P. 133 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13201-13300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cjcjd 13201	The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` ( * ` A ) ) = A )
Theorem	reim0bd 13202	A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( Im ` A ) = 0 )   =>    |- ( ph -> A e. RR )
Theorem	rerebd 13203	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( Re ` A ) = A )   =>    |- ( ph -> A e. RR )
Theorem	cjrebd 13204	A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( * ` A ) = A )   =>    |- ( ph -> A e. RR )
Theorem	cjne0d 13205	A number is nonzero iff its complex conjugate is nonzero. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> A =/= 0 )   =>    |- ( ph -> ( * ` A ) =/= 0 )
Theorem	recjd 13206	Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Re ` ( * ` A ) ) = ( Re ` A ) )
Theorem	imcjd 13207	Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Im ` ( * ` A ) ) = -u ( Im ` A ) )
Theorem	cjmulrcld 13208	A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. ( * ` A ) ) e. RR )
Theorem	cjmulvald 13209	A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. ( * ` A ) ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )
Theorem	cjmulge0d 13210	A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> 0 <_ ( A x. ( * ` A ) ) )
Theorem	renegd 13211	Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Re ` -u A ) = -u ( Re ` A ) )
Theorem	imnegd 13212	Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Im ` -u A ) = -u ( Im ` A ) )
Theorem	cjnegd 13213	Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` -u A ) = -u ( * ` A ) )
Theorem	addcjd 13214	A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A + ( * ` A ) ) = ( 2 x. ( Re ` A ) ) )
Theorem	cjexpd 13215	Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> N e. NN0 )   =>    |- ( ph -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) )
Theorem	readdd 13216	Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A + B ) ) = ( ( Re ` A ) + ( Re ` B ) ) )
Theorem	imaddd 13217	Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Im ` ( A + B ) ) = ( ( Im ` A ) + ( Im ` B ) ) )
Theorem	resubd 13218	Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )
Theorem	imsubd 13219	Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Im ` ( A - B ) ) = ( ( Im ` A ) - ( Im ` B ) ) )
Theorem	remuld 13220	Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) ) )
Theorem	immuld 13221	Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Im ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Im ` B ) ) + ( ( Im ` A ) x. ( Re ` B ) ) ) )
Theorem	cjaddd 13222	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) ) )
Theorem	cjmuld 13223	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) )
Theorem	ipcnd 13224	Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A x. ( * ` B ) ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) ) )
Theorem	cjdivd 13225	Complex conjugate distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B =/= 0 )   =>    |- ( ph -> ( * ` ( A / B ) ) = ( ( * ` A ) / ( * ` B ) ) )
Theorem	rered 13226	A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( Re ` A ) = A )
Theorem	reim0d 13227	The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( Im ` A ) = 0 )
Theorem	cjred 13228	A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( * ` A ) = A )
Theorem	remul2d 13229	Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A x. B ) ) = ( A x. ( Re ` B ) ) )
Theorem	immul2d 13230	Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Im ` ( A x. B ) ) = ( A x. ( Im ` B ) ) )
Theorem	redivd 13231	Real part of a division. Related to remul2 13132. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A =/= 0 )   =>    |- ( ph -> ( Re ` ( B / A ) ) = ( ( Re ` B ) / A ) )
Theorem	imdivd 13232	Imaginary part of a division. Related to remul2 13132. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A =/= 0 )   =>    |- ( ph -> ( Im ` ( B / A ) ) = ( ( Im ` B ) / A ) )
Theorem	crred 13233	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( Re ` ( A + ( _i x. B ) ) ) = A )
Theorem	crimd 13234	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( Im ` ( A + ( _i x. B ) ) ) = B )
5.9.4  Square root; absolute value
Syntax	csqrt 13235	Extend class notation to include square root of a complex number.
