mmtheorems69 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6801-6900 - Page 69 of 107

Type	Label	Description
Statement
Theorem	absmul 6801	Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133.
|- A e. CC   &   |- B e. CC   =>   |- (abs` (A x. B)) = ((abs` A) x. (abs` B))
Theorem	sqabsaddt 6802	Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133.
|- ((A e. CC /\ B e. CC) -> ((abs` (A + B))^2) = ((((abs` A)^2) + ((abs` B)^2)) + (2 x. (Re` (A x. (*` B))))))
Theorem	sqabssubt 6803	Square of absolute value of difference.
|- ((A e. CC /\ B e. CC) -> ((abs` (A - B))^2) = ((((abs` A)^2) + ((abs` B)^2)) - (2 x. (Re` (A x. (*` B))))))
Theorem	sqabsadd 6804	Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133.
|- A e. CC   &   |- B e. CC   =>   |- ((abs` (A + B))^2) = ((((abs` A)^2) + ((abs` B)^2)) + (2 x. (Re` (A x. (*` B)))))
Theorem	sqabssub 6805	Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-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	absval2t 6806	Value of absolute value function. Definition 10.36 of [Gleason] p. 133.
|- (A e. CC -> (abs` A) = (sqr` (((Re` A)^2) + ((Im` A)^2))))
Theorem	abs00t 6807	The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133.
|- (A e. CC -> ((abs` A) = 0 <-> A = 0))
Theorem	absge0t 6808	Absolute value is nonnegative.
|- (A e. CC -> 0 <_ (abs` A))
Theorem	absrpclt 6809	The absolute value of a nonzero number is a positive real. (Contributed by FL, 27-Dec-2007.)
|- ((A e. CC /\ A =/= 0) -> (abs` A) e. RR+)
Theorem	absreimsqt 6810	Square of the absolute value of a number that has been decomposed into real and imaginary parts.
|- ((A e. RR /\ B e. RR) -> ((abs` (A + (i x. B)))^2) = ((A^2) + (B^2)))
Theorem	absreimt 6811	Absolute value of a number that has been decomposed into real and imaginary parts.
|- ((A e. RR /\ B e. RR) -> (abs` (A + (i x. B))) = (sqr` ((A^2) + (B^2))))
Theorem	absmult 6812	Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133.
|- ((A e. CC /\ B e. CC) -> (abs` (A x. B)) = ((abs` A) x. (abs` B)))
Theorem	absdivz 6813	Absolute value distributes over division.
|- A e. CC   &   |- B e. CC   =>   |- (B =/= 0 -> (abs` (A / B)) = ((abs` A) / (abs` B)))
Theorem	absdivt 6814	Absolute value distributes over division.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> (abs` (A / B)) = ((abs` A) / (abs` B)))
Theorem	absid 6815	A nonnegative number is its own absolute value.
|- A e. RR   =>   |- (0 <_ A -> (abs` A) = A)
Theorem	absidt 6816	A nonnegative number is its own absolute value.
|- ((A e. RR /\ 0 <_ A) -> (abs` A) = A)
Theorem	absnidt 6817	A negative number is the negative of its own absolute value.
|- ((A e. RR /\ A <_ 0) -> (abs` A) = -uA)
Theorem	leabst 6818	A real number is less than or equal to its absolute value.
|- (A e. RR -> A <_ (abs` A))
Theorem	absort 6819	The absolute value of a real number is either that number or its negative.
|- (A e. RR -> ((abs` A) = A \/ (abs` A) = -uA))
Theorem	absret 6820	Absolute value of a real number.
|- (A e. RR -> (abs` A) = (sqr` (A^2)))
Theorem	absresqt 6821	Square of the absolute value of a real number.
|- (A e. RR -> ((abs` A)^2) = (A^2))
Theorem	absexpt 6822	Absolute value of natural number exponentiation.
|- ((A e. CC /\ N e. NN0) -> (abs` (A^N)) = ((abs` A)^N))
Theorem	absrelet 6823	The absolute value of a complex number is greater than or equal to the absolute value of its real part.
|- (A e. CC -> (abs` (Re` A)) <_ (abs` A))
Theorem	absimlet 6824	The absolute value of a complex number is greater than or equal to the absolute value of its imaginary part.
|- (A e. CC -> (abs` (Im` A)) <_ (abs` A))
Theorem	absnid 6825	A negative number is the negative of its own absolute value.
|- A e. RR   =>   |- (A <_ 0 -> (abs` A) = -uA)
Theorem	leabs 6826	A real number is less than or equal to its absolute value.
|- A e. RR   =>   |- A <_ (abs` A)
Theorem	absor 6827	The absolute value of a real number is either that number or its negative.
