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Theorem List for Metamath Proof Explorer - 20601-20700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem0cxpd 20601 Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpexpzd 20602 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpefd 20603 Value of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpne0d 20604 Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpp1d 20605 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpnegd 20606 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpmul2zd 20607 Generalize cxpmul2 20580 to negative integers. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpaddd 20608 Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpsubd 20609 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpltd 20610 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpled 20611 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxplead 20612 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremdivcxpd 20613 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremrecxpcld 20614 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpge0d 20615 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxple2ad 20616 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxplt2d 20617 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxple2d 20618 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremmulcxpd 20619 Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxprecd 20620 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremrpcxpcld 20621 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlogcxpd 20622 Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxplt3d 20623 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxple3d 20624 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpmuld 20625 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremdvcxp1 20626* The derivative of a complex power with respect to the first argument. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvcxp2 20627* The derivative of a complex power with respect to the second argument. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvsqr 20628 The derivative of the real square root function. (Contributed by Mario Carneiro, 1-May-2016.)

Theoremcxpcn 20629* Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.)
fld       t

Theoremcxpcn2 20630* Continuity of the complex power function, when the base is real. (Contributed by Mario Carneiro, 1-May-2016.)
fld       t

Theoremcxpcn3lem 20631* Lemma for cxpcn3 20632. (Contributed by Mario Carneiro, 2-May-2016.)
fld       t        t

Theoremcxpcn3 20632* Extend continuity of the complex power function to a base of zero, as long as the exponent has strictly positive real part. (Contributed by Mario Carneiro, 2-May-2016.)
fld       t        t

Theoremresqrcn 20633 Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremsqrcn 20634 Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremcxpaddle 20636 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theoremabscxpbnd 20637 Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremroot1id 20638 Property of an -th root of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremroot1eq1 20639 The only powers of an -th root of unity that equal are the multiples of . In other words, has order in the multiplicative group of nonzero complex numbers. (In fact, these and their powers are the only elements of finite order in the complexes.) (Contributed by Mario Carneiro, 28-Apr-2016.)

Theoremroot1cj 20640 Within the -th roots of unity, the conjugate of the -th root is the -th root. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremcxpeq 20641* Solve an equation involving an -th power. The expression is a way to write the primitive -th root of unity with the smallest positive argument. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremloglesqr 20642 An upper bound on the logarithm. (Contributed by Mario Carneiro, 2-May-2016.)

13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords

Theoremangval 20643* Define the angle function, which takes two complex numbers, treated as vectors from the origin, and returns the angle between them, in the range . To convert from the geometry notation, , the measure of the angle with legs , where is more counterclockwise for positive angles, is represented by . (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremangcan 20644* Cancel a constant multiplier in the angle function. (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremangneg 20645* Cancel a negative sign in the angle function. (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremangvald 20646* The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 20643. (Contributed by David Moews, 28-Feb-2017.)

Theoremangcld 20647* The (signed) angle between two vectors is in . Deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theoremangrteqvd 20648* Two vectors are at a right angle iff their quotient is purely imaginary. (Contributed by David Moews, 28-Feb-2017.)

Theoremcosangneg2d 20649* The cosine of the angle between and is the negative of that between and . If A, B and C are collinear points, this implies that the cosines of DBA and DBC sum to zero, i.e., that DBA and DBC are supplementary. (Contributed by David Moews, 28-Feb-2017.)

Theoremangrtmuld 20650* Perpendicularity of two vectors does not change under rescaling the second. (Contributed by David Moews, 28-Feb-2017.)

Theoremang180lem1 20651* Lemma for ang180 20656. Show that the "revolution number" is an integer, using efeq1 20431 to show that since the product of the three arguments is , the sum of the logarithms must be an integer multiple of away from . (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremang180lem2 20652* Lemma for ang180 20656. Show that the revolution number is strictly between and . Both bounds are established by iterating using the bounds on the imaginary part of the logarithm, logimcl 20467, but the resulting bound gives only for the upper bound. The case is not ruled out here, but it is in some sense an "edge case" that can only happen under very specific conditions; in particular we show that all the angle arguments must lie on the negative real axis, which is a contradiction because clearly if is negative then the other two are positive real. (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremang180lem3 20653* Lemma for ang180 20656. Since ang180lem1 20651 shows that is an integer and ang180lem2 20652 shows that is strictly between and , it follows that , and these two cases correspond to the two possible values for . (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremang180lem4 20654* Lemma for ang180 20656. Reduce the statement to one variable. (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremang180lem5 20655* Lemma for ang180 20656: Reduce the statement to two variables. (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremang180 20656* The sum of angles in a triangle adds up to either or , i.e. 180 degrees. (The sign is due to the two possible orientations of vertex arrangement and our signed notion of angle). (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremlawcoslem1 20657 Lemma for Law of Cosines lawcos 20658. Here we prove the law for a point at the origin and two distinct points U and V, using an expanded version of the signed angle expression on the complex plane. (Contributed by David A. Wheeler, 11-Jun-2015.)

