mmtheorems64 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6301-6400 - Page 64 of 107

Type	Label	Description
Statement
Theorem	ser1f 6301	An infinite series is a function from NN to CC.
|- F:NN-->CC   =>   |- ( + seq1 F):NN-->CC
Theorem	ser1cl1 6302	The partial sums in an infinite series of complex terms are complex.
|- F:NN-->CC   =>   |- (A e. NN -> (( + seq1 F)` A) e. CC)
Theorem	ser1recl 6303	The partial sums in an infinite series of real terms are real.
|- F:NN-->RR   =>   |- (A e. NN -> (( + seq1 F)` A) e. RR)
Theorem	ser1ref 6304	The partial sums of an infinite series of reals is an infinite real sequence.
|- F:NN-->RR   =>   |- ( + seq1 F):NN-->RR
Theorem	ser1cl2 6305	Closure of the value of the B th term of an infinite series.
|- F = {<.x, y>. | (x e. NN /\ y = A)}   &   |- A.x e. NN A e. CC   =>   |- (B e. NN -> (( + seq1 F)` B) e. CC)
Theorem	ser1f2 6306	An infinite series is a function from NN to CC.
|- F = {<.x, y>. | (x e. NN /\ y = A)}   &   |- A.x e. NN A e. CC   =>   |- ( + seq1 F):NN-->CC
Theorem	ser11 6307	The value of the first term in an infinite series.
|- F = {<.x, y>. | (x e. NN /\ y = A)}   &   |- B e. V   &   |- (x = 1 -> A = B)   =>   |- (( + seq1 F)` 1) = B
Theorem	ser1p1 6308	The value of the next term in an infinite series.
|- F = {<.x, y>. | (x e. NN /\ y = A)}   &   |- C e. V   &   |- (x = (B + 1) -> A = C)   =>   |- (B e. NN -> (( + seq1 F)` (B + 1)) = ((( + seq1 F)` B) + C))
Theorem	ser1mono 6309	The partial sums in an infinite series of positive terms form a monotonic sequence.
|- F:NN-->RR   &   |- (x e. NN -> 0 <_ (F` x))   =>   |- (A e. NN -> (( + seq1 F)` A) <_ (( + seq1 F)` (A + 1)))
Theorem	ser1add2 6310	The sum of two infinite series.
|- F:NN-->CC   &   |- G:NN-->CC   &   |- H e. V   &   |- ((k e. NN /\ N e. NN /\ k <_ N) -> (H` k) = ((F` k) + (G` k)))   =>   |- (N e. NN -> (( + seq1 H)` N) = ((( + seq1 F)` N) + (( + seq1 G)` N)))
Theorem	ser1add 6311	The sum of two infinite series.
|- F:NN-->CC   &   |- G:NN-->CC   &   |- H e. V   &   |- ((k e. NN /\ k <_ N) -> (H` k) = ((F` k) + (G` k)))   =>   |- (N e. NN -> (( + seq1 H)` N) = ((( + seq1 F)` N) + (( + seq1 G)` N)))
The "shift" operation
Syntax	cshi 6312	Extend class notation with function shifter.
class shift
Definition	df-shft 6313	Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of CC) and produces a new function on CC. See shftvalt 6318 for its value.
|- shift = {<.<.f, x>., g>. | g = {<.y, z>. | (y e. CC /\ z = (f` (y - x)))}}
Theorem	shftfval 6314	The value of the sequence shifter operation is a function on CC. A is ordinarily an integer.
|- F e. V   =>   |- (A e. B -> (F shift A) = {<.x, y>. | (x e. CC /\ y = (F` (x - A)))})
Theorem	shftfn 6315	Functionality and domain of a sequence shifted by A.
|- F e. V   =>   |- (A e. B -> (F shift A) Fn CC)
Theorem	shftres 6316	Restriction of a shifted sequence.
|- F e. V   =>   |- ((A e. C /\ B (_ CC) -> ((F shift A) |` B) Fn B)
Theorem	shftresvalt 6317	Value of a restricted shifted sequence.
|- F e. V   =>   |- (B e. C -> (((F shift A) |` C)` B) = ((F shift A)` B))
Theorem	shftvalt 6318	Value of a sequence shifted by A.
|- F e. V   =>   |- ((A e. C /\ B e. CC) -> ((F shift A)` B) = (F` (B - A)))
Theorem	shftval2t 6319	Value of a sequence shifted by A - B.
|- F e. V   =>   |- ((A e. CC /\ B e. CC /\ C e. CC) -> ((F shift (A - B))` (A + C)) = (F` (B + C)))
Theorem	shftval3t 6320	Value of a sequence shifted by A - B.
