mmtheorems70 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6901-7000 - Page 70 of 107

Type	Label	Description
Statement
Theorem	facavgt 6901	The product of two factorials is greater than or equal to the factorial of (the floor of) their average.
|- ((M e. NN0 /\ N e. NN0) -> (!` (|_` ((M + N) / 2))) <_ ((!` M) x. (!` N)))
The binomial coefficient operation
Syntax	cbc 6902	Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
class C.
Definition	df-bc 6903	Define the binomial coefficient operation. In the literature, this function is often written as a column vector of the two arguments, or with the arguments as subscripts before and after the letter "C". (N C. K) is read "N choose K." Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 <_ k <_ n does not hold.
|- C. = {<.<.n, k>., m>. | ((n e. NN0 /\ k e. ZZ) /\ m = if((0 <_ k /\ k <_ n), ((!` n) / ((!` (n - k)) x. (!` k))), 0))}
Theorem	bcvalt 6904	Value of the binomial coefficient, N choose K. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 <_ K <_ N does not hold. See bcval2t 6906 for the value in the standard domain.
|- ((N e. NN0 /\ K e. ZZ) -> (N C. K) = if((0 <_ K /\ K <_ N), ((!` N) / ((!` (N - K)) x. (!` K))), 0))
Theorem	bcval3tOLD 6905	Value of the binomial coefficient, N choose K, in its standard domain.
|- (((N e. NN0 /\ K e. ZZ) /\ (0 <_ K /\ K <_ N)) -> (N C. K) = ((!` N) / ((!` (N - K)) x. (!` K))))
Theorem	bcval2t 6906	Value of the binomial coefficient, N choose K, in its standard domain.
|- ((N e. NN0 /\ K e. NN0 /\ K <_ N) -> (N C. K) = ((!` N) / ((!` (N - K)) x. (!` K))))
Theorem	bcval3t 6907	Value of the binomial coefficient, N choose K, in its standard domain.
|- ((N e. NN0 /\ K e. (0...N)) -> (N C. K) = ((!` N) / ((!` (N - K)) x. (!` K))))
Theorem	bcval4t 6908	Value of the binomial coefficient, N choose K, outside of its standard domain. Remark in [Gleason] p. 295.
|- ((N e. NN0 /\ K e. ZZ /\ (K < 0 \/ N < K)) -> (N C. K) = 0)
Theorem	bccmplt 6909	"Complementing" its second argument doesn't change a binary coefficient.
|- ((N e. NN0 /\ K e. NN0 /\ K <_ N) -> (N C. K) = (N C. (N - K)))
Theorem	bcn0t 6910	N choose 0 is 1. Remark in [Gleason] p. 296.
|- (N e. NN0 -> (N C. 0) = 1)
Theorem	bcnnt 6911	N choose N is 1. Remark in [Gleason] p. 296.
|- (N e. NN0 -> (N C. N) = 1)
Theorem	bcnp11t 6912	Binomial coefficient: N + 1 choose 1.
|- (N e. NN0 -> ((N + 1) C. 1) = (N + 1))
Theorem	bcnp1nt 6913	Binomial coefficient: N + 1 choose N.
|- (N e. NN0 -> ((N + 1) C. N) = (N + 1))
Theorem	bcpasc2 6914	Pascal's rule for the binomial coefficient. Equation 2 of [Gleason] p. 295.
|- N e. NN   &   |- K e. NN   &   |- K <_ N   =>   |- ((N C. K) + (N C. (K - 1))) = ((N + 1) C. K)
Theorem	bcpasc2t 6915	Pascal's rule for the binomial coefficient. Equation 2 of [Gleason] p. 295.
|- ((N e. NN /\ K e. NN /\ K <_ N) -> ((N C. K) + (N C. (K - 1))) = ((N + 1) C. K))
Theorem	bcpasc 6916	Pascal's rule for the binomial coefficient, generalized to all integers K. Equation 2 of [Gleason] p. 295.
|- N e. NN0   &   |- K e. ZZ   =>   |- ((N C. K) + (N C. (K - 1))) = ((N + 1) C. K)
Theorem	bcpasct 6917	Pascal's rule for the binomial coefficient, generalized to all integers K. Equation 2 of [Gleason] p. 295.
|- ((N e. NN0 /\ K e. ZZ) -> ((N C. K) + (N C. (K - 1))) = ((N + 1) C. K))
Theorem	bccl2t 6918	A binomial coefficient, in its standard domain, is a natural number.
|- ((N e. NN0 /\ K e. (0...N)) -> (N C. K) e. NN)
Theorem	bcclt 6919	A binomial coefficient, in its extended domain, is a nonnegative integer.
|- ((N e. NN0 /\ K e. ZZ) -> (N C. K) e. NN0)
Theorem	permnnt 6920	The number of permutations of N - R objects from a collection of N objects is a natural number. (Contributed by Jason Orendorff, 24-Jan-2007.)
|- ((N e. NN0 /\ R e. (0...N)) -> ((!` N) / (!` R)) e. NN)
Limits
Syntax	cli 6921	Extend class notation with convergence relation for limits.
class ~~>
Definition	df-clim 6922	Define the limit relation for complex number sequences. See clim 6924 for its relational expression.