class sqr
Syntax	cabs 13236	Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs
Definition	df-sqrt 13237*	Define a function whose value is the square root of a complex number. Since ( y ^ 2 ) = x iff ( -u y ^ 2 ) = x, we ensure uniqueness by restricting the range to numbers with positive real part, or numbers with 0 real part and nonnegative imaginary part. A description can be found under "Principal square root of a complex number" at http://en.wikipedia.org/wiki/Square_root. The square root symbol was introduced in 1525 by Christoff Rudolff. See sqrtcl 13363 for its closure, sqrtval 13239 for its value, sqrtth 13366 and sqsqrti 13377 for its relationship to squares, and sqrt11i 13386 for uniqueness. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 8-Jul-2013.)
|- sqr = ( x e. CC |-> ( iota_ y e. CC ( ( y ^ 2 ) = x /\ 0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) ) )
Definition	df-abs 13238	Define the function for the absolute value (modulus) of a complex number. See abscli 13396 for its closure and absval 13240 or absval2i 13398 for its value. (Contributed by NM, 27-Jul-1999.)
|- abs = ( x e. CC |-> ( sqr ` ( x x. ( * ` x ) ) ) )
Theorem	sqrtval 13239*	Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.)
|- ( A e. CC -> ( sqr ` A ) = ( iota_ x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) )
Theorem	absval 13240	The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
|- ( A e. CC -> ( abs ` A ) = ( sqr ` ( A x. ( * ` A ) ) ) )
Theorem	rennim 13241	A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.)
|- ( A e. RR -> ( _i x. A ) e/ RR+ )
Theorem	cnpart 13242	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 x |-> -u x). (Contributed by Mario Carneiro, 8-Jul-2013.)
|- ( ( A e. CC /\ A =/= 0 ) -> ( ( 0 <_ ( Re ` A ) /\ ( _i x. A ) e/ RR+ ) <-> -. ( 0 <_ ( Re ` -u A ) /\ ( _i x. -u A ) e/ RR+ ) ) )
Theorem	sqr0lem 13243	Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. CC /\ ( ( A ^ 2 ) = 0 /\ 0 <_ ( Re ` A ) /\ ( _i x. A ) e/ RR+ ) ) <-> A = 0 )
Theorem	sqrt0 13244	Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( sqr ` 0 ) = 0
Theorem	sqrlem1 13245*	Lemma for 01sqrex 13252. (Contributed by Mario Carneiro, 10-Jul-2013.)
|- S = { x e. RR+ | ( x ^ 2 ) <_ A }   &    |- B = sup ( S , RR , < )   =>    |- ( ( A e. RR+ /\ A <_ 1 ) -> A. y e. S y <_ 1 )
Theorem	sqrlem2 13246*	Lemma for 01sqrex 13252. (Contributed by Mario Carneiro, 10-Jul-2013.)
|- S = { x e. RR+ | ( x ^ 2 ) <_ A }   &    |- B = sup ( S , RR , < )   =>    |- ( ( A e. RR+ /\ A <_ 1 ) -> A e. S )
Theorem	sqrlem3 13247*	Lemma for 01sqrex 13252. (Contributed by Mario Carneiro, 10-Jul-2013.)
|- S = { x e. RR+ | ( x ^ 2 ) <_ A }   &    |- B = sup ( S , RR , < )   =>    |- ( ( A e. RR+ /\ A <_ 1 ) -> ( S C_ RR /\ S =/= (/) /\ E. z e. RR A. y e. S y <_ z ) )
Theorem	sqrlem4 13248*	Lemma for 01sqrex 13252. (Contributed by Mario Carneiro, 10-Jul-2013.)
|- S = { x e. RR+ | ( x ^ 2 ) <_ A }   &    |- B = sup ( S , RR , < )   =>    |- ( ( A e. RR+ /\ A <_ 1 ) -> ( B e. RR+ /\ B <_ 1 ) )
Theorem	sqrlem5 13249*	Lemma for 01sqrex 13252. (Contributed by Mario Carneiro, 10-Jul-2013.)
