Theorem List for Intuitionistic Logic Explorer - 9401-9500 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Definition | df-xadd 9401* |
Define addition over extended real numbers. (Contributed by Mario
Carneiro, 20-Aug-2015.)
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Definition | df-xmul 9402* |
Define multiplication over extended real numbers. (Contributed by Mario
Carneiro, 20-Aug-2015.)
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Theorem | ltxr 9403 |
The 'less than' binary relation on the set of extended reals.
Definition 12-3.1 of [Gleason] p. 173.
(Contributed by NM,
14-Oct-2005.)
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Theorem | elxr 9404 |
Membership in the set of extended reals. (Contributed by NM,
14-Oct-2005.)
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Theorem | xrnemnf 9405 |
An extended real other than minus infinity is real or positive infinite.
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xrnepnf 9406 |
An extended real other than plus infinity is real or negative infinite.
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xrltnr 9407 |
The extended real 'less than' is irreflexive. (Contributed by NM,
14-Oct-2005.)
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Theorem | ltpnf 9408 |
Any (finite) real is less than plus infinity. (Contributed by NM,
14-Oct-2005.)
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Theorem | 0ltpnf 9409 |
Zero is less than plus infinity (common case). (Contributed by David A.
Wheeler, 8-Dec-2018.)
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Theorem | mnflt 9410 |
Minus infinity is less than any (finite) real. (Contributed by NM,
14-Oct-2005.)
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Theorem | mnflt0 9411 |
Minus infinity is less than 0 (common case). (Contributed by David A.
Wheeler, 8-Dec-2018.)
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Theorem | mnfltpnf 9412 |
Minus infinity is less than plus infinity. (Contributed by NM,
14-Oct-2005.)
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Theorem | mnfltxr 9413 |
Minus infinity is less than an extended real that is either real or plus
infinity. (Contributed by NM, 2-Feb-2006.)
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Theorem | pnfnlt 9414 |
No extended real is greater than plus infinity. (Contributed by NM,
15-Oct-2005.)
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Theorem | nltmnf 9415 |
No extended real is less than minus infinity. (Contributed by NM,
15-Oct-2005.)
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Theorem | pnfge 9416 |
Plus infinity is an upper bound for extended reals. (Contributed by NM,
30-Jan-2006.)
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Theorem | 0lepnf 9417 |
0 less than or equal to positive infinity. (Contributed by David A.
Wheeler, 8-Dec-2018.)
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Theorem | nn0pnfge0 9418 |
If a number is a nonnegative integer or positive infinity, it is greater
than or equal to 0. (Contributed by Alexander van der Vekens,
6-Jan-2018.)
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Theorem | mnfle 9419 |
Minus infinity is less than or equal to any extended real. (Contributed
by NM, 19-Jan-2006.)
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Theorem | xrltnsym 9420 |
Ordering on the extended reals is not symmetric. (Contributed by NM,
15-Oct-2005.)
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Theorem | xrltnsym2 9421 |
'Less than' is antisymmetric and irreflexive for extended reals.
(Contributed by NM, 6-Feb-2007.)
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Theorem | xrlttr 9422 |
Ordering on the extended reals is transitive. (Contributed by NM,
15-Oct-2005.)
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Theorem | xrltso 9423 |
'Less than' is a weakly linear ordering on the extended reals.
(Contributed by NM, 15-Oct-2005.)
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Theorem | xrlttri3 9424 |
Extended real version of lttri3 7715. (Contributed by NM, 9-Feb-2006.)
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Theorem | xrltle 9425 |
'Less than' implies 'less than or equal' for extended reals. (Contributed
by NM, 19-Jan-2006.)
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Theorem | xrltled 9426 |
'Less than' implies 'less than or equal to' for extended reals.
Deduction form of xrltle 9425. (Contributed by Glauco Siliprandi,
11-Dec-2019.)
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Theorem | xrleid 9427 |
'Less than or equal to' is reflexive for extended reals. (Contributed by
NM, 7-Feb-2007.)
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Theorem | xrleidd 9428 |
'Less than or equal to' is reflexive for extended reals. Deduction form
of xrleid 9427. (Contributed by Glauco Siliprandi,
26-Jun-2021.)