|- ~~> = {<.f, y>. | (y e. CC /\ A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((f` k) e. CC /\ (abs` ((f` k) - y)) < x))))}
Theorem	climrel 6923	The limit relation is a relation.
|- Rel ~~>
Theorem	clim 6924	Express the predicate: The limit of complex number sequence F is A, or F converges to A. This means that for any real x, no matter how small, there always exists an integer j such that the absolute difference of any later complex number in the sequence and the limit is less than x.
|- ((F e. C /\ A e. D) -> (F ~~> A <-> (A e. CC /\ A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - A)) < x))))))
Theorem	climcl 6925	Closure of the limit of a sequence of complex numbers.
|- ((A e. C /\ F ~~> A) -> A e. CC)
Finite and infinite sums
Syntax	csu 6926	Extend class notation to include finite summations. (An underscore was added the ASCII token in order to facilitate text searches, since "sum" is is a commonly used word in comments.)
class sum_k e. A B
Definition	df-sum 6927	Define the sum of a series with an index set of integers A. k is normally a free variable in B, i.e. B can be thought of as B(k). The definition is meaningful when A is a finite set of sequential integers (representing a finite sum over them) or a set of upper integers (representing an infinite sum, when the sum converges). The left-hand side of the union expresses the finite sum case, and the right-hand side expresses the infinite sum case. In either case, the other side of the union equals the empty set. Examples: sum_k e. (2...4) k means 2 + 3 + 4 = 9, and sum_k e. NN (1 / (2^k)) means 1/2 + 1/4 + 1/8 + ... = 1. Note: The restrictions to ZZ force the class abstractions to be sets.
|- sum_k e. A B = ({x | E.mE.n e. (ZZ>` m)(A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n))} u. U.{x | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) ~~> x)})
Theorem	sumex 6928	A sum is a set.
|- sum_k e. A B e. V
Theorem	sumeq1 6929	Equality theorem for a sum.
|- (A = B -> sum_k e. A C = sum_k e. B C)
Theorem	hbsum1 6930	Bound-variable hypothesis builder for sum.
|- (x e. A -> A.k x e. A)   =>   |- (x e. sum_k e. A B -> A.k x e. sum_k e. A B)
Theorem	hbsum 6931	Bound-variable hypothesis builder for sum: if x is (effectively) not free in A and B, it is not free in sum_k e. AB.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. sum_k e. A B -> A.x y e. sum_k e. A B)
Theorem	sumeq2 6932	Equality theorem for sum.
|- (A.k e. A B = C -> sum_k e. A B = sum_k e. A C)
Theorem	cbvsum 6933	Change bound variable in a sum.
|- (x e. B -> A.k x e. B)   &   |- (x e. C -> A.j x e. C)   &   |- (j = k -> B = C)   =>   |- sum_j e. A B = sum_k e. A C
Theorem	sumeq1i 6934	Equality inference for sum.
|- A = B   =>   |- sum_k e. A C = sum_k e. B C
Theorem	sumeq2i 6935	Equality inference for sum.
|- (k e. A -> B = C)   =>   |- sum_k e. A B = sum_k e. A C
Theorem	sumeq12i 6936	Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
|- A = B   &   |- (k e. A -> C = D)   =>   |- sum_k e. A C = sum_k e. B D
Theorem	sumeq1d 6937	Equality deduction for sum.
|- (ph -> A = B)   =>   |- (ph -> sum_k e. A C = sum_k e. B C)
Theorem	sumeq2d 6938	Equality deduction for sum. Note that unlike sumeq2dv 6939, k may occur in ph.
|- (ph -> A.k e. A B = C)   =>   |- (ph -> sum_k e. A B = sum_k e. A C)
Theorem	sumeq2dv 6939	Equality deduction for sum.
|- ((ph /\ k e. A) -> B = C)   =>   |- (ph -> sum_k e. A B = sum_k e. A C)
Theorem	sumeq2sdv 6940	Equality deduction for sum.
|- (ph -> B = C)   =>   |- (ph -> sum_k e. A B = sum_k e. A C)
Theorem	2sumeq2dv 6941	Equality deduction for double sum.
|- ((ph /\ j e. A /\ k e. B) -> C = D)   =>   |- (ph -> sum_j e. A sum_k e. B C = sum_j e. A sum_k e. B D)
Theorem	sumeq12dv 6942	Equality deduction for sum.
|- (ph -> A = B)   &   |- ((ph /\ k e. A) -> C = D)   =>   |- (ph -> sum_k e. A C = sum_k e. B D)
Theorem	sumeq12rdv 6943	Equality deduction for sum.
|- (ph -> A = B)   &   |- ((ph /\ k e. B) -> C = D)   =>   |- (ph -> sum_k e. A C = sum_k e. B D)
Theorem	sumeqfv 6944	Convert a sum of function values to a sum of classes A(k).
|- A e. V   &   |- F = {<.k, y>. | (k e. B /\ y = A)}   =>   |- (C (_ B -> sum_k e. C (F` k) = sum_k e. C A)
Finite sums (cont.)
Theorem	dffsum 6945	Special case of series sum over a finite index set.
|- (N e. (ZZ>` M) -> sum_k e. (M...N)A = ((<.M, +