|- S = { x e. RR+ | ( x ^ 2 ) <_ A }   &    |- B = sup ( S , RR , < )   &    |- T = { y | E. a e. S E. b e. S y = ( a x. b ) }   =>    |- ( ( A e. RR+ /\ A <_ 1 ) -> ( ( T C_ RR /\ T =/= (/) /\ E. v e. RR A. u e. T u <_ v ) /\ ( B ^ 2 ) = sup ( T , RR , < ) ) )
Theorem	sqrlem6 13250*	Lemma for 01sqrex 13252. (Contributed by Mario Carneiro, 10-Jul-2013.)
|- S = { x e. RR+ | ( x ^ 2 ) <_ A }   &    |- B = sup ( S , RR , < )   &    |- T = { y | E. a e. S E. b e. S y = ( a x. b ) }   =>    |- ( ( A e. RR+ /\ A <_ 1 ) -> ( B ^ 2 ) <_ A )
Theorem	sqrlem7 13251*	Lemma for 01sqrex 13252. (Contributed by Mario Carneiro, 10-Jul-2013.)
|- S = { x e. RR+ | ( x ^ 2 ) <_ A }   &    |- B = sup ( S , RR , < )   &    |- T = { y | E. a e. S E. b e. S y = ( a x. b ) }   =>    |- ( ( A e. RR+ /\ A <_ 1 ) -> ( B ^ 2 ) = A )
Theorem	01sqrex 13252*	Existence of a square root for reals in the interval ( 0 , 1 ]. (Contributed by Mario Carneiro, 10-Jul-2013.)
|- ( ( A e. RR+ /\ A <_ 1 ) -> E. x e. RR+ ( x <_ 1 /\ ( x ^ 2 ) = A ) )
Theorem	resqrex 13253*	Existence of a square root for positive reals. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> E. x e. RR ( 0 <_ x /\ ( x ^ 2 ) = A ) )
Theorem	sqrmo 13254*	Uniqueness for the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.) (Revised by NM, 17-Jun-2017.)
|- ( A e. CC -> E* x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )
Theorem	resqreu 13255*	Existence and uniqueness for the real square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> E! x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )
Theorem	resqrtcl 13256	Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` A ) e. RR )
Theorem	resqrtthlem 13257	Lemma for resqrtth 13258. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( ( sqr ` A ) ^ 2 ) = A /\ 0 <_ ( Re ` ( sqr ` A ) ) /\ ( _i x. ( sqr ` A ) ) e/ RR+ ) )
Theorem	resqrtth 13258	Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqr ` A ) ^ 2 ) = A )
Theorem	remsqsqrt 13259	Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqr ` A ) x. ( sqr ` A ) ) = A )
Theorem	sqrtge0 13260	The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> 0 <_ ( sqr ` A ) )
Theorem	sqrtgt0 13261	The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.)
|- ( ( A e. RR /\ 0 < A ) -> 0 < ( sqr ` A ) )
Theorem	sqrtmul 13262	Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( sqr ` ( A x. B ) ) = ( ( sqr ` A ) x. ( sqr ` B ) ) )
Theorem	sqrtle 13263	Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A <_ B <-> ( sqr ` A ) <_ ( sqr ` B ) ) )
Theorem	sqrtlt 13264	Square root is strictly monotonic. Closed form of sqrtlti 13391. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A < B <-> ( sqr ` A ) < ( sqr ` B ) ) )
Theorem	sqrt11 13265	The square root function is one-to-one. (Contributed by Scott Fenton, 11-Jun-2013.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqr ` A ) = ( sqr ` B ) <-> A = B ) )
Theorem	sqrt00 13266	A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqr ` A ) = 0 <-> A = 0 ) )
Theorem	rpsqrtcl 13267	The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008.)
