mmtheorems88 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8701-8800 - Page 88 of 107

Type	Label	Description
Statement
Theorem	eff1o 8701	The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
|- (exp |` {x e. CC | (Im` x) e. (-upi[,)pi)}):{x e. CC | (Im` x) e. (-upi[,)pi)}-1-1-onto->(CC \ {0})
The natural logarithm on complex numbers
Syntax	clog 8702	Extend class notation with the natural logarithm function on complex numbers.
class log
Definition	df-log 8703	Define the natural logarithm function on complex numbers. See http://en.wikipedia.org/wiki/Natural_logarithm ("The natural logarithm function can also be defined as the inverse function of the exponential function").
|- log = `'(exp |` {x e. CC | (Im` x) e. (-upi[,)pi)})
Theorem	logrn 8704	The range of the natural logarithm function, also the principal domain of the exponential function. This allows us to write the longer class abstraction as simply ran log. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ran log = {x e. CC | (Im` x) e. (-upi[,)pi)}
Theorem	dflog2 8705	The natural logarithm function in terms of the exponential function restricted to its principal domain. (Contributed by Paul Chapman, 21-Apr-2008.)
|- log = `'(exp |` ran log)
Theorem	resslogrn 8706	The range of the natural logarithm function includes the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
|- RR (_ ran log
Theorem	eff1o2 8707	The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
|- (exp |` ran log):ran log-1-1-onto->(CC \ {0})
Theorem	logf1o 8708	The natural logarithm function maps the nonzero complex numbers one-to-one onto its range. (Contributed by Paul Chapman, 21-Apr-2008.)
|- log:(CC \ {0})-1-1-onto->ran log
Theorem	dfrelog 8709	The natural logarithm function on the positive reals in terms of the real exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- (log |` RR+) = `'(exp |` RR)
Theorem	relogf1o 8710	The natural logarithm function maps the positive reals one-to-one onto the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
|- (log |` RR+):RR+-1-1-onto->RR
Theorem	logclt 8711	Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ((A e. CC /\ A =/= 0) -> (log` A) e. ran log)
Theorem	relogclt 8712	Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (A e. RR+ -> (log` A) e. RR)
Theorem	eflogt 8713	Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ((A e. CC /\ A =/= 0) -> (exp` (log` A)) = A)
Theorem	reeflogt 8714	Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (A e. RR+ -> (exp` (log` A)) = A)
Theorem	logeft 8715	Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- (A e. ran log -> (log` (exp` A)) = A)
Theorem	relogeft 8716	Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (A e. RR -> (log` (exp` A)) = A)
Theorem	logeftb 8717	Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ((A e. CC /\ A =/= 0 /\ B e. ran log) -> ((log` A) = B <-> (exp` B) = A))
Theorem	relogeftb 8718	Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ B e. RR) -> ((log` A) = B <-> (exp` B) = A))
Theorem	log1 8719	The natural logarithm of 1. One case of Property 1a of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (log` 1) = 0
Theorem	loge 8720	The natural logarithm of e. One case of Property 1b of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (log` e) = 1
Theorem	pilog 8721	Relationship between pi and the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- pi = (i x. (log` -u1))
Theorem	relogoprlem 8722	Lemma for relogmult 8723 and relogdivt 8724. Remark of [Cohen] p. 301 ("The proof of Property 3 is quite similar to the proof given for Property 2").
Theorem	relogmult 8723	The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ B e. RR+) -> (log` (A x. B)) = ((log` A) + (log` B)))
Theorem	relogdivt 8724	The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ B e. RR+) -> (log` (A / B)) = ((log` A) - (log` B)))
Theorem	explogt 8725	Exponentiation of a nonzero complex number to a nonnegative integer power. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ((A e. CC /\ A =/= 0 /\ N e. NN0) -> (A^N) = (exp` (N x. (log` A))))
Theorem	reexplogt 8726	Exponentiation of a positive real number to a nonnegative integer power. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ N e. NN0) -> (A^N) = (exp` (N x. (log` A))))
Theorem	relogexpt 8727	The natural logarithm of positive A raised to an nonnegative integer power. Property 4 of [Cohen] p. 301-302, restricted to natural logarithms and nonnegative-integer powers N. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ N e. NN0) -> (log` (A^N)) = (N x. (log` A)))
Theorem	relogiso 8728	The natural logarithm function on positive reals determines an isomorphism from the positive reals onto the reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (log |` RR+) Isom < , < (RR+, RR)
Theorem	logltbt 8729	The natural logarithm function on positive reals is strictly monotonic. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ B e. RR+) -> (A < B <-> (log` A) < (log` B)))
Syntax	clogOLD 8730	Extend class notation with the natural logarithm function on positive reals.
