mmtheorems79 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7801-7900 - Page 79 of 175

Type	Label	Description
Statement
Theorem	seqzcl 7801	Closure of the value of the arbitrary-based recursive sequence builder.
|- S e. _V   &   |- F e. _V   =>   |- ((N e. (ZZ>=` M) /\ F:(ZZ>=` M)-->C /\ S:(C X. C)-->C) -> ((<.M, S>. seq F)` N) e. C)
Theorem	seqzresval 7802	A restriction of its characteristic function that doesn't change the value of the seq function.
|- S e. _V   &   |- F e. _V   =>   |- (N e. (ZZ>=` M) -> ((<.M, S>. seq (F |` (M...N)))` N) = ((<.M, S>. seq F)` N))
Theorem	seqzres 7803	The seq function is unchanged by restricting its characteristic function to the seq function's domain.
|- S e. _V   &   |- F e. _V   =>   |- (M e. ZZ -> (<.M, S>. seq (F |` (ZZ>=` M))) = (<.M, S>. seq F))
Theorem	seqzres2 7804	The seq function is unchanged by substituting its characteristic function with a restricted class builder based on that function.
|- S e. _V   &   |- F e. _V   =>   |- (M e. ZZ -> (<.M, S>. seq ({<.k, y>. | y = (F` k)} |` ZZ)) = (<.M, S>. seq F))
Theorem	serzcl1i 7805	The partial sums in an infinite series of complex terms are complex.
|- F:(ZZ>=` M)-->CC   =>   |- (N e. (ZZ>=` M) -> ((<.M, + >. seq F)` N) e. CC)
Theorem	dfseq0 7806	Alternate version of df-seq0 7777.
|- seq0 = {<.<.f, g>., h>. | h = (<.0, f>. seq g)}
Theorem	ser0cl1i 7807	The partial sums in an infinite 0-based series of complex terms are complex.
|- F:NN0-->CC   =>   |- (N e. NN0 -> (( + seq0 F)` N) e. CC)
Theorem	ser0fi 7808	A 0-based infinite series is a function from NN0 to CC.
|- F:NN0-->CC   =>   |- ( + seq0 F):NN0-->CC
Theorem	ser00i 7809	The value of the first term in a 0-based infinite series.
|- F = {<.k, y>. | (k e. NN0 /\ y = A)}   &   |- B e. _V   &   |- (k = 0 -> A = B)   =>   |- (( + seq0 F)` 0) = B
Theorem	ser0p1i 7810	The value of the next term in a 0-based infinite series.
|- F = {<.k, y>. | (k e. NN0 /\ y = A)}   &   |- B e. _V   &   |- (k = (N + 1) -> A = B)   =>   |- (N e. NN0 -> (( + seq0 F)` (N + 1)) = ((( + seq0 F)` N) + B))
Integer powers
Syntax	cexp 7811	Extend class notation to include exponentiation of a complex number to an integer power.
class ^
Definition	df-exp 7812	Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 7814 and expp1 7817 provide a the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. See expnnval 7815 for a description of how the recursive sequence builder is used. 10-Jun-2005: The definition was extended to include zero exponents, so that 0^0 = 1 per the convention of Definition 10-4.1 of [Gleason] p. 134. (Based on definition contributed by Raph Levien, 15-Oct-2004.)
|- ^ = {<.<.x, y>., z>. | ((x e. CC /\ y e. NN0) /\ z = if(y = 0, 1, (( x. seq1 (NN X. {x}))` y)))}
Theorem	expval 7813	Value of exponentiation to nonnegative integer powers.
|- ((A e. CC /\ N e. NN0) -> (A^N) = if(N = 0, 1, (( x. seq1 (NN X. {A}))` N)))
Theorem	exp0 7814	Value of a complex number raised to the 0th power. Note that under our definition, 0^0 = 1, following the convention used by Gleason. Part of Definition 10-4.1 of [Gleason] p. 134.
