mmtheorems66 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6501-6600 - Page 66 of 107

Type	Label	Description
Statement
Theorem	seq00 6501	Value of the 0-based recursive sequence builder at 0.
|- S e. V   &   |- F e. V   =>   |- ((S seq0 F)` 0) = (F` 0)
Theorem	seq0p1 6502	Value of the 0-based recursive sequence builder at a successor.
|- S e. V   &   |- F e. V   =>   |- (N e. NN0 -> ((S seq0 F)` (N + 1)) = (((S seq0 F)` N)S(F` (N + 1))))
Theorem	seq01 6503	Value of the 0-based recursive sequence builder at 1.
|- S e. V   &   |- F e. V   =>   |- ((S seq0 F)` 1) = ((F` 0)S(F` 1))
Theorem	seqzval2t 6504	Value of the arbitrary-based recursive sequence builder operation.
|- S e. V   &   |- F e. V   =>   |- ((M e. ZZ /\ N e. ZZ /\ M <_ N) -> ((<.M, S>. seq F)` N) = (((S seq0 (F shift -uM)) shift M)` N))
Theorem	seqzfveq 6505	Equality theorem for the recursive sequence builder.
|- S e. V   &   |- F e. V   &   |- G e. V   =>   |- ((N e. (ZZ>` M) /\ A.k e. (M...N)(F` k) = (G` k)) -> ((<.M, S>. seq F)` N) = ((<.M, S>. seq G)` N))
Theorem	seqzeq 6506	Equality theorem for the recursive sequence builder.
|- S e. V   &   |- F e. V   &   |- G e. V   =>   |- ((M e. ZZ /\ A.k e. (ZZ>` M)(F` k) = (G` k)) -> (<.M, S>. seq F) = (<.M, S>. seq G))
Theorem	seqzrn2 6507	Range of a sequence generated by the arbitrary-based recursive sequence builder.
|- S e. V   &   |- F e. V   =>   |- (((M e. ZZ /\ (F` M) e. C) /\ ((F |` (ZZ>` (M + 1))):(ZZ>` (M + 1))-->B /\ S:(C X. B)-->C)) -> ran (<.M, S>. seq F) (_ C)
Theorem	seqzrn 6508	Range of the arbitrary-based recursive sequence builder (special case of seqzrn2 6507).
|- S e. V   &   |- F e. V   =>   |- ((M e. ZZ /\ F:(ZZ>` M)-->C /\ S:(C X. C)-->C) -> ran (<.M, S>. seq F) (_ C)
Theorem	seqzcl 6509	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 6510	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 6511	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 6512	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	serzcl1 6513	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 6514	Alternate version of df-seq0 6485.
|- seq0 = {<.<.f, g>., h>. | h = (<.0, f>. seq g)}
Theorem	ser0cl1 6515	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	ser0f 6516	A 0-based infinite series is a function from NN0 to CC.
|- F:NN0-->CC   =>   |- ( + seq0 F):NN0-->CC
Theorem	ser00 6517	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	ser0p1 6518	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 6519	Extend class notation to include exponentiation of a complex number to an integer power.
class ^
Definition	df-exp 6520	Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0t 6522 and expp1t 6525 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 expnnvalt 6523 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	expvalt 6521	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	exp0t 6522	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	expnnvalt 6523	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	exp1t 6524	Value of a complex number raised to the first power.
|- (A e. CC -> (A^1) = A)
Theorem	expp1t 6525	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 6526	Lemma for proving nonnegative integer exponentiation closure laws.
Theorem	nnexpclt 6527	Closure of exponentiation of nonnegative integers.
|- ((A e. NN /\ N e. NN0) -> (A^N) e. NN)
Theorem	nn0expclt 6528	Closure of exponentiation of nonnegative integers.
|- ((A e. NN0 /\ N e. NN0) -> (A^N) e. NN0)
Theorem	zexpclt 6529	Closure of exponentiation of integers.
|- ((A e. ZZ /\ N e. NN0) -> (A^N) e. ZZ)
Theorem	qexpclt 6530	Closure of exponentiation of rationals.
|- ((A e. QQ /\ N e. NN0) -> (A^N) e. QQ)
Theorem	reexpclt 6531	Closure of exponentiation of reals.
|- ((A e. RR /\ N e. NN0) -> (A^N) e. RR)
Theorem	expclt 6532	Closure law for nonnegative integer exponentiation.
|- ((A e. CC /\ N e. NN0) -> (A^N) e. CC)
Theorem	rpexpclt 6533	Closure law for exponentiation of positive reals.
|- ((A e. RR+ /\ N e. NN0) -> (A^N) e. RR+)
Theorem	expm1t 6534	Exponentiation in terms of predecessor exponent.
|- ((A e. CC /\ N e. NN) -> (A^N) = ((A^(N - 1)) x. A))
Theorem	1expt 6535	Value of one raised to a nonnegative integer power.
|- (N e. NN0 -> (1^N) = 1)
Theorem	expeq0t 6536	Natural number exponentiation is 0 iff its mantissa is 0.
|- ((A e. CC /\ N e. NN) -> ((A^N) = 0 <-> A = 0))
Theorem	expne0t 6537	Natural number exponentiation is nonzero iff its mantissa is nonzero.
|- ((A e. CC /\ N e. NN) -> (A =/= 0 <-> (A^N) =/= 0))
Theorem	expne0it 6538	Nonnegative integer exponentiation is nonzero if its mantissa is nonzero.
|- ((A e. CC /\ N e. NN0 /\ A =/= 0) -> (A^N) =/= 0)
Theorem	expgt0t 6539	Nonnegative integer exponentiation with a positive mantissa is positive.
|- ((A e. RR /\ N e. NN0 /\ 0 < A) -> 0 < (A^N))
Theorem	0expt 6540	Value of zero raised to a natural number power.
|- (N e. NN -> (0^N) = 0)
Theorem	expge0t 6541	Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative.
|- ((A e. RR /\ N e. NN0 /\ 0 <_ A) -> 0 <_ (A^N))
Theorem	expgt1t 6542	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	expge1t 6543	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	mulexpt 6544	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	recexpt 6545	Nonnegative integer exponentiation of a reciprocal.
|- ((A e. CC /\ N e. NN0 /\ A =/= 0) -> ((1 / A)^N) = (1 / (A^N)))
Theorem	expaddt 6546	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	expmult 6547	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^(