Theoremlawcos 20658* Law of Cosines. Given three distinct points A, B, and C, prove a relationship between their segment lengths. This theorem is expressed using the complex number plane as a plane, where is the signed angle construct (as used in ang180 20656), is the distance of line segment BC, is the distance of line segment AC, is the distance of line segment AB, and is the distinguished (signed) angle m/_ BCA on the complex plane. We translate triangle ABC to move C to the origin (C-C), B to U=(B-C), and A to V=(A-C), then use lemma lawcoslem1 20657 to prove this algebraically simpler case. The metamath convention is to use a signed angle; in this case the sign doesn't matter because we use the cosine of the angle (see cosneg 12748). The Pythagorean Theorem pythag 20659 is a special case of the law of cosines. The theorem's expression and approach were suggested by Mario Carneiro. (Contributed by David A. Wheeler, 12-Jun-2015.)

Theorempythag 20659* Pythagorean Theorem. Given three distinct points A, B, and C that form a right triangle (with the right angle at C), prove a relationship between their segment lengths. This theorem is expressed using the complex number plane as a plane, where is the signed angle construct (as used in ang180 20656), is the distance of line segment BC, is the distance of line segment AC, is the distance of line segment AB (the hypotenuse), and is the distinguished (signed) right angle m/_ BCA. We use the law of cosines lawcos 20658 to prove this, along with simple trig facts like coshalfpi 20377 and cosneg 12748. (Contributed by David A. Wheeler, 13-Jun-2015.)

Theoremlogreclem 20660 Symmetry of the natural logarithm range by negation. Lemma for logrec 20661. (Contributed by Saveliy Skresanov, 27-Dec-2016.)

Theoremlogrec 20661 Logarithm of a reciprocal changes sign. (Contributed by Saveliy Skresanov, 28-Dec-2016.)

Theoremisosctrlem1 20662 Lemma for isosctr 20665. (Contributed by Saveliy Skresanov, 30-Dec-2016.)

Theoremisosctrlem2 20663 Lemma for isosctr 20665. Corresponds to the case where one vertex is at 0, another at 1 and the third lies on the unit circle. (Contributed by Saveliy Skresanov, 31-Dec-2016.)

Theoremisosctrlem3 20664* Lemma for isosctr 20665. Corresponds to the case where one vertex is at 0. (Contributed by Saveliy Skresanov, 1-Jan-2017.)

Theoremisosctr 20665* Isosceles triangle theorem. (Contributed by Saveliy Skresanov, 1-Jan-2017.)

Theoremssscongptld 20666* If two triangles have equal sides, one angle in one triangle has the same cosine as the corresponding angle in the other triangle. This is a partial form of the SSS congruence theorem.

This theorem is proven by using lawcos 20658 on both triangles to express one side in terms of the other two, and then equating these expressions and reducing this algebraically to get an equality of cosines of angles. (Contributed by David Moews, 28-Feb-2017.)

Theoremaffineequiv 20667 Equivalence between two ways of expressing as an affine combination of and . (Contributed by David Moews, 28-Feb-2017.)

Theoremaffineequiv2 20668 Equivalence between two ways of expressing as an affine combination of and . (Contributed by David Moews, 28-Feb-2017.)

Theoremangpieqvdlem 20669 Equivalence used in the proof of angpieqvd 20672. (Contributed by David Moews, 28-Feb-2017.)

Theoremangpieqvdlem2 20670* Equivalence used in angpieqvd 20672. (Contributed by David Moews, 28-Feb-2017.)

Theoremangpined 20671* If the angle at ABC is , then A is not equal to C. (Contributed by David Moews, 28-Feb-2017.)

Theoremangpieqvd 20672* The angle ABC is iff B is a nontrivial convex combination of A and C, i.e., iff B is in the interior of the segment AC. (Contributed by David Moews, 28-Feb-2017.)

Theoremchordthmlem 20673* If M is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 20666 to observe that, since AMQ and BMQ have equal sides, the angles QMB and QMA must be equal. Since they are supplementary, both must be right. (Contributed by David Moews, 28-Feb-2017.)

Theoremchordthmlem2 20674* If M is the midpoint of AB, AQ = BQ, and P is on the line AB, then QMP is a right angle. This is proven by reduction to the special case chordthmlem 20673, where P = B, and using angrtmuld 20650 to observe that QMP is right iff QMB is. (Contributed by David Moews, 28-Feb-2017.)

