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Theorem List for Metamath Proof Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrennim 11601 A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.)

Theoremcnpart 11602 The specification of restriction to the right half-plane partitions the complex plane without 0 into two disjoint pieces, which are related by a reflection about the origin (under the map ). (Contributed by Mario Carneiro, 8-Jul-2013.)

Theoremsqr0lem 11603 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqr0 11604 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrlem1 11605* Lemma for 01sqrex 11612. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrlem2 11606* Lemma for 01sqrex 11612. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrlem3 11607* Lemma for 01sqrex 11612. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrlem4 11608* Lemma for 01sqrex 11612. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrlem5 11609* Lemma for 01sqrex 11612. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrlem6 11610* Lemma for 01sqrex 11612. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrlem7 11611* Lemma for 01sqrex 11612. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theorem01sqrex 11612* Existence of a square root for reals in the interval . (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremresqrex 11613* Existence of a square root for positive reals. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrmo 11614* Uniqueness for the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremresqreu 11615* Existence and uniqueness for the real square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremresqrcl 11616 Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremresqrthlem 11617 Lemma for resqrth 11618. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremresqrth 11618 Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremremsqsqr 11619 Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrge0 11620 The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 9-Jul-2013.)

Theoremsqrgt0 11621 The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrmul 11622 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrle 11623 Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremsqrlt 11624 Square root is strictly monotonic. Closed form of sqrlti 11750. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremsqr11 11625 The square root function is one to one. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremsqr00 11626 A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremrpsqrcl 11627 The square root of a positive real is a postive real. (Contributed by NM, 22-Feb-2008.)

Theoremsqrdiv 11628 Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.)

Theoremsqrneglem 11629 The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrneg 11630 The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrsq2 11631 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrsq 11632 Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrmsq 11633 Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqr1 11634 The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)

Theoremsqr4 11635 The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)

Theoremsqr9 11636 The square root of 9 is 3. (Contributed by NM, 11-May-2004.)

Theoremsqr2gt1lt2 11637 The square root of 2 is bounded by 1 and 2. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrm1 11638 The imaginary unit is the square root of negative 1. A lot of people like to call this the "definition" of , but the definition of df-sqr 11597 has already been crafted with being mentioned explicitly, and in any case it doesn't make too much sense to define a value based on a function evaluated outside its domain. A more appropriate view is to take ax-i2m1 8685 or i2 11081 as the "definition", and simply postulate the existence of a number satisfying this property. This is the approach we take here. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremabsneg 11639 Absolute value of negative. (Contributed by NM, 27-Feb-2005.)

Theoremabscl 11640 Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)

Theoremabscj 11641 The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 28-Apr-2005.)

Theoremabsvalsq 11642 Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)

Theoremabsvalsq2 11643 Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.)

Theoremsqabsadd 11644 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.)

Theoremsqabssub 11645 Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)

Theoremabsval2 11646 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.)

Theoremabs0 11647 The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsi 11648 The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

Theoremabsge0 11649 Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsrpcl 11650 The absolute value of a nonzero number is a positive real. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabs00 11651 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 26-Sep-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabs00ad 11652 A complex number is zero iff its absolute value is zero. Deduction form of abs00 11651. (Contributed by David Moews, 28-Feb-2017.)

Theoremabs00bd 11653 If a complex number is zero, its absolute value is zero. Converse of abs00d 11805. One-way deduction form of abs00 11651. (Contributed by David Moews, 28-Feb-2017.)

Theoremabsreimsq 11654 Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.)

Theoremabsreim 11655 Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.)

Theoremabsmul 11656 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsdiv 11657 Absolute value distributes over division. (Contributed by NM, 27-Apr-2005.)

Theoremabsid 11658 A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabs1 11659 The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)

Theoremabsnid 11660 A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.)

Theoremleabs 11661 A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)

Theoremabsor 11662 The absolute value of a real number is either that number or its negative. (Contributed by NM, 27-Feb-2005.)

Theoremabsre 11663 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)

Theoremabsresq 11664 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)

Theoremabsmod0 11665 is divisible by iff its absolute value is. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremabsexp 11666 Absolute value of natural number exponentiation. (Contributed by NM, 5-Jan-2006.)

Theoremabsexpz 11667 Absolute value of integer exponentiation. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremabssq 11668 Square can be moved in and out of absolute value. (Contributed by Scott Fenton, 18-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremsqabs 11669 The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)

Theoremabsrele 11670 The absolute value of a complex number is greater than or equal to the absolute value of its real part. (Contributed by NM, 1-Apr-2005.)

Theoremabsimle 11671 The absolute value of a complex number is greater than or equal to the absolute value of its imaginary part. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremmax0add 11672 The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Mario Carneiro, 24-Aug-2014.)

Theoremabsz 11673 A real number is an integer iff its absolute value is an integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremnn0abscl 11674 The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabslt 11675 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsle 11676 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabssubne0 11677 If the absolute value of a complex number is less than a real, its difference from the real is nonzero. (Contributed by NM, 2-Nov-2007.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabsdiflt 11678 The absolute value of a difference and 'less than' relation. (Contributed by Paul Chapman, 18-Sep-2007.)

Theoremabsdifle 11679 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007.)

Theoremelicc4abs 11680 Membership in a symmetric closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremlenegsq 11681 Comparison to a nonnegative number based on comparison to squares. (Contributed by NM, 16-Jan-2006.)

Theoremreleabs 11682 The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by NM, 1-Apr-2005.)

Theoremrecval 11683 Reciprocal expressed with a real denominator. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremabsidm 11684 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)

Theoremabsgt0 11685 The absolute value of a nonzero number is positive. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremnnabscl 11686 The absolute value of a nonzero integer is a positive integer. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremabssub 11687 Swapping order of subtraction doesn't change the absolute value. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabssubge0 11688 Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)

Theoremabssuble0 11689 Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)

Theoremabsmax 11690 The maximum of two numbers using absolute value. (Contributed by NM, 7-Aug-2008.)

Theoremabstri 11691 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by NM, 7-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabs3dif 11692 Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)

Theoremabs2dif 11693 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2dif2 11694 Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremabs2difabs 11695 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs1m 11696* For any complex number, there exists a unit-magnitude multiplier that produces its absolute value. Part of proof of Theorem 13-2.12 of [Gleason] p. 195. (Contributed by NM, 26-Mar-2005.)

Theoremrecan 11697* Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)

Theoremabsf 11698 Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremabs3lem 11699 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremabslem2 11700 Lemma involving absolute values. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

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