|- A e. RR   =>   |- ((abs` A) = A \/ (abs` A) = -uA)
Theorem	absre 6828	Absolute value of a real number.
|- A e. RR   =>   |- (abs` A) = (sqr` (A^2))
Theorem	abslt 6829	Absolute value and 'less than' relation.
|- A e. RR   &   |- B e. RR   =>   |- ((abs` A) < B <-> (-uB < A /\ A < B))
Theorem	absle 6830	Absolute value and 'less than or equal to' relation.
|- A e. RR   &   |- B e. RR   =>   |- ((abs` A) <_ B <-> (-uB <_ A /\ A <_ B))
Theorem	absltOLD 6831	Absolute value and 'less than' relation.
|- A e. RR   &   |- B e. RR   =>   |- ((abs` A) < B <-> (A < B /\ -uA < B))
Theorem	absleOLD 6832	Absolute value and 'less than or equal to' relation.
|- A e. RR   &   |- B e. RR   =>   |- ((abs` A) <_ B <-> (A <_ B /\ -uA <_ B))
Theorem	abs0 6833	The absolute value of 0.
|- (abs` 0) = 0
Theorem	absi 6834	The absolute value of the imaginary unit.
|- (abs` i) = 1
Theorem	nn0absclt 6835	The absolute value of an integer is a nonnegative integer.
|- (A e. ZZ -> (abs` A) e. NN0)
Theorem	absltt 6836	Absolute value and 'less than' relation.
|- ((A e. RR /\ B e. RR) -> ((abs` A) < B <-> (-uB < A /\ A < B)))
Theorem	abslttOLD 6837	Absolute value and 'less than' relation.
|- ((A e. RR /\ B e. RR) -> ((abs` A) < B <-> (A < B /\ -uA < B)))
Theorem	abslet 6838	Absolute value and 'less than or equal to' relation.
|- ((A e. RR /\ B e. RR) -> ((abs` A) <_ B <-> (-uB <_ A /\ A <_ B)))
Theorem	abssubne0t 6839	If the absolute value of a complex number is less than a real, its difference from the real is nonzero.
|- ((A e. CC /\ B e. RR /\ (abs` A) < B) -> (B - A) =/= 0)
Theorem	absdifltt 6840	The absolute value of a difference and 'less than' relation. (Contributed by Paul Chapman, 18-Sep-2007.)
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((abs` (A - B)) < C <-> ((B - C) < A /\ A < (B + C))))
Theorem	absdiflet 6841	The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007.)
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((abs` (A - B)) <_ C <-> ((B - C) <_ A /\ A <_ (B + C))))
Theorem	lenegsqt 6842	Comparison to a nonnegative number based on comparison to squares.
|- ((A e. RR /\ B e. RR /\ 0 <_ B) -> ((A <_ B /\ -uA <_ B) <-> (A^2) <_ (B^2)))
Theorem	releabst 6843	The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133.
|- (A e. CC -> (Re` A) <_ (abs` A))
Theorem	recvalz 6844	Reciprocal expressed with a real denominator.
|- A e. CC   =>   |- (A =/= 0 -> (1 / A) = ((*` A) / ((abs` A)^2)))
Theorem	cjdiv 6845	Complex conjugate distributes over division.
|- A e. CC   &   |- B e. CC   =>   |- (B =/= 0 -> (*` (A / B)) = ((*` A) / (*` B)))
Theorem	cjdivt 6846	Complex conjugate distributes over division.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> (*` (A / B)) = ((*` A) / (*` B)))
Theorem	releabs 6847	The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133.
|- A e. CC   =>   |- (Re` A) <_ (abs` A)
Theorem	abstri 6848	Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133.
|- A e. CC   &   |- B e. CC   =>   |- (abs` (A + B)) <_ ((abs` A) + (abs` B))
Theorem	absidmt 6849	The absolute value function is idempotent.
|- (A e. CC -> (abs` (abs` A)) = (abs` A))
Theorem	absgt0t 6850	The absolute value of a non-zero number is positive.
|- (A e. CC -> (A =/= 0 <-> 0 < (abs` A)))
Theorem	abssubt 6851	Swapping order of subtraction doesn't change the absolute value.
|- ((A e. CC /\ B e. CC) -> (abs` (A - B)) = (abs` (B - A)))
Theorem	abssubge0t 6852	Absolute value of a nonnegative difference.
|- ((A e. RR /\ B e. RR /\ A <_ B) -> (abs` (B - A)) = (B - A))
Theorem	abssuble0t 6853	Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)
|- ((A e. RR /\ B e.