Theorem List for Intuitionistic Logic Explorer - 9601-9700 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | nmnfgt 9601 |
An extended real is greater than minus infinite iff they are not equal.
(Contributed by Jim Kingdon, 17-Apr-2023.)
|
|
|
Theorem | xrrebnd 9602 |
An extended real is real iff it is strictly bounded by infinities.
(Contributed by NM, 2-Feb-2006.)
|
|
|
Theorem | xrre 9603 |
A way of proving that an extended real is real. (Contributed by NM,
9-Mar-2006.)
|
|
|
Theorem | xrre2 9604 |
An extended real between two others is real. (Contributed by NM,
6-Feb-2007.)
|
|
|
Theorem | xrre3 9605 |
A way of proving that an extended real is real. (Contributed by FL,
29-May-2014.)
|
|
|
Theorem | ge0gtmnf 9606 |
A nonnegative extended real is greater than negative infinity.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | ge0nemnf 9607 |
A nonnegative extended real is greater than negative infinity.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xrrege0 9608 |
A nonnegative extended real that is less than a real bound is real.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | z2ge 9609* |
There exists an integer greater than or equal to any two others.
(Contributed by NM, 28-Aug-2005.)
|
|
|
Theorem | xnegeq 9610 |
Equality of two extended numbers with in front of them.
(Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xnegpnf 9611 |
Minus . Remark
of [BourbakiTop1] p. IV.15. (Contributed
by FL,
26-Dec-2011.)
|
|
|
Theorem | xnegmnf 9612 |
Minus . Remark
of [BourbakiTop1] p. IV.15. (Contributed
by FL,
26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | rexneg 9613 |
Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by
FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xneg0 9614 |
The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xnegcl 9615 |
Closure of extended real negative. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xnegneg 9616 |
Extended real version of negneg 8012. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xneg11 9617 |
Extended real version of neg11 8013. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xltnegi 9618 |
Forward direction of xltneg 9619. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xltneg 9619 |
Extended real version of ltneg 8224. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xleneg 9620 |
Extended real version of leneg 8227. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xlt0neg1 9621 |
Extended real version of lt0neg1 8230. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xlt0neg2 9622 |
Extended real version of lt0neg2 8231. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xle0neg1 9623 |
Extended real version of le0neg1 8232. (Contributed by Mario Carneiro,
9-Sep-2015.)
|
|
|
Theorem | xle0neg2 9624 |
Extended real version of le0neg2 8233. (Contributed by Mario Carneiro,
9-Sep-2015.)
|
|
|
Theorem | xrpnfdc 9625 |
An extended real is or is not plus infinity. (Contributed by Jim Kingdon,
13-Apr-2023.)
|
DECID |
|
Theorem | xrmnfdc 9626 |
An extended real is or is not minus infinity. (Contributed by Jim
Kingdon, 13-Apr-2023.)
|
DECID |
|
Theorem | xaddf 9627 |
The extended real addition operation is closed in extended reals.
(Contributed by Mario Carneiro, 21-Aug-2015.)
|
|
|
Theorem | xaddval 9628 |
Value of the extended real addition operation. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xaddpnf1 9629 |
Addition of positive infinity on the right. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xaddpnf2 9630 |
Addition of positive infinity on the left. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xaddmnf1 9631 |
Addition of negative infinity on the right. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xaddmnf2 9632 |
Addition of negative infinity on the left. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
|
|
Theorem | pnfaddmnf 9633 |
Addition of positive and negative infinity. This is often taken to be a
"null" value or out of the domain, but we define it (somewhat
arbitrarily)
to be zero so that the resulting function is total, which simplifies
proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | mnfaddpnf 9634 |
Addition of negative and positive infinity. This is often taken to be a
"null" value or out of the domain, but we define it (somewhat
arbitrarily)
to be zero so that the resulting function is total, which simplifies
proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | rexadd 9635 |
The extended real addition operation when both arguments are real.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | rexsub 9636 |
Extended real subtraction when both arguments are real. (Contributed by
Mario Carneiro, 23-Aug-2015.)
|
|
|
Theorem | rexaddd 9637 |
The extended real addition operation when both arguments are real.