class logOLD
Definition	df-logOLD 8731	Define the natural logarithm function on positive real numbers. See http://en.wikipedia.org/wiki/Natural_logarithm ("The natural logarithm function can also be defined as the inverse function of the exponential function").
|- logOLD = `'(exp |` RR)
Theorem	dflog2OLD 8732	Alternate version of df-logOLD 8731. (Contributed by Steve Rodriguez, 22-Oct-2007.)
|- logOLD = {<.x, y>. | (x e. (0(,) +oo) /\ y = (`'(exp |` RR)` x))}
Theorem	logvaltOLD 8733	Value of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 22-Oct-2007.)
|- (logOLD` A) = (`'(exp |` RR)` A)
Theorem	logcltOLD 8734	Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (A e. RR+ -> (logOLD` A) e. RR)
Theorem	eflogtOLD 8735	Relationship between the natural logarithm and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (A e. RR+ -> (exp` (logOLD` A)) = A)
Theorem	logeftOLD 8736	Relationship between the natural logarithm and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (A e. RR -> (logOLD` (exp` A)) = A)
Theorem	logeftbOLD 8737	Relationship between the natural logarithm and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ B e. RR) -> ((logOLD` A) = B <-> (exp` B) = A))
Theorem	log1OLD 8738	The natural logarithm of 1. One case of Property 1a of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (logOLD` 1) = 0
Theorem	logeOLD 8739	The natural logarithm of e. One case of Property 1b of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (logOLD` e) = 1
Theorem	logoprlemOLD 8740	Lemma for logmultOLD 8741 and logdivtOLD 8742. Remark of [Cohen] p. 301 ("The proof of Property 3 is quite similar to the proof given for Property 2").
Theorem	logmultOLD 8741	The natural logarithm of a product is a sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ B e. RR+) -> (logOLD` (A x. B)) = ((logOLD` A) + (logOLD` B)))
Theorem	logdivtOLD 8742	The natural logarithm of a quotient is a difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ B e. RR+) -> (logOLD` (A / B)) = ((logOLD` A) - (logOLD` B)))
Theorem	logexptOLD 8743	The natural logarithm of positive A raised to a power. Property 4 of [Cohen] p. 301-302, restricted to natural logarithms and nonnegative-integer powers N. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ N e. NN0) -> (logOLD` (A^N)) = (N x. (logOLD` A)))
Theorem	explogtOLD 8744	Exponentiation of a positive real to a nonnegative integer power. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ N e. NN0) -> (A^N) = (exp` (N x. (logOLD` A))))
Theorem	logisoOLD 8745	The natural logarithm on positive reals determines an isomorphism from positive reals onto reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- logOLD Isom < , < (RR+, RR)
Theorem	logltbtOLD 8746	The natural logarithm function on positive reals is strictly monotonic. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ B e. RR+) -> (A < B <-> (logOLD` A) < (logOLD` B)))
ZFC Set Theory plus Grothendieck's Axiom
Introduce Grothendieck's Axiom
Axiom	ax-groth 8747	Grothendieck's Axiom. For every set x there is an inaccessible cardinal y such that y is not in x. The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics." This version of the axiom is used by the Mizar project (http://www.mizar.org/JFM/Axiomatics/tarski.html). Unlike the ZFC axioms, this axiom is very long when expressed in terms of primitive symbols - see grothprim 8753. An open problem is finding a shorter equivalent.
|- E.y(x e. y /\ A.z e. y (A.w(w (_ z -> w e. y) /\ E.w e. y A.v(v (_ z -> v e. w)) /\ A.z(z (_ y -> (z ~~ y \/ z e. y)))
Theorem	axgroth2 8748	Alternate version of Grothendieck's Axiom.
|- E.y(x e. y /\ A.z e. y (A.w(w (_ z -> w e. y) /\ E.w e. y A.v(v (_ z -> v e. w)) /\ A.z(z (_ y -> (y ~<_ z \/ z e. y)))
Theorem	axgroth3 8749	Alternate version of Grothendieck's Axiom. ax-ac 4734 is used to derive this version.
