mmtheorems75 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7401-7500 - Page 75 of 123

Type	Label	Description
Statement
Theorem	isum1clim 7401	An infinite sum equals the value its series converges to.
|- F e. V   &   |- A e. V   =>   |- (( + seq1 F) ~~> A -> sum_k e. NN (F` k) = A)
Theorem	isumclim2f 7402	Version of isumclim2 7403 with a bound-variable hypothesis instead of a distinct variable condition.
|- (y e. F -> A.k y e. F)   &   |- F e. V   =>   |- ((M e. ZZ /\ E.x(<.M, + >. seq F) ~~> x) -> (<.M, + >. seq F) ~~> sum_k e. (ZZ>=` M)(F` k))
Theorem	isumclim2 7403	A converging series converges to its infinite sum.
|- F e. V   =>   |- ((M e. ZZ /\ E.x(<.M, + >. seq F) ~~> x) -> (<.M, + >. seq F) ~~> sum_k e. (ZZ>=` M)(F` k))
Theorem	isumclim3 7404	A converging series converges to its infinite sum.
|- A e. V   &   |- F = {<.k, y>. | (k e. (ZZ>=` M) /\ y = A)}   =>   |- ((M e. ZZ /\ E.x(<.M, + >. seq F) ~~> x) -> (<.M, + >. seq F) ~~> sum_k e. (ZZ>=` M)A)
Theorem	isumclim4 7405	An infinite sum equals the value its series converges to.
|- A e. V   &   |- F = {<.k, y>. | (k e. (ZZ>=` M) /\ y = A)}   &   |- B e. V   =>   |- ((M e. ZZ /\ (<.M, + >. seq F) ~~> B) -> sum_k e. (ZZ>=` M)A = B)
Theorem	isumclim5 7406	The sequence of partial finite sums of a converging infinite series converge to the infinite sum of the series. Note that j must not occur in A.
|- A e. V   &   |- F = {<.j, y>. | (j e. (ZZ>=` M) /\ y = sum_k e. (M...j)A)}   =>   |- ((M e. ZZ /\ E.x F ~~> x) -> F ~~> sum_k e. (ZZ>=` M)A)
Theorem	isumnul 7407	The sum of a non-convergent infinite series evaluates to the empty set. (Contributed by Paul Chapman, 04-Nov-2007.)
|- F e. V   =>   |- ((M e. ZZ /\ -. E.x(<.M, + >. seq F) ~~> x) -> sum_k e. (ZZ>=` M)(F` k) = (/))
Theorem	isumshfti 7408	Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.)
|- F e. V   &   |- K e. ZZ   &   |- M e. ZZ   &   |- N = (M + K)   =>   |- sum_k e. (ZZ>=` M)(F` k) = sum_k e. (ZZ>=` N)(F` (k - K))
Theorem	isumshft2i 7409	Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.)
|- F e. V   &   |- K e. ZZ   &   |- M e. ZZ   &   |- N = (M + K)   =>   |- sum_k e. (ZZ>=` N)(F` k) = sum_k e. (ZZ>=` M)(F` (K + k))
Theorem	isum1p 7410	The infinite sum of a converging infinite series equals the first term plus the infinite sum of the rest of it.
|- F e. V   =>   |- ((M e. ZZ /\ A.k e. (ZZ>=` M)(F` k) e. CC /\ E.x(<.(M + 1), + >. seq F) ~~> x) -> sum_k e. (ZZ>=` M)(F` k) = ((F` M) + sum_k e. (ZZ>=` (M + 1))(F` k)))
Theorem	isumnn0nn 7411	Sum from 0 to infinity in terms of sum from 1 to infinity.
|- F e. V   =>   |- ((A.k e. NN0 (F` k) e. CC /\ E.x( + seq1 F) ~~> x) -> sum_k e. NN0 (F` k) = ((F` 0) + sum_k e. NN (F` k)))
Theorem	isumnn0nnai 7412	Sum from 0 to infinity in terms of sum from 1 to infinity of a class term A(k).
|- A e. V   &   |- F e. V   &   |- (y e. F -> A.k y e. F)   &   |- (k e. NN0 -> (F` k) = A)   =>   |- ((A.k e. NN0 A e. CC /\ E.x( + seq1 F) ~~> x) -> sum_k e. NN0 A = ([_0 / k]_A + sum_k e. NN A))
Theorem	isumcl 7413	The sum of a converging infinite series is a complex number.