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Theorem | xrletri3 9429 |
Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
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Theorem | xrlelttr 9430 |
Transitive law for ordering on extended reals. (Contributed by NM,
19-Jan-2006.)
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Theorem | xrltletr 9431 |
Transitive law for ordering on extended reals. (Contributed by NM,
19-Jan-2006.)
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Theorem | xrletr 9432 |
Transitive law for ordering on extended reals. (Contributed by NM,
9-Feb-2006.)
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Theorem | xrlttrd 9433 |
Transitive law for ordering on extended reals. (Contributed by Mario
Carneiro, 23-Aug-2015.)
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Theorem | xrlelttrd 9434 |
Transitive law for ordering on extended reals. (Contributed by Mario
Carneiro, 23-Aug-2015.)
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Theorem | xrltletrd 9435 |
Transitive law for ordering on extended reals. (Contributed by Mario
Carneiro, 23-Aug-2015.)
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Theorem | xrletrd 9436 |
Transitive law for ordering on extended reals. (Contributed by Mario
Carneiro, 23-Aug-2015.)
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Theorem | xrltne 9437 |
'Less than' implies not equal for extended reals. (Contributed by NM,
20-Jan-2006.)
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Theorem | nltpnft 9438 |
An extended real is not less than plus infinity iff they are equal.
(Contributed by NM, 30-Jan-2006.)
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Theorem | npnflt 9439 |
An extended real is less than plus infinity iff they are not equal.
(Contributed by Jim Kingdon, 17-Apr-2023.)
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Theorem | xgepnf 9440 |
An extended real which is greater than plus infinity is plus infinity.
(Contributed by Thierry Arnoux, 18-Dec-2016.)
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Theorem | ngtmnft 9441 |
An extended real is not greater than minus infinity iff they are equal.
(Contributed by NM, 2-Feb-2006.)
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Theorem | nmnfgt 9442 |
An extended real is greater than minus infinite iff they are not equal.
(Contributed by Jim Kingdon, 17-Apr-2023.)
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Theorem | xrrebnd 9443 |
An extended real is real iff it is strictly bounded by infinities.
(Contributed by NM, 2-Feb-2006.)
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Theorem | xrre 9444 |
A way of proving that an extended real is real. (Contributed by NM,
9-Mar-2006.)
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Theorem | xrre2 9445 |
An extended real between two others is real. (Contributed by NM,
6-Feb-2007.)
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Theorem | xrre3 9446 |
A way of proving that an extended real is real. (Contributed by FL,
29-May-2014.)
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Theorem | ge0gtmnf 9447 |
A nonnegative extended real is greater than negative infinity.
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | ge0nemnf 9448 |
A nonnegative extended real is greater than negative infinity.
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xrrege0 9449 |
A nonnegative extended real that is less than a real bound is real.
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | z2ge 9450* |
There exists an integer greater than or equal to any two others.
(Contributed by NM, 28-Aug-2005.)
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Theorem | xnegeq 9451 |
Equality of two extended numbers with  in front of them.
(Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro,
20-Aug-2015.)
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Theorem | xnegpnf 9452 |
Minus . Remark
of [BourbakiTop1] p. IV.15. (Contributed
by FL,
26-Dec-2011.)
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Theorem | xnegmnf 9453 |
Minus . Remark
of [BourbakiTop1] p. IV.15. (Contributed
by FL,
26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
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Theorem | rexneg 9454 |
Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by
FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
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Theorem | xneg0 9455 |
The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xnegcl 9456 |
Closure of extended real negative. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xnegneg 9457 |
Extended real version of negneg 7883. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xneg11 9458 |
Extended real version of neg11 7884. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xltnegi 9459 |
Forward direction of xltneg 9460. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xltneg 9460 |
Extended real version of ltneg 8091. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xleneg 9461 |
Extended real version of leneg 8094. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xlt0neg1 9462 |
Extended real version of lt0neg1 8097. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xlt0neg2 9463 |
Extended real version of lt0neg2 8098. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xle0neg1 9464 |
Extended real version of le0neg1 8099. (Contributed by Mario Carneiro,
9-Sep-2015.)
