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Statement List for Metamath Proof Explorer - 6701-6800 - Page 68 of 107

Type	Label	Description
Statement
Definition	df-im 6701	Define a function whose value is the imaginary part of a complex number. See imvalt 6705 for its value, imcl 6716 for its closure, and replimt 6710 for its use in decomposing a complex number.
Definition	df-cj 6702	Define the complex conjugate function. See cjcl 6717 for its closure and cjvalt 6713 for its value.
Definition	df-abs 6703	Define the function for the absolute value (modulus) of a complex number. See abscl 6792 for its closure and absvalt 6712 or absval2 6794 for its value.
Theorem	revalt 6704	The value of the real part of a complex number.
Theorem	imvalt 6705	The value of the imaginary part of a complex number.
Theorem	reclt 6706	The real part of a complex number is real.
Theorem	imclt 6707	The imaginary part of a complex number is real.
Theorem	ref 6708	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.)
Theorem	imf 6709	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.)
Theorem	replimt 6710	Reconstruct a complex number from its real and imaginary parts.
Theorem	replimtOLD 6711	Reconstruct a complex number from its real and imaginary parts.
Theorem	absvalt 6712	The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133.
Theorem	cjvalt 6713	Value of the conjugate of a complex number. The value is the real part minus times the imaginary part. Definition 10-3.2 of [Gleason] p. 132.
Theorem	cjclt 6714	The conjugate of a complex number is a complex number (closure law).
Theorem	recl 6715	The real part of a complex number is real (closure law).
Theorem	imcl 6716	The imaginary part of a complex number is real (closure law).
Theorem	cjcl 6717	Closure law for complex conjugate.
Theorem	replim 6718	Construct a complex number from its real and imaginary parts.
Theorem	replimOLD 6719	Construct a complex number from its real and imaginary parts.
Theorem	crret 6720	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
Theorem	crretOLD 6721	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
Theorem	crimt 6722	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
Theorem	crimtOLD 6723	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
Theorem	crre 6724	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
Theorem	crreOLD 6725	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
Theorem	crim 6726	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
Theorem	crimOLD 6727	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
Theorem	imret 6728	The imaginary part of a complex number in terms of the real part function.
Theorem	reim0t 6729	The imaginary part of a real number is 0.
Theorem	reim0bt 6730	A number is real iff its imaginary part is 0.
Theorem	cjcj 6731	The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133.
Theorem	reim0b 6732	A number is real iff its imaginary part is 0.
Theorem	rereb 6733	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133.
Theorem	cjreb 6734	A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133.
Theorem	recj 6735	Real part of a complex conjugate.
Theorem	imcj 6736	Imaginary part of a complex conjugate.
Theorem	readd 6737	Real part distributes over addition.
Theorem	imadd 6738	Imaginary part distributes over addition.
Theorem	remul 6739	Real part of a product.
Theorem	immul 6740	Imaginary part of a product.
Theorem	cjadd 6741	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133.
Theorem	cjmul 6742	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133.
Theorem	ipcn 6743	Standard inner product on complex numbers.
Theorem	cjmulrcl 6744	A complex number times its conjugate is real.
Theorem	cjmulval 6745	A complex number times its conjugate.
Theorem	cjmulge0 6746	A complex number times its conjugate is nonnegative.
Theorem	reneg 6747	Real part of negative.
Theorem	negreb 6748	The negative of a real is real.
Theorem	imneg 6749	Imaginary part of negative.
Theorem	cjneg 6750	Complex conjugate of negative.
Theorem	addcj 6751	A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133.
Theorem	reret 6752	A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Paul Chapman, 7-Sep-2007.)
Theorem	cjrebt 6753	A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133.
Theorem	cjmulrclt 6754	A complex number times its conjugate is real.
Theorem	cjmulvalt 6755	A complex number times its conjugate.
Theorem	cjmulge0t 6756	A complex number times its conjugate is nonnegative.
Theorem	renegt 6757	Real part of negative.
Theorem	readdt 6758	Real part distributes over addition.
Theorem	resubt 6759	Real part distributes over subtraction.
Theorem	imnegt 6760	The imaginary part of a negative number.
Theorem	imaddt 6761	Imaginary part distributes over addition.
Theorem	imsubt