|- (A e. CC -> (A^0) = 1)
Theorem	expnnval 7815	Value of exponentiation to natural number powers. NN X. {A} is the constant function with value A. The seq1 operation produces the sequence A, A x. A, (A x. A) x. A,... that we evaluate at index B.
|- ((A e. CC /\ B e. NN) -> (A^B) = (( x. seq1 (NN X. {A}))` B))
Theorem	exp1 7816	Value of a complex number raised to the first power.
|- (A e. CC -> (A^1) = A)
Theorem	expp1 7817	Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134.
|- ((A e. CC /\ N e. NN0) -> (A^(N + 1)) = ((A^N) x. A))
Theorem	expcllem 7818	Lemma for proving nonnegative integer exponentiation closure laws.
Theorem	nnexpcl 7819	Closure of exponentiation of nonnegative integers.
|- ((A e. NN /\ N e. NN0) -> (A^N) e. NN)
Theorem	nn0expcl 7820	Closure of exponentiation of nonnegative integers.
|- ((A e. NN0 /\ N e. NN0) -> (A^N) e. NN0)
Theorem	zexpcl 7821	Closure of exponentiation of integers.
|- ((A e. ZZ /\ N e. NN0) -> (A^N) e. ZZ)
Theorem	qexpcl 7822	Closure of exponentiation of rationals.
|- ((A e. QQ /\ N e. NN0) -> (A^N) e. QQ)
Theorem	reexpcl 7823	Closure of exponentiation of reals.
|- ((A e. RR /\ N e. NN0) -> (A^N) e. RR)
Theorem	expcl 7824	Closure law for nonnegative integer exponentiation.
|- ((A e. CC /\ N e. NN0) -> (A^N) e. CC)
Theorem	rpexpcl 7825	Closure law for exponentiation of positive reals.
|- ((A e. RR+ /\ N e. NN0) -> (A^N) e. RR+)
Theorem	expm1t 7826	Exponentiation in terms of predecessor exponent.
|- ((A e. CC /\ N e. NN) -> (A^N) = ((A^(N - 1)) x. A))
Theorem	1exp 7827	Value of one raised to a nonnegative integer power.
|- (N e. NN0 -> (1^N) = 1)
Theorem	expeq0 7828	Natural number exponentiation is 0 iff its mantissa is 0.
|- ((A e. CC /\ N e. NN) -> ((A^N) = 0 <-> A = 0))
Theorem	expne0 7829	Natural number exponentiation is nonzero iff its mantissa is nonzero.
|- ((A e. CC /\ N e. NN) -> ((A^N) =/= 0 <-> A =/= 0))
Theorem	expne0i 7830	Nonnegative integer exponentiation is nonzero if its mantissa is nonzero.
|- ((A e. CC /\ A =/= 0 /\ N e. NN0) -> (A^N) =/= 0)
Theorem	expgt0 7831	Nonnegative integer exponentiation with a positive mantissa is positive.
|- ((A e. RR /\ N e. NN0 /\ 0 < A) -> 0 < (A^N))
Theorem	0exp 7832	Value of zero raised to a natural number power.
|- (N e. NN -> (0^N) = 0)
Theorem	expge0 7833	Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative.
|- ((A e. RR /\ N e. NN0 /\ 0 <_ A) -> 0 <_ (A^N))
Theorem	expgt1 7834	Natural number exponentiation with a mantissa greater than 1 is greater than 1.
|- ((A e. RR /\ N e. NN /\ 1 < A) -> 1 < (A^N))
Theorem	expge1 7835	Nonnegative integer exponentiation with a mantissa greater than or equal to 1 is greater than or equal to 1.
|- ((A e. RR /\ N e. NN0 /\ 1 <_ A) -> 1 <_ (A^N))
Theorem	mulexp 7836	Natural number exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents.
