Theorem List for Intuitionistic Logic Explorer - 6501-6600 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | piord 6501 |
A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
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Theorem | niex 6502 |
The class of positive integers is a set. (Contributed by NM,
15-Aug-1995.)
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Theorem | 0npi 6503 |
The empty set is not a positive integer. (Contributed by NM,
26-Aug-1995.)
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Theorem | elni2 6504 |
Membership in the class of positive integers. (Contributed by NM,
27-Nov-1995.)
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Theorem | 1pi 6505 |
Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
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Theorem | addpiord 6506 |
Positive integer addition in terms of ordinal addition. (Contributed by
NM, 27-Aug-1995.)
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Theorem | mulpiord 6507 |
Positive integer multiplication in terms of ordinal multiplication.
(Contributed by NM, 27-Aug-1995.)
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Theorem | mulidpi 6508 |
1 is an identity element for multiplication on positive integers.
(Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro,
17-Nov-2014.)
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Theorem | ltpiord 6509 |
Positive integer 'less than' in terms of ordinal membership. (Contributed
by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
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Theorem | ltsopi 6510 |
Positive integer 'less than' is a strict ordering. (Contributed by NM,
8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.)
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Theorem | pitric 6511 |
Trichotomy for positive integers. (Contributed by Jim Kingdon,
21-Sep-2019.)
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Theorem | pitri3or 6512 |
Trichotomy for positive integers. (Contributed by Jim Kingdon,
21-Sep-2019.)
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Theorem | ltdcpi 6513 |
Less-than for positive integers is decidable. (Contributed by Jim
Kingdon, 12-Dec-2019.)
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DECID |
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Theorem | ltrelpi 6514 |
Positive integer 'less than' is a relation on positive integers.
(Contributed by NM, 8-Feb-1996.)
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Theorem | dmaddpi 6515 |
Domain of addition on positive integers. (Contributed by NM,
26-Aug-1995.)
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Theorem | dmmulpi 6516 |
Domain of multiplication on positive integers. (Contributed by NM,
26-Aug-1995.)
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Theorem | addclpi 6517 |
Closure of addition of positive integers. (Contributed by NM,
18-Oct-1995.)
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Theorem | mulclpi 6518 |
Closure of multiplication of positive integers. (Contributed by NM,
18-Oct-1995.)
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Theorem | addcompig 6519 |
Addition of positive integers is commutative. (Contributed by Jim
Kingdon, 26-Aug-2019.)
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Theorem | addasspig 6520 |
Addition of positive integers is associative. (Contributed by Jim
Kingdon, 26-Aug-2019.)
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Theorem | mulcompig 6521 |
Multiplication of positive integers is commutative. (Contributed by Jim
Kingdon, 26-Aug-2019.)
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Theorem | mulasspig 6522 |
Multiplication of positive integers is associative. (Contributed by Jim
Kingdon, 26-Aug-2019.)
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Theorem | distrpig 6523 |
Multiplication of positive integers is distributive. (Contributed by Jim
Kingdon, 26-Aug-2019.)
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Theorem | addcanpig 6524 |
Addition cancellation law for positive integers. (Contributed by Jim
Kingdon, 27-Aug-2019.)
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Theorem | mulcanpig 6525 |
Multiplication cancellation law for positive integers. (Contributed by
Jim Kingdon, 29-Aug-2019.)
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Theorem | addnidpig 6526 |
There is no identity element for addition on positive integers.
(Contributed by NM, 28-Nov-1995.)
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Theorem | ltexpi 6527* |
Ordering on positive integers in terms of existence of sum.
(Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro,
14-Jun-2013.)
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Theorem | ltapig 6528 |
Ordering property of addition for positive integers. (Contributed by Jim
Kingdon, 31-Aug-2019.)
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Theorem | ltmpig 6529 |
Ordering property of multiplication for positive integers. (Contributed
by Jim Kingdon, 31-Aug-2019.)
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Theorem | 1lt2pi 6530 |
One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
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Theorem | nlt1pig 6531 |
No positive integer is less than one. (Contributed by Jim Kingdon,
31-Aug-2019.)
