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Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremefopnlem2 20501 Lemma for efopn 20502. (Contributed by Mario Carneiro, 2-May-2015.)
fld

Theoremefopn 20502 The exponential map is an open map. (Contributed by Mario Carneiro, 23-Apr-2015.)
fld

Theoremlogtayllem 20503* Lemma for logtayl 20504. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremlogtayl 20504* The Taylor series for . (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremlogtaylsum 20505* The Taylor series for , as an infinite sum. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremlogtayl2 20506* Power series expression for the logarithm. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremlogccv 20507 The natural logarithm function on the reals is a strictly concave function. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremcxpval 20508 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpef 20509 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theorem0cxp 20510 Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpexpz 20511 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpexp 20512 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremlogcxp 20513 Logarithm of a complex power. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxp0 20514 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxp1 20515 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theorem1cxp 20516 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremecxp 20517 Write the exponential function as an exponent to the power . (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpcl 20518 Closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremrecxpcl 20519 Real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremrpcxpcl 20520 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpne0 20521 Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpeq0 20522 Complex exponentiation is zero iff the mantissa is zero and the exponent is nonzero. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremcxpadd 20523 Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpp1 20524 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpneg 20525 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpsub 20526 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremcxpge0 20527 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremmulcxplem 20528 Lemma for mulcxp 20529. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremmulcxp 20529 Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxprec 20530 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremdivcxp 20531 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theoremcxpmul 20532 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpmul2 20533 Product of exponents law for complex exponentiation. Variation on cxpmul 20532 with more general conditions on and when is an integer. (Contributed by Mario Carneiro, 9-Aug-2014.)

Theoremcxproot 20534 The complex power function allows us to write n-th roots via the idiom . (Contributed by Mario Carneiro, 6-May-2015.)

Theoremcxpmul2z 20535 Generalize cxpmul2 20533 to negative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremabscxp 20536 Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremabscxp2 20537 Absolute value of a power, when the exponent is real. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxplt 20538 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxple 20539 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxplea 20540 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 10-Sep-2014.)

Theoremcxple2 20541 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theoremcxplt2 20542 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxple2a 20543 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxplt3 20544 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremcxple3 20545 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremcxpsqrlem 20546 Lemma for cxpsqr 20547. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpsqr 20547 The complex exponential function with exponent exactly matches the complex square root function (the branch cut is in the same place for both functions), and thus serves as a suitable generalization to other -th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremlogsqr 20548 Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)

Theoremcxp0d 20549 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxp1d 20550 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)

Theorem1cxpd 20551 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpcld 20552 Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpmul2d 20553 Product of exponents law for complex exponentiation. Variation on cxpmul 20532 with more general conditions on and when is an integer. (Contributed by Mario Carneiro, 30-May-2016.)

Theorem0cxpd 20554 Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpexpzd 20555 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpefd 20556 Value of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpne0d 20557 Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpp1d 20558 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpnegd 20559 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpmul2zd 20560 Generalize cxpmul2 20533 to negative integers. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpaddd 20561 Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpsubd 20562 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpltd 20563 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpled 20564 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxplead 20565 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremdivcxpd 20566 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremrecxpcld 20567 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpge0d 20568 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxple2ad 20569 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxplt2d 20570 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxple2d 20571 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremmulcxpd 20572 Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxprecd 20573 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremrpcxpcld 20574 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlogcxpd 20575 Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxplt3d 20576 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxple3d 20577 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpmuld 20578 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremdvcxp1 20579* The derivative of a complex power with respect to the first argument. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvcxp2 20580* The derivative of a complex power with respect to the second argument. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvsqr 20581 The derivative of the real square root function. (Contributed by Mario Carneiro, 1-May-2016.)

Theoremcxpcn 20582* Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.)
fld       t

Theoremcxpcn2 20583* Continuity of the complex power function, when the base is real. (Contributed by Mario Carneiro, 1-May-2016.)
fld       t

Theoremcxpcn3lem 20584* Lemma for cxpcn3 20585. (Contributed by Mario Carneiro, 2-May-2016.)
fld       t        t

Theoremcxpcn3 20585* 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 20586 Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremsqrcn 20587 Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremcxpaddle 20589 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theoremabscxpbnd 20590 Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremroot1id 20591 Property of an -th root of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremroot1eq1 20592 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 20593 Within the -th roots of unity, the conjugate of the -th root is the -th root. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremcxpeq 20594* 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 20595 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 20596* 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 20597* Cancel a constant multiplier in the angle function. (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremangneg 20598* Cancel a negative sign in the angle function. (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremangvald 20599* The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 20596. (Contributed by David Moews, 28-Feb-2017.)

Theoremangcld 20600* The (signed) angle between two vectors is in . Deduction form. (Contributed by David Moews, 28-Feb-2017.)

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