|- ((A e. CC /\ B e. CC /\ N e. NN0) -> ((A x. B)^N) = ((A^N) x. (B^N)))
Theorem	exprec 7837	Nonnegative integer exponentiation of a reciprocal.
|- ((A e. CC /\ A =/= 0 /\ N e. NN0) -> ((1 / A)^N) = (1 / (A^N)))
Theorem	exprecOLD 7838	Nonnegative integer exponentiation of a reciprocal.
|- ((A e. CC /\ N e. NN0 /\ A =/= 0) -> ((1 / A)^N) = (1 / (A^N)))
Theorem	expadd 7839	Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135.
|- ((A e. CC /\ M e. NN0 /\ N e. NN0) -> (A^(M + N)) = ((A^M) x. (A^N)))
Theorem	expmul 7840	Product of exponents law for natural number exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents.
|- ((A e. CC /\ M e. NN0 /\ N e. NN0) -> (A^(M x. N)) = ((A^M)^N))
Theorem	expsub 7841	Exponent subtraction law for nonnegative integer exponentiation.
|- ((((A e. CC /\ A =/= 0) /\ M e. NN0 /\ N e. NN0) /\ N <_ M) -> (A^(M - N)) = ((A^M) / (A^N)))
Theorem	expsubOLD 7842	Exponent subtraction law for nonnegative integer exponentiation.
|- (((A e. CC /\ M e. NN0 /\ N e. NN0) /\ (A =/= 0 /\ N <_ M)) -> (A^(M - N)) = ((A^M) / (A^N)))
Theorem	expm1 7843	Value of a complex number raised to a nonnegative integer power minus one.
|- ((A e. CC /\ A =/= 0 /\ N e. NN) -> (A^(N - 1)) = ((A^N) / A))
Theorem	expdiv 7844	Nonnegative integer exponentiation of a quotient.
|- ((A e. CC /\ (B e. CC /\ B =/= 0) /\ N e. NN0) -> ((A / B)^N) = ((A^N) / (B^N)))
Theorem	expordi 7845	Ordering relationship for exponentiation.
|- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (1 < A /\ M < N)) -> (A^M) < (A^N))
Theorem	expcan 7846	Cancellation law for exponentiation.
|- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 1 < A) -> ((A^M) = (A^N) <-> M = N))
Theorem	expord 7847	Ordering law for exponentiation.
|- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 1 < A) -> (M < N <-> (A^M) < (A^N)))
Theorem	expwordi 7848	Weak ordering relationship for exponentiation.
|- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (1 <_ A /\ M <_ N)) -> (A^M) <_ (A^N))
Theorem	expord2 7849	The power of a positive number smaller than 1 decreases as its exponent increases.
|- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A < 1)) -> (M < N <-> (A^N) < (A^M)))
Theorem	expword2i 7850	Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.)
|- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A <_ 1 /\ M < N)) -> (A^N) <_ (A^M))
Theorem	expmwordi 7851	Weak mantissa ordering relationship for exponentiation.
|- (((A e. RR /\ B e. RR /\ N e. NN0) /\ (0 <_ A /\ A <_ B)) -> (A^N) <_ (B^N))
Theorem	exple1 7852	Nonnegative integer exponentiation with a mantissa between 0 and 1 inclusive is less than or equal to 1. (Contributed by Paul Chapman, 29-Dec-2007.)
|- (((A e. RR /\ 0 <_ A /\ A <_ 1) /\ N e. NN0) -> (A^N) <_ 1)
Theorem	expubnd 7853	An upper bound on A^N when 2 <_ A.
|- ((A e. RR /\ N e. NN0 /\ 2 <_ A) -> (A^N) <_ ((2^N) x. ((A - 1)^N)))
Theorem	sqval 7854	Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
|- (A e. CC -> (A^2) = (A x. A))
Theorem	sqneg 7855	The square of the negative of a number.)
|- (A e. CC -> (-uA^2) = (A^2))
Theorem	sqcl 7856	Closure of square.
