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Type | Label | Description |
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Statement | ||
Theorem | cxpne0 23701 | Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | cxpeq0 23702 | Complex exponentiation is zero iff the mantissa is zero and the exponent is nonzero. (Contributed by Mario Carneiro, 23-Apr-2015.) |
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Theorem | cxpadd 23703 | Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | cxpp1 23704 | Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | cxpneg 23705 | Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | cxpsub 23706 | Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 22-Sep-2014.) |
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Theorem | cxpge0 23707 | Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | mulcxplem 23708 | Lemma for mulcxp 23709. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | mulcxp 23709 | Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | cxprec 23710 | Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | divcxp 23711 | Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.) |
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Theorem | cxpmul 23712 | Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | cxpmul2 23713 |
Product of exponents law for complex exponentiation. Variation on
cxpmul 23712 with more general conditions on ![]() ![]() ![]() |
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Theorem | cxproot 23714 |
The complex power function allows us to write n-th roots via the idiom
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | cxpmul2z 23715 | Generalize cxpmul2 23713 to negative integers. (Contributed by Mario Carneiro, 23-Apr-2015.) |
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Theorem | abscxp 23716 | Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.) |
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Theorem | abscxp2 23717 | Absolute value of a power, when the exponent is real. (Contributed by Mario Carneiro, 15-Sep-2014.) |
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Theorem | cxplt 23718 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | cxple 23719 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | cxplea 23720 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 10-Sep-2014.) |
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Theorem | cxple2 23721 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.) |
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Theorem | cxplt2 23722 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.) |
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Theorem | cxple2a 23723 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.) |
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Theorem | cxplt3 23724 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.) |
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Theorem | cxple3 23725 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.) |
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Theorem | cxpsqrtlem 23726 | Lemma for cxpsqrt 23727. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | cxpsqrt 23727 |
The complex exponential function with exponent ![]() ![]() ![]() ![]() |
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Theorem | logsqrt 23728 | Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.) |
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Theorem | cxp0d 23729 | Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxp1d 23730 | Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | 1cxpd 23731 | Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxpcld 23732 | Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxpmul2d 23733 |
Product of exponents law for complex exponentiation. Variation on
cxpmul 23712 with more general conditions on ![]() ![]() ![]() |
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Theorem | 0cxpd 23734 | Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxpexpzd 23735 | Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxpefd 23736 | Value of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxpne0d 23737 | Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxpp1d 23738 | Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 30-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cxpnegd 23739 | Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxpmul2zd 23740 | Generalize cxpmul2 23713 to negative integers. (Contributed by Mario Carneiro, 30-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cxpaddd 23741 | Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxpsubd 23742 | Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cxpltd 23743 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxpled 23744 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxplead 23745 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | divcxpd 23746 | Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 30-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | recxpcld 23747 | Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxpge0d 23748 | Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxple2ad 23749 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxplt2d 23750 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxple2d 23751 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | mulcxpd 23752 | Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cxprecd 23753 | Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | rpcxpcld 23754 | Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | logcxpd 23755 | Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cxplt3d 23756 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | cxple3d 23757 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cxpmuld 23758 | Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | dvcxp1 23759* | The derivative of a complex power with respect to the first argument. (Contributed by Mario Carneiro, 24-Feb-2015.) |
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Theorem | dvcxp2 23760* | The derivative of a complex power with respect to the second argument. (Contributed by Mario Carneiro, 24-Feb-2015.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dvsqrt 23761 | The derivative of the real square root function. (Contributed by Mario Carneiro, 1-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dvcncxp1 23762* | Derivative of complex power with respect to first argument on the complex plane. (Contributed by Brendan Leahy, 18-Dec-2018.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dvcnsqrt 23763* | Derivative of square root function. (Contributed by Brendan Leahy, 18-Dec-2018.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cxpcn 23764* | Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cxpcn2 23765* | Continuity of the complex power function, when the base is real. (Contributed by Mario Carneiro, 1-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cxpcn3lem 23766* | Lemma for cxpcn3 23767. (Contributed by Mario Carneiro, 2-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cxpcn3 23767* | 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrtcn 23768 | Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | sqrtcn 23769 | Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.) |
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Theorem | cxpaddlelem 23770 | Lemma for cxpaddle 23771. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | cxpaddle 23771 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | abscxpbnd 23772 | Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) |
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Theorem | root1id 23773 |
Property of an ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | root1eq1 23774 |
The only powers of an ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | root1cj 23775 |
Within the ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cxpeq 23776* |
Solve an equation involving an ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | loglesqrt 23777 | An upper bound on the logarithm. (Contributed by Mario Carneiro, 2-May-2016.) |
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Theorem | logreclem 23778 | Symmetry of the natural logarithm range by negation. Lemma for logrec 23779. (Contributed by Saveliy Skresanov, 27-Dec-2016.) |
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Theorem | logrec 23779 | Logarithm of a reciprocal changes sign. (Contributed by Saveliy Skresanov, 28-Dec-2016.) |
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Define "log using an arbitrary base" function and then prove some of
its
properties. Note that logb is generalized to an arbitrary base
and
arbitrary parameter in Metamath doesn't care what letters are used to represent classes. Usually classes begin with the letter "A", but here we use "B" and "X" to more clearly distinguish between "base" and "other parameter of log".