|- F e. V   =>   |- ((M e. ZZ /\ E.x(<.M, + >. seq F) ~~> x) -> sum_k e. (ZZ>=` M)(F` k) e. CC)
Theorem	isumrecl 7414	The sum of a converging real infinite series is a real number.
|- F e. V   =>   |- ((M e. ZZ /\ A.k e. (ZZ>=` M)(F` k) e. RR /\ E.x(<.M, + >. seq F) ~~> x) -> sum_k e. (ZZ>=` M)(F` k) e. RR)
Theorem	iserzgt0 7415	The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.)
|- F e. V   =>   |- ((N e. (ZZ>=` M) /\ A.k e. (ZZ>=` M)((F` k) e. RR /\ 0 < (F` k)) /\ E.x(<.M, + >. seq F) ~~> x) -> 0 < sum_k e. (ZZ>=` N)(F` k))
Theorem	isummulc1 7416	Distribute a constant multiplier into an infinite sum (function value version). See isummulc1ai 7418 for class term version.
|- F e. V   =>   |- (((M e. ZZ /\ C e. CC) /\ (A.k e. (ZZ>=` M)(F` k) e. CC /\ E.x(<.M, + >. seq F) ~~> x)) -> (C x. sum_k e. (ZZ>=` M)(F` k)) = sum_k e. (ZZ>=` M)(C x. (F` k)))
Theorem	isummulc1iALT 7417	Older isummulc1 7416 proved without using iserzmulc1 7339.
|- F e. V   =>   |- (((M e. ZZ /\ C e. CC) /\ (A.k e. (ZZ>=` M)(F` k) e. CC /\ E.x(<.M, + >. seq F) ~~> x)) -> (C x. sum_k e. (ZZ>=` M)(F` k)) = sum_k e. (ZZ>=` M)(C x. (F` k)))
Theorem	isummulc1ai 7418	Distribute a constant multiplier into an infinite sum of a class term A(k). See isummulc1 7416 for function value version.
|- A e. V   &   |- F e. V   &   |- (y e. F -> A.k y e. F)   &   |- (k e. (ZZ>=` M) -> (F` k) = A)   =>   |- (((M e. ZZ /\ C e. CC) /\ (A.k e. (ZZ>=` M)A e. CC /\ E.x(<.M, + >. seq F) ~~> x)) -> (C x. sum_k e. (ZZ>=` M)A) = sum_k e. (ZZ>=` M)(C x. A))
Theorem	isumcmpii 7419	Comparison of two infinite sums. (Contributed by Paul Chapman, 13-Nov-2007.)
|- A e. V   &   |- B e. V   &   |- F = {<.j, y>. | (j e. (ZZ>=` M) /\ y = A)}   &   |- G = {<.j, y>. | (j e. (ZZ>=` M) /\ y = B)}   &   |- (j e. (ZZ>=` M) -> (A e. RR /\ B e. RR /\ A <_ B))   =>   |- ((M e. ZZ /\ (E.x(<.M, + >. seq F) ~~> x /\ E.x(<.M, + >. seq G) ~~> x)) -> sum_j e. (ZZ>=` M)A <_ sum_j e. (ZZ>=` M)B)
Theorem	isumspliti 7420	Split off the first N terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2007.)
|- M e. ZZ   &   |- N e. (ZZ>=` M)   &   |- F:(ZZ>=` M)-->CC   &   |- E.x(<.M, + >. seq F) ~~> x   =>   |- sum_k e. (ZZ>=` M)(F` k) = (sum_k e. (M...N)(F` k) + sum_k e. (ZZ>=` (N + 1))(F` k))
Theorem	isum0spliti 7421	Split off the first N terms of a 0-based infinite sum. (Contributed by Paul Chapman, 9-Feb-2007.)
|- N e. NN0   &   |- F:NN0-->CC   &   |- E.x( + seq0 F) ~~> x   =>   |- sum_k e. NN0 (F` k) = (sum_k e. (0...N)(F` k) + sum_k e. (ZZ>=` (N + 1))(F` k))
Miscellaneous converging sequences
Theorem	reccnv 7422	The sequence of reciprocals of natural numbers converges to zero.
|- F e. V   =>   |- (A.k e. NN (F` k) = (1 / k) -> F ~~> 0)
Theorem	infcvgaux1i 7423	Auxilliary theorem for applications of infcvgi 7428. Hypothesis for several supremum theorems.