|- (A e. CC -> (A^2) e. CC)
Theorem	sqmul 7857	Distribution of square over multiplication.
|- ((A e. CC /\ B e. CC) -> ((A x. B)^2) = ((A^2) x. (B^2)))
Theorem	sqeq0 7858	A number is zero iff its square is zero.
|- (A e. CC -> ((A^2) = 0 <-> A = 0))
Theorem	sqvali 7859	Value of square. Inference version.
|- A e. CC   =>   |- (A^2) = (A x. A)
Theorem	sqcli 7860	Closure of square.
|- A e. CC   =>   |- (A^2) e. CC
Theorem	sqeq0i 7861	A number is zero iff its square is zero.
|- A e. CC   =>   |- ((A^2) = 0 <-> A = 0)
Theorem	sqmuli 7862	Distribution of square over multiplication.
|- A e. CC   &   |- B e. CC   =>   |- ((A x. B)^2) = ((A^2) x. (B^2))
Theorem	sqdivi 7863	Distribution of square over division.
|- A e. CC   &   |- B e. CC   &   |- B =/= 0   =>   |- ((A / B)^2) = ((A^2) / (B^2))
Theorem	sqrecii 7864	Square of reciprocal.
|- A e. CC   &   |- A =/= 0   =>   |- ((1 / A)^2) = (1 / (A^2))
Theorem	sqne0 7865	A number is nonzero iff its square is nonzero.
|- (A e. CC -> ((A^2) =/= 0 <-> A =/= 0))
Theorem	resqcl 7866	Closure of the square of a real number.
|- (A e. RR -> (A^2) e. RR)
Theorem	sqgt0 7867	The square of a nonzero real is positive.
|- ((A e. RR /\ A =/= 0) -> 0 < (A^2))
Theorem	resqcli 7868	Closure of square in reals.
|- A e. RR   =>   |- (A^2) e. RR
Theorem	lt2sqi 7869	The square function on nonnegative reals is strictly monotonic.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> (A < B <-> (A^2) < (B^2)))
Theorem	le2sqi 7870	The square function on nonnegative reals is monotonic.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> (A <_ B <-> (A^2) <_ (B^2)))
Theorem	sq11i 7871	The square function is one-to-one for nonnegative reals.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> ((A^2) = (B^2) <-> A = B))
Theorem	sqgt0i 7872	The square of a nonzero real is positive.
|- A e. RR   =>   |- (A =/= 0 -> 0 < (A^2))
Theorem	sqge0i 7873	A square of a real is nonnegative.
|- A e. RR   =>   |- 0 <_ (A^2)
Theorem	sq11 7874	The square function is one-to-one for nonnegative reals.
|- (((A e. RR /\ 0 <_ A) /\ (B e. RR /\ 0 <_ B)) -> ((A^2) = (B^2) <-> A = B))
Theorem	lt2sq 7875	The square function on nonnegative reals is strictly monotonic.
|- (((A e. RR /\ 0 <_ A) /\ (B e. RR /\ 0 <_ B)) -> (A < B <-> (A^2) < (B^2)))
Theorem	le2sq 7876	The square function on nonnegative reals is monotonic.
|- (((A e. RR /\ 0 <_ A) /\ (B e. RR /\ 0 <_ B)) -> (A <_ B <-> (A^2) <_ (B^2)))
Theorem	le2sq2 7877	The square of a 'less than or equal to' ordering.
|- (((A e. RR /\ 0 <_ A) /\ (B e. RR /\ A <_ B)) -> (A^2) <_ (B^2))
Theorem	sqge0 7878	A square of a real is nonnegative.
|- (A e. RR -> 0 <_ (A^2))
Theorem	sumsqne0i 7879	The sum of two squares is nonzero iff one of its terms is nonzero.