|- E.y(x e. y /\ A.z e. y (A.w(w (_ z -> w e. y) /\ E.w e. y A.v(v (_ z -> v e. w)) /\ A.z(z (_ y -> ((y \ z) ~<_ z \/ z e. y)))
Theorem	axgroth4 8750	Alternate version of Grothendieck's Axiom. ax-ac 4734 is used to derive this version.
|- E.y(x e. y /\ A.z e. y E.v e. y A.w(w (_ z -> w e. (y i^i v)) /\ A.z(z (_ y -> ((y \ z) ~<_ z \/ z e. y)))
Theorem	grothinf 8751	Grothendieck's Axiom implies the Axiom of Infinity (in the form of omex 4617). Note that our proof does not depend on the Axiom of Infinity.
|- om e. V
Theorem	grothprimlem 8752	Lemma for grothprim 8753. Expand the membership of an unordered pair into primitives.
Theorem	grothprim 8753	Grothendieck's Axiom ax-groth 8747 expanded into set theory primitives using 163 symbols. An open problem is whether a shorter equivalent exists (when expanded to primitives).
|- E.y(x e. y /\ A.z((z e. y -> E.v(v e. y /\ A.w(A.u(u e. w -> u e. z) -> (w e. y /\ w e. v)))) /\ E.w((w e. z -> w e. y) -> (A.v((v e. z -> E.tA.u(E.g(g e. w /\ A.h(h e. g <-> (h = v \/ h = u))) -> u = t)) /\ (v e. y -> (v e. z \/ E.u(u e. z /\ E.g(g e. w /\ A.h(h e. g <-> (h = u \/ h = v))))))) \/ z e. y))))
Humor
April Fool's theorem
Theorem	avril1 8754	Poisson d'Avril's Theorem. This theorem is noted for its Selbstdokumentieren property, which means, literally, "self-documenting" and recalls the principle of quidquid germanus dictum sit, altum viditur, often used in set theory. Starting with the seemingly simple yet profound fact that any object x equals itself (proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we demonstrate that the power set of the real numbers, as a relation on the value of the imaginary unit, does not conjoin with an empty relation on the product of the additive and multiplicative identity elements, leading to this startling conclusion that has left even seasoned professional mathematicians scratching their heads. (Contributed by Prof. Loof Lirpa, 1-Apr-2005.) A reply to skeptics can be found at http://us.metamath.org/mpegif/mmnotes.txt, under the 1-Apr-06 entry.
|- -. (AP~RR(i` 1) /\ F(/)(0 x. 1))
Theorem	2bornot2b 8755	The law of excluded middle. Act III, Theorem 1 of Shakespeare, Hamlet, Prince of Denmark (1602). Its author leaves its proof as an exercise for the reader - "To be, or not to be: that is the question" - starting a trend that has become standard in modern-day textbooks, serving to make the frustrated reader feel inferior, or in some cases to mask the fact that the author does not know its solution. (Contributed by Prof. Loof Lirpa, 1-Apr-2006.)
|- (2 x. B \/ -. 2 x. B)
Theorem	helloworld 8756	The classic "Hello world" benchmark has been translated into 314 computer programming languages - see http://www.roesler-ac.de/wolfram/hello.htm. However, for many years it eluded a proof that it is more than just a conjecture, even though a wily mathematician once claimed, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Using an IBM 709 mainframe, a team of mathematicians led by Prof. Loof Lirpa, at the New College of Tahiti, were finally able put it rest with a remarkably short proof only 4 lines long. (Contributed by Prof. Loof Lirpa, 1-Apr-2007.)
|- -. (h e. (LL0) /\ W(/)(R.1d))
Theorem	1p1e2 8757	One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel O'Cat, along with the latest supercomputer technology, Prof. Loof Lirpa and colleagues were able to shorten Whitehead and Russell's 360-page proof that 1+1=2 in Principia Mathematica to this remarkable proof only two steps long, thus establishing a new world's record for this famous theorem. (Contributed by Prof. Loof Lirpa, 1-Apr-2008.)
|- (1 + 1) = 2
Hilbert Space Explorer
Syntax	chil 8758	Extend class notation with Hilbert vector space.
class H~
Syntax	cva 8759	Extend class notation with vector addition in Hilbert space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition + caddc 5227.
class +h
Syntax	csm 8760	Extend class notation with scalar multiplication in Hilbert space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.
class .h
Syntax	c0v 8761	Extend class notation with zero vector in Hilbert space.
class 0h
Syntax	cmv 8762	Extend class notation wit