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Theorem | xle0neg2 9465 |
Extended real version of le0neg2 8100. (Contributed by Mario Carneiro,
9-Sep-2015.)
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Theorem | xrpnfdc 9466 |
An extended real is or is not plus infinity. (Contributed by Jim Kingdon,
13-Apr-2023.)
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Theorem | xrmnfdc 9467 |
An extended real is or is not minus infinity. (Contributed by Jim
Kingdon, 13-Apr-2023.)
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 DECID   |
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Theorem | xaddf 9468 |
The extended real addition operation is closed in extended reals.
(Contributed by Mario Carneiro, 21-Aug-2015.)
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Theorem | xaddval 9469 |
Value of the extended real addition operation. (Contributed by Mario
Carneiro, 20-Aug-2015.)
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Theorem | xaddpnf1 9470 |
Addition of positive infinity on the right. (Contributed by Mario
Carneiro, 20-Aug-2015.)
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Theorem | xaddpnf2 9471 |
Addition of positive infinity on the left. (Contributed by Mario
Carneiro, 20-Aug-2015.)
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Theorem | xaddmnf1 9472 |
Addition of negative infinity on the right. (Contributed by Mario
Carneiro, 20-Aug-2015.)
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Theorem | xaddmnf2 9473 |
Addition of negative infinity on the left. (Contributed by Mario
Carneiro, 20-Aug-2015.)
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Theorem | pnfaddmnf 9474 |
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.)
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Theorem | mnfaddpnf 9475 |
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.)
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Theorem | rexadd 9476 |
The extended real addition operation when both arguments are real.
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | rexsub 9477 |
Extended real subtraction when both arguments are real. (Contributed by
Mario Carneiro, 23-Aug-2015.)
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Theorem | rexaddd 9478 |
The extended real addition operation when both arguments are real.
Deduction version of rexadd 9476. (Contributed by Glauco Siliprandi,
24-Dec-2020.)
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Theorem | xnegcld 9479 |
Closure of extended real negative. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | xrex 9480 |
The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
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Theorem | xaddnemnf 9481 |
Closure of extended real addition in the subset
 .
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xaddnepnf 9482 |
Closure of extended real addition in the subset
 .
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xnegid 9483 |
Extended real version of negid 7880. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xaddcl 9484 |
The extended real addition operation is closed in extended reals.
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xaddcom 9485 |
The extended real addition operation is commutative. (Contributed by NM,
26-Dec-2011.)
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Theorem | xaddid1 9486 |
Extended real version of addid1 7771. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xaddid2 9487 |
Extended real version of addid2 7772. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xaddid1d 9488 |
is a right identity for
extended real addition. (Contributed by
Glauco Siliprandi, 17-Aug-2020.)
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Theorem | xnn0lenn0nn0 9489 |
An extended nonnegative integer which is less than or equal to a
nonnegative integer is a nonnegative integer. (Contributed by AV,
24-Nov-2021.)
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  NN0*    |
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Theorem | xnn0le2is012 9490 |
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.)
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  NN0*
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Theorem | xnn0xadd0 9491 |
The sum of two extended nonnegative integers is iff each of the two
extended nonnegative integers is . (Contributed by AV,
14-Dec-2020.)
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  NN0* NN0*            |
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Theorem | xnegdi 9492 |
Extended real version of negdi 7890. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xaddass 9493 |
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 9494, where is not present. (Contributed by Mario
Carneiro, 20-Aug-2015.)
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Theorem | xaddass2 9494 |
Associativity of extended real addition. See xaddass 9493 for notes on the
hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xpncan 9495 |
Extended real version of pncan 7839. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xnpcan 9496 |
Extended real version of npcan 7842. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xleadd1a 9497 |
Extended real version of leadd1 8059; note that the converse implication is
not true, unlike the real version (for example but
  
     ).
(Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xleadd2a 9498 |
Commuted form of xleadd1a 9497. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | xleadd1 9499 |
Weakened version of xleadd1a 9497 under which the reverse implication is
true. (Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xltadd1 9500 |
Extended real version of ltadd1 8058. (Contributed by Mario Carneiro,
23-Aug-2015.) (Revised by Jim Kingdon, 16-Apr-2023.)
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