Deduction version of rexadd 9635. (Contributed by Glauco Siliprandi,
24-Dec-2020.)
|
|
|
Theorem | xnegcld 9638 |
Closure of extended real negative. (Contributed by Mario Carneiro,
28-May-2016.)
|
|
|
Theorem | xrex 9639 |
The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
|
|
|
Theorem | xaddnemnf 9640 |
Closure of extended real addition in the subset
.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xaddnepnf 9641 |
Closure of extended real addition in the subset
.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xnegid 9642 |
Extended real version of negid 8009. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xaddcl 9643 |
The extended real addition operation is closed in extended reals.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xaddcom 9644 |
The extended real addition operation is commutative. (Contributed by NM,
26-Dec-2011.)
|
|
|
Theorem | xaddid1 9645 |
Extended real version of addid1 7900. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xaddid2 9646 |
Extended real version of addid2 7901. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xaddid1d 9647 |
is a right identity for
extended real addition. (Contributed by
Glauco Siliprandi, 17-Aug-2020.)
|
|
|
Theorem | xnn0lenn0nn0 9648 |
An extended nonnegative integer which is less than or equal to a
nonnegative integer is a nonnegative integer. (Contributed by AV,
24-Nov-2021.)
|
NN0* |
|
Theorem | xnn0le2is012 9649 |
An extended nonnegative integer which is less than or equal to 2 is either
0 or 1 or 2. (Contributed by AV, 24-Nov-2021.)
|
NN0*
|
|
Theorem | xnn0xadd0 9650 |
The sum of two extended nonnegative integers is iff each of the two
extended nonnegative integers is . (Contributed by AV,
14-Dec-2020.)
|
NN0* NN0* |
|
Theorem | xnegdi 9651 |
Extended real version of negdi 8019. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xaddass 9652 |
Associativity of extended real addition. The correct condition here is
"it is not the case that both and appear as one of
,
i.e. ", but this
condition is difficult to work with, so we break the theorem into two
parts: this one, where is not present in , and
xaddass2 9653, where is not present. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xaddass2 9653 |
Associativity of extended real addition. See xaddass 9652 for notes on the
hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xpncan 9654 |
Extended real version of pncan 7968. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xnpcan 9655 |
Extended real version of npcan 7971. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xleadd1a 9656 |
Extended real version of leadd1 8192; note that the converse implication is
not true, unlike the real version (for example but
).
(Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xleadd2a 9657 |
Commuted form of xleadd1a 9656. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
|
|
Theorem | xleadd1 9658 |
Weakened version of xleadd1a 9656 under which the reverse implication is
true. (Contributed by Mario Carneiro, 20-Aug-2015.)
|
|
|
Theorem | xltadd1 9659 |
Extended real version of ltadd1 8191. (Contributed by Mario Carneiro,
23-Aug-2015.) (Revised by Jim Kingdon, 16-Apr-2023.)
|
|
|
Theorem | xltadd2 9660 |
Extended real version of ltadd2 8181. (Contributed by Mario Carneiro,
23-Aug-2015.)
|
|
|
Theorem | xaddge0 9661 |
The sum of nonnegative extended reals is nonnegative. (Contributed by
Mario Carneiro, 21-Aug-2015.)
|
|
|
Theorem | xle2add 9662 |
Extended real version of le2add 8206. (Contributed by Mario Carneiro,
23-Aug-2015.)
|
|
|
Theorem | xlt2add 9663 |
Extended real version of lt2add 8207. Note that ltleadd 8208, which has
weaker assumptions, is not true for the extended reals (since
fails). (Contributed by Mario
Carneiro,
23-Aug-2015.)
|
|
|
Theorem | xsubge0 9664 |
Extended real version of subge0 8237. (Contributed by Mario Carneiro,
24-Aug-2015.)
|
|
|
Theorem | xposdif 9665 |
Extended real version of posdif 8217. (Contributed by Mario Carneiro,
24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.)
|
|
|
Theorem | xlesubadd 9666 |
Under certain conditions, the conclusion of lesubadd 8196 is true even in the
extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.)
|
|
|
Theorem | xaddcld 9667 |
The extended real addition operation is closed in extended reals.
(Contributed by Mario Carneiro, 28-May-2016.)
|
|
|
Theorem | xadd4d 9668 |
Rearrangement of 4 terms in a sum for extended addition, analogous to
add4d 7931. (Contributed by Alexander van der Vekens,
21-Dec-2017.)
|
|
|
Theorem | xnn0add4d 9669 |
Rearrangement of 4 terms in a sum for extended addition of extended
nonnegative integers, analogous to xadd4d 9668. (Contributed by AV,
12-Dec-2020.)
|
NN0* NN0* NN0* NN0* |
|
Theorem | xleaddadd 9670 |
Cancelling a factor of two in (expressed as addition rather than
as a factor to avoid extended real multiplication). (Contributed by Jim
Kingdon, 18-Apr-2023.)