|- R = {x | E.y e. X x = -uA}   &   |- (y e. X -> A e. RR)   &   |- Z e. X   &   |- E.z e. RR A.w e. R w <_ z   =>   |- (R (_ RR /\ R =/= (/) /\ E.z e. RR A.w e. R w <_ z)
Theorem	infcvgaux2i 7424	Auxilliary theorem for applications of infcvgi 7428.
|- R = {x | E.y e. X x = -uA}   &   |- (y e. X -> A e. RR)   &   |- Z e. X   &   |- E.z e. RR A.w e. R w <_ z   &   |- S = -usup(R, RR, < )   &   |- (y = C -> A = B)   =>   |- (C e. X -> S <_ B)
Theorem	infcvglem1 7425	Lemma for infcvgi 7428. Use ac6s 4902 to show the existence of a sequence f with values extracted from R.
Theorem	infcvglem2 7426	Lemma for infcvgi 7428. Show that G converges to the infimum.
Theorem	infcvglem3 7427	Lemma for infcvgi 7428. Using climsqueeze 7343, show that sequence F, constructed from f, converges to the supremum.
Theorem	infcvgi 7428	Extract a sequence that converges to the infimum S of a set of reals A(y). The sequence F is built using values of a sequence f that converges when its values are mapped to reals via A(y). Equation 4 of [Kreyszig] p. 144.
|- R = {x | E.y e. X x = -uA}   &   |- (y e. X -> A e. RR)   &   |- Z e. X   &   |- E.z e. RR A.w e. R w <_ z   &   |- S = -usup(R, RR, < )   &   |- F e. V   &   |- (y = (f` k) -> A = B)   &   |- (k e. NN -> (F` k) = B)   =>   |- E.f(f:NN-->X /\ F ~~> S)
Arithmetic series
Theorem	arisumilem 7429	Lemma for arisumi 7430.
Theorem	arisumi 7430	Arithmetic series sum of the first (N + 1) non-negative integers. (Contributed by FL, 10-Dec-2006.)
|- N e. NN   =>   |- sum_k e. (0...N)k = ((((N + 1)^2) - (N + 1)) / 2)
Geometric series
Theorem	expcnvlem1 7431	Lemma for expcnv 7437. Convert an antecedent from a comparison with a real into comparison with a natural number.
Theorem	expcnvlem2 7432	Lemma for expcnv 7437. Compute an upper bound for exponentiation using Bernoulli's inequality bernneq 6849.
Theorem	expcnvlem3 7433	Lemma for expcnv 7437. Apply weak deduction theorem.
Theorem	expcnvlem4 7434	Lemma for expcnv 7437. Combine expcnvlem1 7431 and expcnvlem3 7433.
Theorem	expcnvlem5 7435	Lemma for expcnv 7437. Apply weak deduction theorem.
Theorem	expcnvlem6 7436	Lemma for expcnv 7437. Add in the case of A = 0.
Theorem	expcnv 7437	A sequence of powers of a complex number A with absolute value smaller than 1 converges to zero.
|- F e. V   =>   |- ((A e. CC /\ A.k e. NN (F` k) = (A^k) /\ (abs` A) < 1) -> F ~~> 0)
Theorem	explecnv 7438	A sequence of terms converges to zero when it is less than powers of a number A whose absolute value is smaller than 1.
|- F e. V   =>   |- ((A e. RR /\ (abs` A) < 1 /\ A.k e. NN ((F` k) e. CC /\ (abs` (F` k)) <_ (A^k))) -> F ~~> 0)
Theorem	geoseri 7439	The value of the finite geometric series 1 + A^1 + A^2 +... + A^N.
|- F = {<.k, y>. | (k e. NN0 /\ y = (A^k))}   &   |- A e. CC   =>   |- ((N e. NN0 /\ A =/= 1) -> (( + seq0 F)` N) = ((1 - (A^(N + 1))) / (1 - A)))
Theorem	geolimilem 7440	Lemma for geolimi 7441.
