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	effoiOLD 8701	The exponential function maps the set S, of complex numbers with imaginary part in a closed-below, open-above real interval of length 2 x. pi starting at A, onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
|- A e. RR   &   |- D = (A[,)(A + (2 x. pi)))   &   |- S = {v e. CC | (Im` v) e. D}   &   |- F = {<.z, w>. | (z e. D /\ w = (exp` (i x. z)))}   &   |- C = {v e. CC | (abs` v) = 1}   =>   |- (exp |` S):S-onto->(CC \ {0})
Theorem	eff1oi 8702	The exponential function maps the set S, of complex numbers with imaginary part in a closed-below, open-above real interval of length 2 x. pi starting at A, one-to-one onto the nonzero complex numbers. A would normally be fixed at 0 or -upi, according to choice of principal domain for the exponential function. (Contributed by Paul Chapman, 16-Apr-2008.)
|- A e. RR   &   |- D = (A[,)(A + (2 x. pi)))   &   |- S = {v e. CC | (Im` v) e. D}   =>   |- (exp |` S):S-1-1-onto->(CC \ {0})
Theorem	efper 8703	The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ((A e. CC /\ K e. ZZ) -> (exp` (A + ((i x. (2 x. pi)) x. K))) = (exp` A))
Theorem	eff1o 8704	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 8705	Extend class notation with the natural logarithm function on complex numbers.
class log
Definition	df-log 8706	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 8707	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 8708	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 8709	The range of the natural logarithm function includes the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
|- RR (_ ran log
Theorem	eff1o2 8710	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 8711	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 8712	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 8713	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 8714	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 8715	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 8716	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 8717	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 8718	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 8719	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 8720	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 8721	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 8722	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 8723	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 8724	Relationship between pi and the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- pi = (i x. (log` -u1))
Theorem	relogoprlem 8725	Lemma for relogmult 8726 and relogdivt 8727. Remark of [Cohen] p. 301 ("The proof of Property 3 is quite similar to the proof given for Property 2").
Theorem	relogmult 8726	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 8727	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 8728	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 8729	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 8730	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 8731	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 8732	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 8733	Extend class notation with the natural logarithm function on positive reals.
class logOLD
Definition	df-logOLD 8734	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 8735	Alternate version of df-logOLD 8734. (Contributed by Steve Rodriguez, 22-Oct-2007.)
|- logOLD = {<.x, y>. | (x e. (0(,) +oo) /\ y = (`'(exp |` RR)` x))}
Theorem	logvaltOLD 8736	Value of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 22-Oct-2007.)
|- (logOLD` A) = (`'(exp |` RR)` A)
Theorem	logcltOLD 8737	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 8738	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 8739	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 8740	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 8741	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 8742	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 8743	Lemma for logmultOLD 8744 and logdivtOLD 8745. Remark of [Cohen] p. 301 ("The proof of Property 3 is quite similar to the proof given for Property 2").
Theorem	logmultOLD 8744	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 8745	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 8746	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 8747	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 8748	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 8749	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 8750	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 8756. 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 8751	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 8752	Alternate version of Grothendieck's Axiom. ax-ac 4735 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 8753	Alternate version of Grothendieck's Axiom. ax-ac 4735 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 8754	Grothendieck's Axiom implies the Axiom of Infinity (in the form of omex 4618). Note that our proof does not depend on the Axiom of Infinity.
|- om e. V
Theorem	grothprimlem 8755	Lemma for grothprim 8756. Expand the membership of an unordered pair into primitives.
Theorem	grothprim 8756	Grothendieck's Axiom ax-groth 8750 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))