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	shftresvalt 6301	Value of a restricted shifted sequence.
|- F e. V   =>   |- (B e. C -> (((F shift A) |` C)` B) = ((F shift A)` B))
Theorem	shftvalt 6302	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 6303	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 6304	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 6305	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 6306	Value of a shifted sequence.
|- F e. V   =>   |- ((A e. CC /\ B e. CC) -> ((F shift A)` (B + A)) = (F` B))
Theorem	shftf 6307	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 6308	Composite shift operations.
|- F e. V   =>   |- ((A e. CC /\ B e. CC) -> ((F shift A) shift B) = (F shift (A + B)))
Theorem	shftcan2t 6309	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 6310	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 6311	Identity law for the shift operation.
|- F e. V   =>   |- (A e. CC -> ((F shift 0)` A) = (F` A))
Theorem	seq1shftid 6312	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 6313	Extend class notation with the set of open intervals of extended reals.
class (,)
Syntax	cioc 6314	Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
Syntax	cico 6315	Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
Syntax	cicc 6316	Extend class notation with the set of closed intervals of extended reals.
class [,]
Definition	df-ioo 6317	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 6318	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 6319	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 6320	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 6321	The set of open intervals of extended reals exists.
|- (,) e. V
Theorem	ioovalt 6322	Value of the open interval function.
|- ((A e. RR* /\ B e. RR*) -> (A(,)B) = {x e. RR* | (A < x /\ x < B)})
Theorem	iooval2t 6323	Value of the open interval function.
|- ((A e. RR* /\ B e. RR*) -> (A(,)B) = {x e. RR | (A < x /\ x < B)})
Theorem	ioo0t 6324	An empty open interval of extended reals.
|- ((A e. RR* /\ B e. RR*) -> ((A(,)B) = (/) <-> B <_ A))
Theorem	ioon0t 6325	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 6326	The open interval function's value is empty outside of its domain.
|- (-. (A e. RR* /\ B e. RR*) -> (A(,)B) = (/))
Theorem	iooid 6327	An open interval with identical lower and upper bounds is empty.
|- (A(,)A) = (/)
Theorem	iooint 6328	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 6329	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 6330	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 6331	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 6332	Value of the closed-below, open-above interval function.
|- ((A e. RR* /\ B e. RR*) -> (A[,)B) = {x e. RR* | (A <_ x /\ x < B)})
Theorem	iccvalt 6333	Value of the closed interval function.
|- ((A e. RR* /\ B e. RR*) -> (A[,]B) = {x e. RR* | (A <_ x /\ x <_ B)})
Theorem	elioo1t 6334	Membership in an open interval of extended reals.
|- ((A e. RR* /\ B e. RR*) -> (C e. (A(,)B) <-> (C e. RR* /\ A < C /\ C < B)))
Theorem	elioo2t 6335	Membership in an open interval of extended reals.
|- ((A e. RR* /\ B e. RR*) -> (C e. (A(,)B) <-> (C e. RR /\ A < C /\ C < B)))
Theorem	elioo3g 6336	Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show A e. RR* and B e. RR*.
|- (B e. D -> (C e. (A(,)B) <-> ((A e. RR* /\ B e. RR* /\ C e. RR*) /\ (A < C /\ C < B))))
Theorem	elioo4g 6337	Membership in an open interval of extended reals.
|- (B e. D -> (C e. (A(,)B) <-> ((A e. RR* /\ B e. RR* /\ C e. RR) /\ (A < C /\ C < B))))
Theorem	elioc1t 6338	Membership in an open-below, closed-above interval of extended reals.
|- ((A e. RR* /\ B e. RR*) -> (C e. (A(,]B) <-> (C e. RR* /\ A < C /\ C <_ B)))
Theorem	elico1t 6339	Membership in a closed-below, open-above interval of extended reals.
|- ((A e. RR* /\ B e. RR*) -> (C e. (A[,)B) <-> (C e. RR* /\ A <_ C /\ C < B)))
Theorem	elicc1t 6340	Membership in a closed interval of extended reals.
|- ((A e. RR* /\ B e. RR*) -> (C e. (A[,]B) <-> (C e. RR* /\ A <_ C /\ C <_ B)))
Theorem	elioc2t 6341	Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.)
|- ((A e. RR /\ B e. RR) -> (C e. (A(,]B) <-> (C e. RR /\ A < C /\ C <_ B)))
Theorem	elico2t 6342	Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.)
|- ((A e. RR /\ B e. RR) -> (C e. (A[,)B) <-> (C e. RR /\ A <_ C /\ C < B)))
Theorem	elicc2t 6343	Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.)
|- ((A e. RR /\ B e. RR) -> (C e. (A[,]B) <-> (C e. RR /\ A <_ C /\ C <_ B)))
Theorem	ioomax 6344	The open interval from minus to plus infinity.
|- ( -oo(,) +oo) = RR
Theorem	ioopos 6345	The set of positive reals expressed as an open interval.
|- (0(,) +oo) = {x e. RR | 0 < x}
Theorem	ioorp 6346	The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (0(,) +oo) = RR+
Theorem	ioossre 6347	An open interval is a set of reals.
|- (A(,)B) (_ RR
Theorem	iccssret 6348	A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007. Proof shortened by Paul Chapman, 21-Jan-2008.)
|- ((A e. RR /\ B e. RR) -> (A[,]B) (_ RR)
Theorem	ioossicc 6349	An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
|- (A(,)B) (_ (A[,]B)
Theorem	iccsupr 6350	A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum (see suprcl 6021). (Contributed by Paul Chapman, 21-Jan-2008.)
|- (((A e. RR /\ B e. RR) /\ S (_ (A[,]B) /\ C e. S) -> (S (_ RR /\ S =/= (/) /\ E.x e. RR A.y e. S y <_ x))
Theorem	repos 6351	Two ways of saying that a real number is positive.
|- (A e. (0(,) +oo) <-> (A e. RR /\ 0 < A))
Theorem	ioof 6352	The set of open intervals of extended reals maps to subsets of reals.
|- (,):(RR* X. RR*)-->P~RR
Theorem	iccf 6353	The set of closed intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.)
|- [,]:(RR* X. RR*)-->P~RR*
Theorem	unirnioo 6354	The union of the range of the open interval function.
|- U.ran (,) = RR
Theorem	dfioo2 6355	Alternate definition of 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)})}
Theorem	lbicc2t 6356	The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ((A e. RR /\ B e. RR /\ A <_ B) -> A e. (A[,]