Theorem	geolimi 7441	The partial sums in the infinite series 1 + A^1 + A^2... converge to (1 / (1 - A)).
|- F = {<.k, y>. | (k e. NN0 /\ y = (A^k))}   &   |- A e. CC   &   |- (abs` A) < 1   =>   |- ( + seq0 F) ~~> (1 / (1 - A))
Theorem	geolim 7442	The partial sums in the infinite series 1 + A^1 + A^2... converge to (1 / (1 - A)).
|- F = {<.k, y>. | (k e. NN0 /\ y = (A^k))}   =>   |- ((A e. CC /\ (abs` A) < 1) -> ( + seq0 F) ~~> (1 / (1 - A)))
Theorem	geolim1i 7443	The partial sums in the geometric series A^1 + A^2... converge to (A / (1 - A)).
|- F = {<.k, y>. | (k e. NN /\ y = (A^k))}   &   |- A e. CC   &   |- (abs` A) < 1   =>   |- ( + seq1 F) ~~> (A / (1 - A))
Theorem	geolim1 7444	The partial sums in the infinite series A^1 + A^2... converge to (A / (1 - A)).
|- F = {<.k, y>. | (k e. NN /\ y = (A^k))}   =>   |- ((A e. CC /\ (abs` A) < 1) -> ( + seq1 F) ~~> (A / (1 - A)))
Theorem	georeclim 7445	The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.)
|- F = {<.j, y>. | (j e. NN0 /\ y = ((1 / A)^j))}   =>   |- ((A e. CC /\ 1 < (abs` A)) -> ( + seq0 F) ~~> (A / (A - 1)))
Theorem	geosumi 7446	The value of the finite geometric series 1 + A^1 + A^2 +... + A^N.
|- A e. CC   =>   |- ((N e. NN0 /\ A =/= 1) -> sum_k e. (0...N)(A^k) = ((1 - (A^(N + 1))) / (1 - A)))
Theorem	geoisum 7447	The infinite sum of 1 + A^1 + A^2... is (1 / (1 - A)).
|- ((A e. CC /\ (abs` A) < 1) -> sum_k e. NN0 (A^k) = (1 / (1 - A)))
Theorem	geoisumr 7448	The infinite sum of reciprocals 1 + (1 / A)^1 + (1 / A)^2 ... is A / (A - 1). (Contributed by rpenner, 3-Nov-2007.)
|- ((A e. CC /\ 1 < (abs` A)) -> sum_k e. NN0 ((1 / A)^k) = (A / (A - 1)))
Theorem	geoisum1 7449	The infinite sum of A^1 + A^2... is (A / (1 - A)).
|- ((A e. CC /\ (abs` A) < 1) -> sum_k e. NN (A^k) = (A / (1 - A)))
Theorem	geoisum1c 7450	The infinite sum of A x. (R^1) + A x. (R^2)... is (A x. R) / (1 - R).
|- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> sum_k e. NN (A x. (R^k)) = ((A x. R) / (1 - R)))
Theorem	0.999... 7451	The recurring decimal 0.999..., which is defined as the infinite sum 0.9 + 0.09 + 0.009 + ... i.e. 9 / 10^1 + 9 / 10^2 + 9 / 10^3 + ..., is exactly equal to 1, according to ZF set theory. Interestingly, about 40% of the people responding to a poll at http://forum.physorg.com/index.php?showtopic=13177 disagree.
|- sum_k e. NN (9 / (10^k)) = 1
Ratio test for infinite series convergence
Theorem	cvgratlem1ALT 7452	Lemma for cvgrati 7460. Establish, by induction, an exponential upper bound for the terms of a real series, given that the ratio of successive terms is less than some positive constant A beyond a starting index B.
Theorem	cvgratlem2ALT 7453	Lemma for cvgrati 7460. Using expsub 6793, restate cvgratlem1ALT 7452 with an absolute index C instead of just an offset from the starting index B.
Theorem	cvgratlem3ALT 7454	Lemma for cvgrati 7460. Restate cvgratlem2ALT 7453 (which was for a real function) in terms of the absolute values of the terms of a complex function F, with the help of an auxiliary function G.
Theorem	cvgratlem1 7455	Lemma for cvgrati 7460. Establish, by induction, an exponential upper bound for the terms of a real series, given that the ratio of successive terms is less than some positive constant A beyond a starting index B.
Theorem	cvgratlem2 7456	Lemma for cvgrati 7460. Using expsubOLD 6794, restate cvgratlem1 7455 with an absolute index C instead of just an offset from the starting index B.
Theorem	cvgratlem3 7457	Lemma for cvgrati 7460. Restate cvgratlem2 7456 (which was for a real function) in terms of the absolute values of the terms of a complex function F, with the help of an auxiliary function G.