|- ( A e. RR+ -> ( sqr ` A ) e. RR+ )
Theorem	sqrtdiv 13268	Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( A / B ) ) = ( ( sqr ` A ) / ( sqr ` B ) ) )
Theorem	sqrtneglem 13269	The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A /\ 0 <_ ( Re ` ( _i x. ( sqr ` A ) ) ) /\ ( _i x. ( _i x. ( sqr ` A ) ) ) e/ RR+ ) )
Theorem	sqrtneg 13270	The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` -u A ) = ( _i x. ( sqr ` A ) ) )
Theorem	sqrtsq2 13271	Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqr ` A ) = B <-> A = ( B ^ 2 ) ) )
Theorem	sqrtsq 13272	Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` ( A ^ 2 ) ) = A )
Theorem	sqrtmsq 13273	Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` ( A x. A ) ) = A )
Theorem	sqrt1 13274	The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)
|- ( sqr ` 1 ) = 1
Theorem	sqrt4 13275	The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)
|- ( sqr ` 4 ) = 2
Theorem	sqrt9 13276	The square root of 9 is 3. (Contributed by NM, 11-May-2004.)
|- ( sqr ` 9 ) = 3
Theorem	sqrt2gt1lt2 13277	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.)
|- ( 1 < ( sqr ` 2 ) /\ ( sqr ` 2 ) < 2 )
Theorem	sqrtm1 13278	The imaginary unit is the square root of negative 1. A lot of people like to call this the "definition" of _i, but the definition of sqr df-sqrt 13237 has already been crafted with _i 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 9553 or i2 12320 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.)
|- _i = ( sqr ` -u 1 )
Theorem	absneg 13279	Absolute value of negative. (Contributed by NM, 27-Feb-2005.)
|- ( A e. CC -> ( abs ` -u A ) = ( abs ` A ) )
Theorem	abscl 13280	Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)
|- ( A e. CC -> ( abs ` A ) e. RR )
Theorem	abscj 13281	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.)
|- ( A e. CC -> ( abs ` ( * ` A ) ) = ( abs ` A ) )
Theorem	absvalsq 13282	Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)
|- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) )
Theorem	absvalsq2 13283	Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.)
|- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )
Theorem	sqabsadd 13284	Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A + B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) + ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )
Theorem	sqabssub 13285	Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) - ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )
Theorem	absval2 13286	Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.)
|- ( A e. CC -> ( abs ` A ) = ( sqr ` ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) )
Theorem	abs0 13287	The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( abs ` 0 ) = 0
Theorem	absi 13288	The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
|- ( abs ` _i ) = 1
Theorem	absge0 13289	Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( A e. CC -> 0 <_ ( abs ` A ) )
Theorem	absrpcl 13290	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.)
|- ( ( A e. CC /\ A =/= 0 ) -> ( abs ` A ) e. RR+ )
Theorem	abs00 13291	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.)
|- ( A e. CC -> ( ( abs ` A ) = 0 <-> A = 0 ) )
Theorem	abs00ad 13292	A complex number is zero iff its absolute value is zero. Deduction form of abs00 13291. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( ( abs ` A ) = 0 <-> A = 0 ) )
Theorem	abs00bd 13293	If a complex number is zero, its absolute value is zero. Converse of abs00d 13446. One-way deduction form of abs00 13291. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A = 0 )   =>    |- ( ph -> ( abs ` A ) = 0 )
Theorem	absreimsq 13294	Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` ( A + ( _i x. B ) ) ) ^ 2 ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
Theorem	absreim 13295	Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.)
|- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A + ( _i x. B ) ) ) = ( sqr ` ( ( A ^ 2 ) + ( B ^ 2 ) ) ) )
Theorem	absmul 13296	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.)
|- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) )
Theorem	absdiv 13297	Absolute value distributes over division. (Contributed by NM, 27-Apr-2005.)
|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( abs ` ( A / B ) ) = ( ( abs ` A ) / ( abs ` B ) ) )
Theorem	absid 13298	A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A )
Theorem	abs1 13299	The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
|- ( abs ` 1 ) = 1
Theorem	absnid 13300	A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.)
|- ( ( A e. RR /\ A <_ 0 ) -> ( abs ` A ) = -u A )