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Theorem | indpi 6532* |
Principle of Finite Induction on positive integers. (Contributed by NM,
23-Mar-1996.)
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Theorem | nnppipi 6533 |
A natural number plus a positive integer is a positive integer.
(Contributed by Jim Kingdon, 10-Nov-2019.)
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Definition | df-plpq 6534* |
Define pre-addition on positive fractions. This is a "temporary" set
used in the construction of complex numbers, and is intended to be used
only by the construction. This "pre-addition" operation works
directly
with ordered pairs of integers. The actual positive fraction addition
(df-plqqs 6539) works with the equivalence classes of these
ordered pairs determined by the equivalence relation
(df-enq 6537). (Analogous remarks apply to the other
"pre-" operations
in the complex number construction that follows.) From Proposition
9-2.3 of [Gleason] p. 117. (Contributed
by NM, 28-Aug-1995.)
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Definition | df-mpq 6535* |
Define pre-multiplication on positive fractions. This is a
"temporary"
set used in the construction of complex numbers, and is intended to be
used only by the construction. From Proposition 9-2.4 of [Gleason]
p. 119. (Contributed by NM, 28-Aug-1995.)
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Definition | df-ltpq 6536* |
Define pre-ordering relation on positive fractions. This is a
"temporary" set used in the construction of complex numbers,
and is
intended to be used only by the construction. Similar to Definition 5
of [Suppes] p. 162. (Contributed by NM,
28-Aug-1995.)
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Definition | df-enq 6537* |
Define equivalence relation for positive fractions. This is a
"temporary" set used in the construction of complex numbers,
and is
intended to be used only by the construction. From Proposition 9-2.1 of
[Gleason] p. 117. (Contributed by NM,
27-Aug-1995.)
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Definition | df-nqqs 6538 |
Define class of positive fractions. This is a "temporary" set used
in
the construction of complex numbers, and is intended to be used only by
the construction. From Proposition 9-2.2 of [Gleason] p. 117.
(Contributed by NM, 16-Aug-1995.)
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Definition | df-plqqs 6539* |
Define addition on positive fractions. This is a "temporary" set
used
in the construction of complex numbers, and is intended to be used only
by the construction. From Proposition 9-2.3 of [Gleason] p. 117.
(Contributed by NM, 24-Aug-1995.)
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Definition | df-mqqs 6540* |
Define multiplication on positive fractions. This is a "temporary"
set
used in the construction of complex numbers, and is intended to be used
only by the construction. From Proposition 9-2.4 of [Gleason] p. 119.
(Contributed by NM, 24-Aug-1995.)
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Definition | df-1nqqs 6541 |
Define positive fraction constant 1. This is a "temporary" set used
in
the construction of complex numbers, and is intended to be used only by
the construction. From Proposition 9-2.2 of [Gleason] p. 117.
(Contributed by NM, 29-Oct-1995.)
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Definition | df-rq 6542* |
Define reciprocal on positive fractions. It means the same thing as one
divided by the argument (although we don't define full division since we
will never need it). This is a "temporary" set used in the
construction
of complex numbers, and is intended to be used only by the construction.
From Proposition 9-2.5 of [Gleason] p.
119, who uses an asterisk to
denote this unary operation. (Contributed by Jim Kingdon,
20-Sep-2019.)
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Definition | df-ltnqqs 6543* |
Define ordering relation on positive fractions. This is a
"temporary"
set used in the construction of complex numbers, and is intended to be
used only by the construction. Similar to Definition 5 of [Suppes]
p. 162. (Contributed by NM, 13-Feb-1996.)
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Theorem | dfplpq2 6544* |
Alternate definition of pre-addition on positive fractions.
(Contributed by Jim Kingdon, 12-Sep-2019.)
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Theorem | dfmpq2 6545* |
Alternate definition of pre-multiplication on positive fractions.
(Contributed by Jim Kingdon, 13-Sep-2019.)
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Theorem | enqbreq 6546 |
Equivalence relation for positive fractions in terms of positive
integers. (Contributed by NM, 27-Aug-1995.)