Theorem	cvgratlem4 7458	Lemma for cvgrati 7460. The ratio of successive terms meeting the ratio test criterion is positive.
Theorem	cvgratlem5 7459	Lemma for cvgrati 7460. A complex infinite series F meeting the ratio test criterion converges. We show that the partial sums of F are smaller than the partial sums of a geometric series (which converges by geolimi 7441), so by the comparison test cvgcmp3ce 7394, F also converges.
Theorem	cvgrati 7460	Ratio test for convergence of a complex infinite series. If the ratio A of the absolute values of of successive terms in an infinite sequence F is less than 1 for all terms beyond some index B, then the infinite sum of the terms of F converges to a complex number. Equivalent to first part of Exercise 4 of [Gleason] p. 182.
|- F:NN-->CC   =>   |- (((A e. RR /\ A < 1) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> E.y( + seq1 F) ~~> y)
The product of two finite sums
Theorem	fsum0diaglem1 7461	Lemma for fsum0diag 7463.
Theorem	fsum0diaglem2 7462	Lemma for fsum0diag 7463 that provides its induction hypothesis. Warning: The HTML proof page is 0.8 megabyte in size.
Theorem	fsum0diag 7463	Two ways to express "the sum of A(j) x. B(k) where j + k <_ N."
|- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B) = sum_k e. (0...N)sum_j e. (0...k)(A x. [_(k - j) / k]_B))
Theorem	fsum0diag2 7464	Two ways to express "the sum of A(j) x. B(k) where j + k <_ N."
|- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B) = sum_j e. (0...N)sum_k e. (0...j)([_(j - k) / j]_A x. B))
Theorem	fsum0diag3 7465	Two ways to express "the sum of A(j) x. B(k) where j + k <_ N."
|- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B) = sum_j e. (0...N)(A x. sum_k e. (0...(N - j))B))
Theorem	fsum0diag4 7466	Two ways to express "the sum of A(j) x. B(k) where j + k <_ N."
|- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B) = sum_k e. (0...N)(B x. sum_j e. (0...(N - k))A))
Continuous complex functions
Syntax	ccncf 7467	Extend class notation to include the operation which returns a class of continuous complex functions.
class -cn->
Definition	df-cncf 7468	Define the operation whose value is a class of continuous complex functions.
|- -cn-> = {<.<.a, b>., s>. | ((a (_ CC /\ b (_ CC) /\ s = {f | (f:a-->b /\ A.x e. a A.y e. RR+ E.z e. RR+ A.w e. a ((abs` (x - w)) < z -> (abs` ((f` x) - (f` w))) < y))})}
Theorem	cncfval 7469	The value of the continuous complex function operation is the set of continuous functions from A to B. (Contributed by Paul Chapman, 11-Oct-2007.)
|- ((A (_ CC /\ B (_ CC) -> (A-cn->B) = {f | (f:A-->B /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((f` x) - (f` w))) < y))})
Theorem	elcncf 7470	Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 11-Oct-2007.)
|- ((A (_ CC /\ B (_ CC) -> (F e. (A-cn->B) <-> (F:A-->B /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))))
Theorem	cncff 7471	A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.)
|- ((P (_ CC /\ Q (_ CC /\ F e. (P-cn->Q)) -> F:P-->Q)
Theorem	cncffvelrnOLD 7472	A continuous complex function's value belongs to its codomain. (Contributed by Paul Chapman, 21-Jan-2008.)
|- ((A (_ CC /\ B (_ CC /\ F e. (A-cn->B)) -> (C e. A -> (F` C) e. B))
Theorem	cncffvelrn 7473	A continuous complex function's value belongs to its codomain. (Contributed by Paul Chapman, 21-Jan-2008.)
|- ((A (_ CC /\ B (_ CC /\ F e. (A-cn->B)) -> (C e. A -> (F` C) e. B))
Theorem	negfcncfi 7474	The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.)
|- A (_ CC   &   |- F e. (A-cn->CC)   &   |- G = {<.a, b>. | (a e. A /\ b = -u(F` a))}   =>   |- G e. (A-cn->CC)
Theorem	elcncf1di 7475	Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 26-Nov-2007.)