Theoremchordthmlem3 20675 If M is the midpoint of AB, AQ = BQ, and P is on the line AB, then PQ 2 = QM 2 PM 2 . This follows from chordthmlem2 20674 and the Pythagorean theorem (pythag 20659) in the case where P and Q are unequal to M. If either P or Q equals M, the result is trivial. (Contributed by David Moews, 28-Feb-2017.)

Theoremchordthmlem4 20676 If P is on the segment AB and M is the midpoint of AB, then PA PB = BM 2 PM 2 . If all lengths are reexpressed as fractions of AB, this reduces to the identity 2 2 . (Contributed by David Moews, 28-Feb-2017.)

Theoremchordthmlem5 20677 If P is on the segment AB and AQ = BQ, then PA PB = BQ 2 PQ 2 . This follows from two uses of chordthmlem3 20675 to show that PQ 2 = QM 2 PM 2 and BQ 2 = QM 2 BM 2 , so BQ 2 PQ 2 = (QM 2 BM 2 ) (QM 2 PM 2 ) = BM 2 PM 2 , which equals PA PB by chordthmlem4 20676. (Contributed by David Moews, 28-Feb-2017.)

Theoremchordthm 20678* The intersecting chords theorem. If points A, B, C, and D lie on a circle (with center Q, say), and the point P is on the interior of the segments AB and CD, then the two products of lengths PA PB and PC PD are equal. The Euclidean plane is identified with the complex plane, and the fact that P is on AB and on CD is expressed by the hypothesis that the angles APB and CPD are equal to . The result is proven by using chordthmlem5 20677 twice to show that PA PB and PC PD both equal BQ 2 PQ 2 . This is similar to the proof of the theorem given in Euclid's Elements, where it is Proposition III.35. (Contributed by David Moews, 28-Feb-2017.)

13.3.6  Solutions of quadratic, cubic, and quartic equations

Theoremquad2 20679 The quadratic equation, without specifying the particular branch to the square root. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theorem1cubrlem 20681 The cube roots of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theorem1cubr 20682 The cube roots of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremdcubic1lem 20683 Lemma for dcubic1 20685 and dcubic2 20684: simplify the cubic equation under the substitution . (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdcubic2 20684* Reverse direction of dcubic 20686. Given a solution to the "substitution" quadratic equation , show that is in the desired form. (Contributed by Mario Carneiro, 25-Apr-2015.)

Theoremdcubic1 20685 Forward direction of dcubic 20686: the claimed formula produces solutions to the cubic equation. (Contributed by Mario Carneiro, 25-Apr-2015.)

Theoremdcubic 20686* Solutions to the depressed cubic, a special case of cubic 20689. (The definitions of here differ from mcubic 20687 by scale factors of , , and respectively, to simplify the algebra and presentation.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremmcubic 20687* Solutions to a monic cubic equation, a special case of cubic 20689. (Contributed by Mario Carneiro, 24-Apr-2015.)
;

Theoremcubic2 20688* The solution to the general cubic equation, for arbitrary choices and of the square and cube roots. (Contributed by Mario Carneiro, 23-Apr-2015.)
;

Theoremcubic 20689* The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 3864 to convert the existential quantifier to a triple disjunction. (Contributed by Mario Carneiro, 26-Apr-2015.)
;

Theorembinom4 20690 Work out a quartic binomial. (You would think that by this point it would be faster to use binom 12609, but it turns out to be just as much work to put it into this form after clearing all the sums and calculating binomial coefficients.) (Contributed by Mario Carneiro, 6-May-2015.)

Theoremdquartlem1 20691 Lemma for dquart 20693. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremdquartlem2 20692 Lemma for dquart 20693. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremdquart 20693 Solve a depressed quartic equation. To eliminate , which is the square root of a solution to the resolvent cubic equation, apply cubic 20689 or one of its variants. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremquart1cl 20694 Closure lemmas for quart 20701. (Contributed by Mario Carneiro, 7-May-2015.)
; ;;

Theoremquart1lem 20695 Lemma for quart1 20696. (Contributed by Mario Carneiro, 6-May-2015.)
; ;;                      ;;

Theoremquart1 20696 Depress a quartic equation. (Contributed by Mario Carneiro, 6-May-2015.)
; ;;

Theoremquartlem1 20697 Lemma for quart 20701. (Contributed by Mario Carneiro, 6-May-2015.)
;        ; ;        ;

Theoremquartlem2 20698 Closure lemmas for quart 20701. (Contributed by Mario Carneiro, 7-May-2015.)
; ;;        ;        ; ;

Theoremquartlem3 20699 Closure lemmas for quart 20701. (Contributed by Mario Carneiro, 7-May-2015.)
; ;;        ;        ; ;

Theoremquartlem4 20700 Closure lemmas for quart 20701. (Contributed by Mario Carneiro, 7-May-2015.)
; ;;        ;        ; ;

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