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Theorem | enqbreq2 6547 |
Equivalence relation for positive fractions in terms of positive integers.
(Contributed by Mario Carneiro, 8-May-2013.)
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Theorem | enqer 6548 |
The equivalence relation for positive fractions is an equivalence
relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM,
27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
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Theorem | enqeceq 6549 |
Equivalence class equality of positive fractions in terms of positive
integers. (Contributed by NM, 29-Nov-1995.)
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Theorem | enqex 6550 |
The equivalence relation for positive fractions exists. (Contributed by
NM, 3-Sep-1995.)
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Theorem | enqdc 6551 |
The equivalence relation for positive fractions is decidable.
(Contributed by Jim Kingdon, 7-Sep-2019.)
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DECID
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Theorem | enqdc1 6552 |
The equivalence relation for positive fractions is decidable.
(Contributed by Jim Kingdon, 7-Sep-2019.)
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DECID |
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Theorem | nqex 6553 |
The class of positive fractions exists. (Contributed by NM,
16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
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Theorem | 0nnq 6554 |
The empty set is not a positive fraction. (Contributed by NM,
24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
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Theorem | ltrelnq 6555 |
Positive fraction 'less than' is a relation on positive fractions.
(Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro,
27-Apr-2013.)
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Theorem | 1nq 6556 |
The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
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Theorem | addcmpblnq 6557 |
Lemma showing compatibility of addition. (Contributed by NM,
27-Aug-1995.)
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Theorem | mulcmpblnq 6558 |
Lemma showing compatibility of multiplication. (Contributed by NM,
27-Aug-1995.)
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Theorem | addpipqqslem 6559 |
Lemma for addpipqqs 6560. (Contributed by Jim Kingdon, 11-Sep-2019.)
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Theorem | addpipqqs 6560 |
Addition of positive fractions in terms of positive integers.
(Contributed by NM, 28-Aug-1995.)
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Theorem | mulpipq2 6561 |
Multiplication of positive fractions in terms of positive integers.
(Contributed by Mario Carneiro, 8-May-2013.)
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Theorem | mulpipq 6562 |
Multiplication of positive fractions in terms of positive integers.
(Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro,
8-May-2013.)
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Theorem | mulpipqqs 6563 |
Multiplication of positive fractions in terms of positive integers.
(Contributed by NM, 28-Aug-1995.)
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Theorem | ordpipqqs 6564 |
Ordering of positive fractions in terms of positive integers.
(Contributed by Jim Kingdon, 14-Sep-2019.)
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Theorem | addclnq 6565 |
Closure of addition on positive fractions. (Contributed by NM,
29-Aug-1995.)
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Theorem | mulclnq 6566 |
Closure of multiplication on positive fractions. (Contributed by NM,
29-Aug-1995.)
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Theorem | dmaddpqlem 6567* |
Decomposition of a positive fraction into numerator and denominator.
Lemma for dmaddpq 6569. (Contributed by Jim Kingdon, 15-Sep-2019.)
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Theorem | nqpi 6568* |
Decomposition of a positive fraction into numerator and denominator.
Similar to dmaddpqlem 6567 but also shows that the numerator and
denominator are positive integers. (Contributed by Jim Kingdon,
20-Sep-2019.)
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Theorem | dmaddpq 6569 |
Domain of addition on positive fractions. (Contributed by NM,
24-Aug-1995.)
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Theorem | dmmulpq 6570 |
Domain of multiplication on positive fractions. (Contributed by NM,
24-Aug-1995.)
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Theorem | addcomnqg 6571 |
Addition of positive fractions is commutative. (Contributed by Jim
Kingdon, 15-Sep-2019.)
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Theorem | addassnqg 6572 |
Addition of positive fractions is associative. (Contributed by Jim
Kingdon, 16-Sep-2019.)
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Theorem | mulcomnqg 6573 |
Multiplication of positive fractions is commutative. (Contributed by
Jim Kingdon, 17-Sep-2019.)