|- F e. V   =>   |- ((A e. CC /\ B e. CC) -> ((F shift (A - B))` A) = (F` B))
Theorem	shftval4t 6321	Value of a sequence shifted by -uA.
|- F e. V   =>   |- ((A e. CC /\ B e. CC) -> ((F shift -uA)` B) = (F` (A + B)))
Theorem	shftval5t 6322	Value of a shifted sequence.
|- F e. V   =>   |- ((A e. CC /\ B e. CC) -> ((F shift A)` (B + A)) = (F` B))
Theorem	shftf 6323	Functionality of a restricted shifted sequence.
|- F e. V   =>   |- ((A e. D /\ B (_ CC /\ A.x e. B (F` (x - A)) e. C) -> ((F shift A) |` B):B-->C)
Theorem	2shft 6324	Composite shift operations.
|- F e. V   =>   |- ((A e. CC /\ B e. CC) -> ((F shift A) shift B) = (F shift (A + B)))
Theorem	shftcan2t 6325	Cancellation law for the shift operation.
|- F e. V   =>   |- ((A e. CC /\ B e. CC) -> (((F shift -uA) shift A)` B) = (F` B))
Theorem	shftcan1t 6326	Cancellation law for the shift operation.
|- F e. V   =>   |- ((A e. CC /\ B e. CC) -> (((F shift A) shift -uA)` B) = (F` B))
Theorem	shftidt 6327	Identity law for the shift operation.
|- F e. V   =>   |- (A e. CC -> ((F shift 0)` A) = (F` A))
Theorem	seq1shftid 6328	Identity law for the shift operation in a 1-based sequence builder.
|- S e. V   &   |- F e. V   =>   |- (S seq1 (F shift 0)) = (S seq1 F)
Real number intervals
Syntax	cioo 6329	Extend class notation with the set of open intervals of extended reals.
class (,)
Syntax	cioc 6330	Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
Syntax	cico 6331	Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
Syntax	cicc 6332	Extend class notation with the set of closed intervals of extended reals.
class [,]
Definition	df-ioo 6333	Define the set of open intervals of extended reals.
|- (,) = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})}
Definition	df-ioc 6334	Define the set of open-below, closed-above intervals of extended reals.
|- (,] = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w <_ y)})}
Definition	df-ico 6335	Define the set of closed-below, open-above intervals of extended reals.
|- [,) = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x <_ w /\ w < y)})}
Definition	df-icc 6336	Define the set of closed intervals of extended reals.
|- [,] = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x <_ w /\ w <_ y)})}
Theorem	iooex 6337	The set of open intervals of extended reals exists.
|- (,) e. V
Theorem	ioovalt 6338	Value of the open interval function.
|- ((A e. RR* /\ B e. RR*) -> (A(,)B) = {x e. RR* | (A < x /\ x < B)})
Theorem	iooval2t 6339	Value of the open interval function.
|- ((A e. RR* /\ B e. RR*) -> (A(,)B) = {x e. RR | (A < x /\ x < B)})
Theorem	ioo0t 6340	An empty open interval of extended reals.
|- ((A e. RR* /\ B e. RR*) -> ((A(,)B) = (/) <-> B <_ A))
Theorem	ioon0t 6341	An open interval of extended reals is nonempty iff the lower argument is less than the upper argument.
|- ((A e. RR* /\ B e. RR*) -> ((A(,)B) =/= (/) <-> A < B))
Theorem	ndmioo 6342	The open interval function's value is empty outside of its domain.
|- (-. (A e. RR* /\ B e. RR*) -> (A(,)B) = (/))
Theorem	iooid 6343	An open interval with identical lower and upper bounds is empty.
|- (A(,)A) = (/)
Theorem	iooint 6344	Intersection of two open intervals of extended reals.
|- (((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ D e. RR*)) -> ((A(,)B) i^i (C(,)D)) = (if(A <_ C, C, A)(,)if(B <_ D, B, D)))
Theorem	iooss1 6345	Subset relationship for open intervals of extended reals.
|- (((A e. RR* /\ B e. RR* /\ C e. RR*) /\ A <_ B) -> (B(,)C) (_ (A(,)C))
Theorem	iooss2 6346	Subset relationship for open intervals of extended reals.
|- (((A e. RR* /\ B e. RR* /\ C e. RR*) /\ B <_ C) -> (A(,)B) (_ (A(,)C))
Theorem	iocvalt 6347	Value of the open-below, closed-above interval function.
|- ((A e. RR* /\ B e. RR*) -> (A(,]B) = {x e. RR* | (A < x /\ x <_ B)})
Theorem	icovalt 6348	Value of the closed-below, open-above interval function.
|- ((A e. RR* /\ B e. RR*) -> (A[,)