|
|
|
4.5.3 Real number intervals
|
|
Syntax | cioo 9671 |
Extend class notation with the set of open intervals of extended reals.
|
|
|
Syntax | cioc 9672 |
Extend class notation with the set of open-below, closed-above intervals
of extended reals.
|
|
|
Syntax | cico 9673 |
Extend class notation with the set of closed-below, open-above intervals
of extended reals.
|
|
|
Syntax | cicc 9674 |
Extend class notation with the set of closed intervals of extended
reals.
|
|
|
Definition | df-ioo 9675* |
Define the set of open intervals of extended reals. (Contributed by NM,
24-Dec-2006.)
|
|
|
Definition | df-ioc 9676* |
Define the set of open-below, closed-above intervals of extended reals.
(Contributed by NM, 24-Dec-2006.)
|
|
|
Definition | df-ico 9677* |
Define the set of closed-below, open-above intervals of extended reals.
(Contributed by NM, 24-Dec-2006.)
|
|
|
Definition | df-icc 9678* |
Define the set of closed intervals of extended reals. (Contributed by
NM, 24-Dec-2006.)
|
|
|
Theorem | ixxval 9679* |
Value of the interval function. (Contributed by Mario Carneiro,
3-Nov-2013.)
|
|
|
Theorem | elixx1 9680* |
Membership in an interval of extended reals. (Contributed by Mario
Carneiro, 3-Nov-2013.)
|
|
|
Theorem | ixxf 9681* |
The set of intervals of extended reals maps to subsets of extended
reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro,
16-Nov-2013.)
|
|
|
Theorem | ixxex 9682* |
The set of intervals of extended reals exists. (Contributed by Mario
Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
|
|
|
Theorem | ixxssxr 9683* |
The set of intervals of extended reals maps to subsets of extended
reals. (Contributed by Mario Carneiro, 4-Jul-2014.)
|
|
|
Theorem | elixx3g 9684* |
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
and .
(Contributed by Mario Carneiro,
3-Nov-2013.)
|
|
|
Theorem | ixxssixx 9685* |
An interval is a subset of its closure. (Contributed by Paul Chapman,
18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
|
|
|
Theorem | ixxdisj 9686* |
Split an interval into disjoint pieces. (Contributed by Mario
Carneiro, 16-Jun-2014.)
|
|
|
Theorem | ixxss1 9687* |
Subset relationship for intervals of extended reals. (Contributed by
Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro,
28-Apr-2015.)
|
|
|
Theorem | ixxss2 9688* |
Subset relationship for intervals of extended reals. (Contributed by
Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro,
28-Apr-2015.)
|
|
|
Theorem | ixxss12 9689* |
Subset relationship for intervals of extended reals. (Contributed by
Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro,
28-Apr-2015.)
|
|
|
Theorem | iooex 9690 |
The set of open intervals of extended reals exists. (Contributed by NM,
6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
|
|
|
Theorem | iooval 9691* |
Value of the open interval function. (Contributed by NM, 24-Dec-2006.)
(Revised by Mario Carneiro, 3-Nov-2013.)
|
|
|
Theorem | iooidg 9692 |
An open interval with identical lower and upper bounds is empty.
(Contributed by Jim Kingdon, 29-Mar-2020.)
|
|
|
Theorem | elioo3g 9693 |
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
and .
(Contributed by NM, 24-Dec-2006.)
(Revised by Mario Carneiro, 3-Nov-2013.)
|
|
|
Theorem | elioo1 9694 |
Membership in an open interval of extended reals. (Contributed by NM,
24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
|
|
|
Theorem | elioore 9695 |
A member of an open interval of reals is a real. (Contributed by NM,
17-Aug-2008.) (Revised by Mario Carneiro, 3-Nov-2013.)
|
|
|
Theorem | lbioog 9696 |
An open interval does not contain its left endpoint. (Contributed by
Jim Kingdon, 30-Mar-2020.)
|
|
|
Theorem | ubioog 9697 |
An open interval does not contain its right endpoint. (Contributed by
Jim Kingdon, 30-Mar-2020.)
|
|
|
Theorem | iooval2 9698* |
Value of the open interval function. (Contributed by NM, 6-Feb-2007.)
(Revised by Mario Carneiro, 3-Nov-2013.)
|
|
|
Theorem | iooss1 9699 |
Subset relationship for open intervals of extended reals. (Contributed
by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)
|
|
|
Theorem | iooss2 9700 |
Subset relationship for open intervals of extended reals. (Contributed
by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
|
|