|- A e. RR   &   |- B e. RR   =>   |- ((A =/= 0 \/ B =/= 0) <-> ((A^2) + (B^2)) =/= 0)
Theorem	sq0 7880	The square of 0 is 0.
|- (0^2) = 0
Theorem	sq0i 7881	If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
|- (A = 0 -> (A^2) = 0)
Theorem	sq1 7882	The square of 1 is 1.
|- (1^2) = 1
Theorem	sq2 7883	The square of 2 is 4.
|- (2^2) = 4
Theorem	sq3 7884	The square of 3 is 9.
|- (3^2) = 9
Theorem	cu2 7885	The cube of 2 is 8.
|- (2^3) = 8
Theorem	expnass 7886	A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)
|- ((3^3)^3) < (3^(3^3))
Theorem	sqlecan 7887	Cancel one factor of a square in a <_ comparison. Unlike lemul1 7011, the common factor A may be zero.
|- (((A e. RR /\ 0 <_ A) /\ (B e. RR /\ 0 <_ B)) -> ((A^2) <_ (B x. A) <-> A <_ B))
Theorem	subsq 7888	Factor the difference of two squares.
|- ((A e. CC /\ B e. CC) -> ((A^2) - (B^2)) = ((A + B) x. (A - B)))
Theorem	subsq2 7889	Express the difference of the squares of two numbers as a polynomial in the difference of the numbers.
|- ((A e. CC /\ B e. CC) -> ((A^2) - (B^2)) = (((A - B)^2) + ((2 x. B) x. (A - B))))
Theorem	binom2i 7890	The square of a binomial.
|- A e. CC   &   |- B e. CC   =>   |- ((A + B)^2) = (((A^2) + (2 x. (A x. B))) + (B^2))
Theorem	binom2aiOLD 7891	Product of sum and difference.
|- A e. CC   &   |- B e. CC   =>   |- ((A + B) x. (A - B)) = ((A^2) - (B^2))
Theorem	subsqi 7892	Factor the difference of two squares.
|- A e. CC   &   |- B e. CC   =>   |- ((A^2) - (B^2)) = ((A + B) x. (A - B))
Theorem	sqeqori 7893	The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of [Gleason] p. 311 and its converse.
|- A e. CC   &   |- B e. CC   =>   |- ((A^2) = (B^2) <-> (A = B \/ A = -uB))
Theorem	subsq0i 7894	The two solutions to the difference of squares set equal to zero.
|- A e. CC   &   |- B e. CC   =>   |- (((A^2) - (B^2)) = 0 <-> (A = B \/ A = -uB))
Theorem	sqeqor 7895	The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of [Gleason] p. 311 and its converse. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ((A e. CC /\ B e. CC) -> ((A^2) = (B^2) <-> (A = B \/ A = -uB)))
Theorem	binom2 7896	The square of a binomial. (Contributed by FL, 10-Dec-2006.)
|- ((A e. CC /\ B e. CC) -> ((A + B)^2) = (((A^2) + (2 x. (A x. B))) + (B^2)))
Theorem	sq01 7897	If a complex number equals its square, it must be 0 or 1.
|- (A e. CC -> ((A^2) = A <-> (A = 0 \/ A = 1)))
Theorem	bernneq 7898	Bernoulli's inequality, due to Johan Bernoulli (1667-1748).
|- ((A e. RR /\ N e. NN0 /\ -u1 <_ A) -> (1 + (A x. N)) <_ ((1 + A)^N))
Theorem	bernneqOLD 7899	Bernoulli's inequality, due to Johan Bernoulli (1667-1748).
|- ((A e. RR /\ N e. NN0 /\ -u1 <_ A) -> (1 + (A x. N)) <_ ((1 + A)^N))
Theorem	bernneq2 7900	Variation of Bernoulli's inequality bernneq 7898.
|- ((A e. RR /\ N e. NN0 /\ 0 <_ A) -> (((A - 1) x. N) + 1) <_ (A^N))