There are different ways this could be defined in Metamath. The approach
used here is intentionally similar to existing 2-parameter Metamath functions
(operations): | ||
Syntax | clogb 23780 | Extend class notation to include the logarithm generalized to an arbitrary base. |
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Definition | df-logb 23781* |
Define the logb operator. This is the logarithm generalized to an
arbitrary base. It can be used as ![]() ![]() ![]() ![]() |
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Theorem | logbval 23782 | Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.) |
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Theorem | logbcl 23783 | General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.) |
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Theorem | logbid1 23784 | General logarithm is 1 when base and arg match. Property 1(a) of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by David A. Wheeler, 22-Jul-2017.) |
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Theorem | logb1 23785 |
The logarithm of ![]() ![]() |
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Theorem | elogb 23786 |
The general logarithm of a number to the base being Euler's constant is
the natural logarithm of the number. Put another way, using ![]() ![]() |
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Theorem | logbchbase 23787 | Change of base for logarithms. Property in [Cohen4] p. 367. (Contributed by AV, 11-Jun-2020.) |
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Theorem | relogbval 23788 | Value of the general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
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Theorem | relogbcl 23789 | Closure of the general logarithm with a positive real base on positive reals. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.) |
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Theorem | relogbzcl 23790 | Closure of the general logarithm with integer base on positive reals. (Contributed by Thierry Arnoux, 27-Sep-2017.) (Proof shortened by AV, 9-Jun-2020.) |
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Theorem | relogbreexp 23791 | Power law for the general logarithm for real powers: The logarithm of a positive real number to the power of a real number is equal to the product of the exponent and the logarithm of the base of the power. Property 4 of [Cohen4] p. 361. (Contributed by AV, 9-Jun-2020.) |
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Theorem | relogbzexp 23792 | Power law for the general logarithm for integer powers: The logarithm of a positive real number to the power of an integer is equal to the product of the exponent and the logarithm of the base of the power. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.) |
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Theorem | relogbmul 23793 | The logarithm of the product of two positive real numbers is the sum of logarithms. Property 2 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 29-May-2020.) |
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Theorem | relogbmulexp 23794 | The logarithm of the product of a positive real and a positive real number to the power of a real number is the sum of the logarithm of the first real number and the scaled logarithm of the second real number. (Contributed by AV, 29-May-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | relogbdiv 23795 | The logarithm of the quotient of two positive real numbers is the difference of logarithms. Property 3 of [Cohen4] p. 361. (Contributed by AV, 29-May-2020.) |
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Theorem | relogbexp 23796 | Identity law for general logarithm: the logarithm of a power to the base is the exponent. Property 6 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.) |
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Theorem | nnlogbexp 23797 | Identity law for general logarithm with integer base. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.) |
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Theorem | logbrec 23798 | Logarithm of a reciprocal changes sign. See logrec 23779. Particular case of Property 3 of [Cohen4] p. 361. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
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Theorem | logbleb 23799 | The general logarithm function is monotone/increasing. See logleb 23631. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by AV, 31-May-2020.) |
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Theorem | logblt 23800 | The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 23628. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.) |
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