|- (ph -> F:A-->B)   &   |- (ph -> ((x e. A /\ y e. RR+) -> Z e. RR+))   &   |- (ph -> (((x e. A /\ w e. A) /\ y e. RR+) -> ((abs` (x - w)) < Z -> (abs` ((F` x) - (F` w))) < y)))   =>   |- (ph -> ((A (_ CC /\ B (_ CC) -> F e. (A-cn->B)))
Theorem	elcncf1ii 7476	Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 26-Nov-2007.)
|- F:A-->B   &   |- ((x e. A /\ y e. RR+) -> Z e. RR+)   &   |- (((x e. A /\ w e. A) /\ y e. RR+) -> ((abs` (x - w)) < Z -> (abs` ((F` x) - (F` w))) < y))   =>   |- ((A (_ CC /\ B (_ CC) -> F e. (A-cn->B))
Theorem	rescncf 7477	A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.)
|- ((A (_ CC /\ B (_ CC /\ C (_ A) -> (F e. (A-cn->B) -> (F |` C) e. (C-cn->B)))
Theorem	cncffvrn 7478	Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.)
|- (((A (_ CC /\ B (_ CC /\ C (_ CC) /\ A.x e. A (F` x) e. C) -> (F e. (A-cn->B) -> F e. (A-cn->C)))
Theorem	abscncflem 7479	Lemma for abscncf 7480, recncf 7481, imcncf 7482, and cjcncf 7483.
Theorem	abscncf 7480	Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.)
|- abs e. (CC-cn->RR)
Theorem	recncf 7481	Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.)
|- Re e. (CC-cn->RR)
Theorem	imcncf 7482	Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.)
|- Im e. (CC-cn->RR)
Theorem	cjcncf 7483	Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.)
|- * e. (CC-cn->CC)
Theorem	mulc1cncf 7484	Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
|- F = {<.x, y>. | (x e. CC /\ y = (A x. x))}   =>   |- (A e. CC -> F e. (CC-cn->CC))
Theorem	divccncf 7485	Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
|- F = {<.x, y>. | (x e. CC /\ y = (x / A))}   =>   |- ((A e. CC /\ A =/= 0) -> F e. (CC-cn->CC))
Intermediate value theorem
Theorem	ivthlem1 7486	Lemma for isupivthi 7495.
Theorem	ivthlem2 7487	Lemma for isupivthi 7495.
Theorem	ivthlem3 7488	Lemma for isupivthi 7495.
Theorem	ivthlem4 7489	Lemma for isupivthi 7495.
Theorem	ivthlem5 7490	Lemma for isupivthi 7495.
Theorem	ivthlem6 7491	Lemma for isupivthi 7495: modus tollens case 1.
Theorem	ivthlem7 7492	Lemma for isupivthi 7495: modus tollens case 2.
Theorem	ivthlem8 7493	Lemma for isupivthi 7495.
Theorem	ivthlem9 7494	Lemma for isupivthi 7495.
Theorem	isupivthi 7495	The intermediate value theorem, increasing case with supremum solution. (Contributed by Paul Chapman, 22-Jan-2008.)
|- A e. RR   &   |- B e. RR   &   |- U e. RR   &   |- A < B   &   |- (A[,]B) (_ D   &   |- D (_ CC   &   |- F e. (D-cn->CC)   &   |- (x e. (A[,]B) -> (F` x) e. RR)   &   |- S = {x e. (A[,]B) | (F` x) = U}   &   |- ((F` A) < U /\ U < (F` B))   &   |- C = sup(S, RR, < )   =>   |- (C e. (A(,)B) /\ (F` C) = U)
Theorem	dsupivthlem 7496	Lemma for dsupivthi 7497.
Theorem	dsupivthi 7497	The intermediate value theorem, decreasing case with supremum solution. (Contributed by Paul Chapman, 22-Jan-2008.)
|- A e. RR   &   |- B e. RR   &   |- U e. RR   &   |- A < B   &   |- (A[,]B) (_ D   &   |- D (_ CC   &   |- F e. (D-cn->CC)   &   |- (x e. (A[,]B) -> (F` x) e. RR)   &   |- S = {x e. (A[,]B) | (F` x) = U}   &   |- ((F` B) < U /\ U < (F` A))   &   |- C = sup(S, RR, < )   =>   |- (C e. (A(,)B) /\ (F` C) = U)
The exponential, sine, and cosine functions
Syntax	ce 7498	Extend class notation to include the exponential function.
class exp
Syntax	ceu 7499	Extend class notation to include Euler's constant = 2.7182818....
class e
Syntax	csin 7500	Extend class notation to include the sine function.
class sin