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	shftidt 6301	Identity law for the shift operation.
|- F e. V   =>   |- (A e. CC -> ((F shift 0)` A) = (F` A))
Theorem	seq1shftid 6302	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 6303	Extend class notation with the set of open intervals of extended reals.
class (,)
Syntax	cioc 6304	Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
Syntax	cico 6305	Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
Syntax	cicc 6306	Extend class notation with the set of closed intervals of extended reals.
class [,]
Definition	df-ioo 6307	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 6308	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 6309	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 6310	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 6311	The set of open intervals of extended reals exists.
|- (,) e. V
Theorem	ioovalt 6312	Value of the open interval function.
|- ((A e. RR* /\ B e. RR*) -> (A(,)B) = {x e. RR* | (A < x /\ x < B)})
Theorem	iooval2t 6313	Value of the open interval function.
|- ((A e. RR* /\ B e. RR*) -> (A(,)B) = {x e. RR | (A < x /\ x < B)})
Theorem	ioo0t 6314	An empty open interval of extended reals.
|- ((A e. RR* /\ B e. RR*) -> ((A(,)B) = (/) <-> B <_ A))
Theorem	ioon0t 6315	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 6316	The open interval function's value is empty outside of its domain.
|- (-. (A e. RR* /\ B e. RR*) -> (A(,)B) = (/))
Theorem	iooid 6317	An open interval with identical lower and upper bounds is empty.
|- (A(,)A) = (/)
Theorem	iooint 6318	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 6319	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 6320	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 6321	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 6322	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 6323	Value of the closed interval function.
|- ((A e. RR* /\ B e. RR*) -> (A[,]B) = {x e. RR* | (A <_ x /\ x <_ B)})
Theorem	elioo1t 6324	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 6325	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 6326	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 6327	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 6328	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 6329	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 6330	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 6331	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 6332	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 6333	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 6334	The open interval from minus to plus infinity.
|- ( -oo(,) +oo) = RR
Theorem	ioopos 6335	The set of positive reals expressed as an open interval.
|- (0(,) +oo) = {x e. RR | 0 < x}
Theorem	ioorp 6336	The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (0(,) +oo) = RR+
Theorem	ioossre 6337	An open interval is a set of reals.
|- (A(,)B) (_ RR
Theorem	iccssret 6338	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 6339	An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
|- (A(,)B) (_ (A[,]B)
Theorem	iccsupr 6340	A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum (see suprcl 6011). (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 6341	Two ways of saying that a real number is positive.
|- (A e. (0(,) +oo) <-> (A e. RR /\ 0 < A))
Theorem	ioof 6342	The set of open intervals of extended reals maps to subsets of reals.
|- (,):(RR* X. RR*)-->P~RR
Theorem	iccf 6343	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 6344	The union of the range of the open interval function.
|- U.ran (,) = RR
Theorem	dfioo2 6345	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 6346	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[,]B))
Theorem	ubicc2t 6347	The upper 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) -> B e. (A[,]B))
Theorem	ioonegt 6348	Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (C e. (A(,)B) <-> -uC e. (-uB(,)-uA)))
Theorem	iccnegt 6349	Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (C e. (A[,]B) <-> -uC e. (-uB[,]-uA)))
Theorem	icoshft 6350	A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (X e. (A[,)B) -> (X + C) e. ((A + C)[,)(B + C))))
Theorem	icoshftf1oi 6351	Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.)
|- F = {<.x, y>. | (x e. (A[,)B) /\ y = (x + C))}   &   |- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- F:(A[,)B)-1-1-onto->((A + C)[,)(B + C))
Theorem	icoshftf1olem 6352	Lemma for icoshftf1o 6353.
Theorem	icoshftf1o 6353	Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.)
|- F = {<.x, y>. | (x e. (A[,)B) /\ y = (x + C))}   =>   |- ((A e. RR /\ B e. RR /\ C e. RR) -> F:(A[,)B)-1-1-onto->((A + C)[,)(B + C)))
Theorem	icounlem 6354	Lemma for icoun 6355.
Theorem	icoun 6355	The union of end-to-end closed-below, open-above real intervals. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A <_ B /\ B <_ C)) -> ((A[,)B) u. (B[,)C)) = (A[,)C))
Theorem	snunioolem 6356	Lemma for snunioo 6357.
Theorem	snunioo 6357	The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ((A e. RR /\ B e. RR /\ A < B) -> ({A} u. (A(,)B)) = (A[,)B))
Upper partititions of integers
Syntax	cuz 6358	Extend class notation with the upper integer function. Read "ZZ>` M" as "the set of integers greater than or equal to M."
class ZZ>
Definition	df-uz 6359	Define a function whose value at j is the semi-infinite set of contiguous integers starting at j, which we will also call the upper integers starting at j. Read "ZZ>` M" as "the set of integers greater than or equal to M." See uzvalt 6360 for its value, uzssz 6371 for its relationship to ZZ, nnuz 6380 and nn0uz 6379 for its relationships to NN and NN0, and