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Theorem | mulassnqg 6574 |
Multiplication of positive fractions is associative. (Contributed by
Jim Kingdon, 17-Sep-2019.)
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Theorem | mulcanenq 6575 |
Lemma for distributive law: cancellation of common factor. (Contributed
by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.)
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Theorem | mulcanenqec 6576 |
Lemma for distributive law: cancellation of common factor. (Contributed
by Jim Kingdon, 17-Sep-2019.)
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Theorem | distrnqg 6577 |
Multiplication of positive fractions is distributive. (Contributed by
Jim Kingdon, 17-Sep-2019.)
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Theorem | 1qec 6578 |
The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
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Theorem | mulidnq 6579 |
Multiplication identity element for positive fractions. (Contributed by
NM, 3-Mar-1996.)
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Theorem | recexnq 6580* |
Existence of positive fraction reciprocal. (Contributed by Jim Kingdon,
20-Sep-2019.)
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Theorem | recmulnqg 6581 |
Relationship between reciprocal and multiplication on positive
fractions. (Contributed by Jim Kingdon, 19-Sep-2019.)
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Theorem | recclnq 6582 |
Closure law for positive fraction reciprocal. (Contributed by NM,
6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
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Theorem | recidnq 6583 |
A positive fraction times its reciprocal is 1. (Contributed by NM,
6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
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Theorem | recrecnq 6584 |
Reciprocal of reciprocal of positive fraction. (Contributed by NM,
26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.)
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Theorem | rec1nq 6585 |
Reciprocal of positive fraction one. (Contributed by Jim Kingdon,
29-Dec-2019.)
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Theorem | nqtri3or 6586 |
Trichotomy for positive fractions. (Contributed by Jim Kingdon,
21-Sep-2019.)
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Theorem | ltdcnq 6587 |
Less-than for positive fractions is decidable. (Contributed by Jim
Kingdon, 12-Dec-2019.)
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DECID |
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Theorem | ltsonq 6588 |
'Less than' is a strict ordering on positive fractions. (Contributed by
NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.)
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Theorem | nqtric 6589 |
Trichotomy for positive fractions. (Contributed by Jim Kingdon,
21-Sep-2019.)
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Theorem | ltanqg 6590 |
Ordering property of addition for positive fractions. Proposition
9-2.6(ii) of [Gleason] p. 120.
(Contributed by Jim Kingdon,
22-Sep-2019.)
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Theorem | ltmnqg 6591 |
Ordering property of multiplication for positive fractions. Proposition
9-2.6(iii) of [Gleason] p. 120.
(Contributed by Jim Kingdon,
22-Sep-2019.)
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Theorem | ltanqi 6592 |
Ordering property of addition for positive fractions. One direction of
ltanqg 6590. (Contributed by Jim Kingdon, 9-Dec-2019.)
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Theorem | ltmnqi 6593 |
Ordering property of multiplication for positive fractions. One direction
of ltmnqg 6591. (Contributed by Jim Kingdon, 9-Dec-2019.)
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Theorem | lt2addnq 6594 |
Ordering property of addition for positive fractions. (Contributed by Jim
Kingdon, 7-Dec-2019.)
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Theorem | lt2mulnq 6595 |
Ordering property of multiplication for positive fractions. (Contributed
by Jim Kingdon, 18-Jul-2021.)
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Theorem | 1lt2nq 6596 |
One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
(Revised by Mario Carneiro, 10-May-2013.)
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Theorem | ltaddnq 6597 |
The sum of two fractions is greater than one of them. (Contributed by
NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
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Theorem | ltexnqq 6598* |
Ordering on positive fractions in terms of existence of sum. Definition
in Proposition 9-2.6 of [Gleason] p.
119. (Contributed by Jim Kingdon,
23-Sep-2019.)
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Theorem | ltexnqi 6599* |
Ordering on positive fractions in terms of existence of sum.
(Contributed by Jim Kingdon, 30-Apr-2020.)
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Theorem | halfnqq 6600* |
One-half of any positive fraction is a fraction. (Contributed by Jim
Kingdon, 23-Sep-2019.)
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