ A) → ((√ ‘A) = 0 ↔ A = 0))
Theorem	rpsqrclt 6664	The square root of a positive real is a postive real.
⊢ (A ∈ ℝ+ → (√ ‘A) ∈ ℝ+)
Theorem	sqr1 6665	The square root of 1 is 1.
⊢ (√ ‘1) = 1
Theorem	sqr4 6666	The square root of 4 is 2.
⊢ (√ ‘4) = 2
Theorem	sqr9 6667	The square root of 9 is 3.
⊢ (√ ‘9) = 3
Theorem	sqr2gt1lt2 6668	The square root of 2 is bounded by 1 and 2. (Contributed by Roy F. Longton, 8-Aug-2005.)
⊢ (1 < (√ ‘2) ⋀ (√ ‘2) < 2)
Theorem	sqrsq 6669	Square root of square.
⊢ A ∈ ℝ    ⇒   ⊢ (0 ≤ A → (√ ‘(A↑2)) = A)
Theorem	sqsqr 6670	Square of square root.
⊢ A ∈ ℝ    ⇒   ⊢ (0 ≤ A → ((√ ‘A)↑2) = A)
Theorem	sqrsqt 6671	Square root of square.
⊢ ((A ∈ ℝ ⋀ 0 ≤ A) → (√ ‘(A↑2)) = A)
Theorem	sqsqrt 6672	Square of square root.
⊢ ((A ∈ ℝ ⋀ 0 ≤ A) → ((√ ‘A)↑2) = A)
Irrationality of square root of 2
Theorem	sqr2irrlem1 6673	Lemma for irrationality of square root of 2. Technical lemma used to simplify the main induction step.
Theorem	sqr2irrlem2 6674	Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.
Theorem	sqr2irrlem3 6675	Main theorem for irrationality of square root of 2. There are no natural numbers such that the square of one is twice the square of the other. Uses strong induction.
⊢ ¬ ∃x ∈ ℕ ∃y ∈ ℕ (x↑2) = (2 · (y↑2))
Theorem	sqr2irrlem4 6676	Lemma for irrationality of square root of 2.
Theorem	sqr2irrlem5 6677	Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.
Theorem	sqr2irr 6678	The square root of 2 is irrational.
⊢ (√ ‘2) ∉ ℚ
Theorem	sqr2re 6679	The square root of 2 exists and is a real number.
⊢ (√ ‘2) ∈ ℝ
Imaginary and complex number properties
Theorem	irec 6680	The reciprocal of i.
⊢ (1 / i) = -i
Theorem	i2 6681	i squared.
⊢ (i↑2) = -1
Theorem	i3 6682	i cubed.
⊢ (i↑3) = -i
Theorem	i4 6683	i to the fourth power.
⊢ (i↑4) = 1
Theorem	inelr 6684	The imaginary unit i is not a real number.
⊢ ¬ i ∈ ℝ
Theorem	crulem 6685	Lemma for cru 6686.
Theorem	cru 6686	The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130.
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ C ∈ ℝ    &   ⊢ D ∈ ℝ    ⇒   ⊢ ((A + (i · B)) = (C + (i · D)) ↔ (A = C ⋀ B = D))
Theorem	crut 6687	The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130.
⊢ (((A ∈ ℝ ⋀ B ∈ ℝ) ⋀ (C ∈ ℝ ⋀ D ∈ ℝ)) → ((A + (i · B)) = (C + (i · D)) ↔ (A = C ⋀ B = D)))
Theorem	crutOLD 6688	The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130.
⊢ (((A ∈ ℝ ⋀ B ∈ ℝ) ⋀ (C ∈ ℝ ⋀ D ∈ ℝ)) → ((A + (B · i)) = (C + (D · i)) ↔ (A = C ⋀ B = D)))
Theorem	crne0 6689	The real representation of complex numbers is nonzero iff one of its terms is nonzero.
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((A ≠ 0 ⋁ B ≠ 0) ↔ (A + (i · B)) ≠ 0)
Theorem	crmul 6690	Multiplication rule for complex number representation. Remark in [Apostol] p. 361. In normal use, the arguments are the real components of two complex numbers, but the theorem works for complex components as well.
⊢ A ∈ ℂ    &   ⊢ B ∈ ℂ    &   ⊢ C ∈ ℂ    &   ⊢ D ∈ ℂ    ⇒   ⊢ ((A + (i · B)) · (C + (i · D))) = (((A · C) − (B · D)) + (i · ((A · D) + (B · C))))
Theorem	crrecz 6691	Reciprocal of a complex number in terms of real and imaginary components. Remark in [Apostol] p. 361.
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((A ≠ 0 ⋁ B ≠ 0) → (1 / (A + (i · B))) = ((A − (i · B)) / ((A↑2) + (B↑2))))
Theorem	creur 6692	The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130.
⊢ (A ∈ ℂ → ∃!x ∈ ℝ ∃y ∈ ℝ A = (x + (i · y)))
Theorem	creui 6693	The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130.
⊢ (A ∈ ℂ → ∃!y ∈ ℝ ∃x ∈ ℝ A = (x + (i · y)))
Theorem	rimul 6694	A real number times the imaginary unit is real only if the number is 0.
⊢ ((A ∈ ℝ ⋀ (i · A) ∈ ℝ) → A = 0)
Theorem	nthruc 6695	The sequence ℕ, ℤ, ℚ, ℝ, and ℂ forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to ℤ but not ℕ, one-half belongs to ℚ but not ℤ, the square root of 2 belongs to ℝ but not ℚ, and finally that the imaginary number i belongs to ℂ but not ℝ. See nthruz 6696 for a further refinement.
⊢ ((ℕ ⊂ ℤ ⋀ ℤ ⊂ ℚ) ⋀ (ℚ ⊂ ℝ ⋀ ℝ ⊂ ℂ))
Theorem	nthruz 6696	The sequence ℕ, ℕ0, and ℤ forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to ℕ0 but not ℕ and minus one belongs to ℤ but not ℕ0. This theorem refines the chain of proper subsets nthruc 6695.
⊢ (ℕ ⊂ ℕ0 ⋀ ℕ0 ⊂ ℤ)
Real and imaginary parts; conjugate; absolute value
Syntax	cre 6697	Extend class notation to include real part of a complex number.
class ℜ
Syntax	cim 6698	Extend class notation to include imaginary part of a complex number.
class ℑ
Syntax	ccj 6699	Extend class notation to include complex conjugate function.
class ∗
Syntax	cabs 6700	Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs