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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | npex 7501 | The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) |
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Theorem | elinp 7502* | Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.) |
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Theorem | prop 7503 | A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.) |
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Theorem | elnp1st2nd 7504* |
Membership in positive reals, using ![]() ![]() |
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Theorem | prml 7505* | A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
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Theorem | prmu 7506* | A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
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Theorem | prssnql 7507 | The lower cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.) |
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Theorem | prssnqu 7508 | The upper cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.) |
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Theorem | elprnql 7509 | An element of a positive real's lower cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.) |
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Theorem | elprnqu 7510 | An element of a positive real's upper cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.) |
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Theorem | 0npr 7511 | The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) |
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Theorem | prcdnql 7512 | A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.) |
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Theorem | prcunqu 7513 | An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.) |
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Theorem | prubl 7514 | A positive fraction not in a lower cut is an upper bound. (Contributed by Jim Kingdon, 29-Sep-2019.) |
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Theorem | prltlu 7515 | An element of a lower cut is less than an element of the corresponding upper cut. (Contributed by Jim Kingdon, 15-Oct-2019.) |
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Theorem | prnmaxl 7516* | A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.) |
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Theorem | prnminu 7517* | An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.) |
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Theorem | prnmaddl 7518* | A lower cut has no largest member. Addition version. (Contributed by Jim Kingdon, 29-Sep-2019.) |
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Theorem | prloc 7519 | A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.) |
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Theorem | prdisj 7520 | A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.) |
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Theorem | prarloclemlt 7521 | Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7531. (Contributed by Jim Kingdon, 10-Nov-2019.) |
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Theorem | prarloclemlo 7522* | Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 7531. (Contributed by Jim Kingdon, 10-Nov-2019.) |
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Theorem | prarloclemup 7523 | Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 7531. (Contributed by Jim Kingdon, 10-Nov-2019.) |
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Theorem | prarloclem3step 7524* | Induction step for prarloclem3 7525. (Contributed by Jim Kingdon, 9-Nov-2019.) |
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Theorem | prarloclem3 7525* | Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7531. (Contributed by Jim Kingdon, 27-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | prarloclem4 7526* | A slight rearrangement of prarloclem3 7525. Lemma for prarloc 7531. (Contributed by Jim Kingdon, 4-Nov-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | prarloclemn 7527* | Subtracting two from a positive integer. Lemma for prarloc 7531. (Contributed by Jim Kingdon, 5-Nov-2019.) |
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Theorem | prarloclem5 7528* |
A substitution of zero for ![]() ![]() ![]() |
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Theorem | prarloclem 7529* |
A special case of Lemma 6.16 from [BauerTaylor], p. 32. Given evenly
spaced rational numbers from ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | prarloclemcalc 7530 | Some calculations for prarloc 7531. (Contributed by Jim Kingdon, 26-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | prarloc 7531* |
A Dedekind cut is arithmetically located. Part of Proposition 11.15 of
[BauerTaylor], p. 52, slightly
modified. It states that given a
tolerance ![]() Usually, proofs will be shorter if they use prarloc2 7532 instead. (Contributed by Jim Kingdon, 22-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | prarloc2 7532* |
A Dedekind cut is arithmetically located. This is a variation of
prarloc 7531 which only constructs one (named) point and
is therefore often
easier to work with. It states that given a tolerance ![]() |
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Theorem | ltrelpr 7533 | Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) |
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Theorem | ltdfpr 7534* | More convenient form of df-iltp 7498. (Contributed by Jim Kingdon, 15-Dec-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | genpdflem 7535* | Simplification of upper or lower cut expression. Lemma for genpdf 7536. (Contributed by Jim Kingdon, 30-Sep-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | genpdf 7536* | Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | genipv 7537* | Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.) |
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Theorem | genplt2i 7538* |
Operating on both sides of two inequalities, when the operation is
consistent with ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | genpelxp 7539* | Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | genpelvl 7540* | Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.) |
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Theorem | genpelvu 7541* | Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | genpprecll 7542* | Pre-closure law for general operation on lower cuts. (Contributed by Jim Kingdon, 2-Oct-2019.) |
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Theorem | genppreclu 7543* | Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-2019.) |
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Theorem | genipdm 7544* | Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.) |
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Theorem | genpml 7545* | The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.) |
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Theorem | genpmu 7546* | The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.) |
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Theorem | genpcdl 7547* | Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.) |
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Theorem | genpcuu 7548* | Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.) |
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Theorem | genprndl 7549* | The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.) |
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Theorem | genprndu 7550* | The upper cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.) |
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Theorem | genpdisj 7551* | The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (Contributed by Jim Kingdon, 15-Oct-2019.) |
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Theorem | genpassl 7552* | Associativity of lower cuts. Lemma for genpassg 7554. (Contributed by Jim Kingdon, 11-Dec-2019.) |
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Theorem | genpassu 7553* | Associativity of upper cuts. Lemma for genpassg 7554. (Contributed by Jim Kingdon, 11-Dec-2019.) |
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Theorem | genpassg 7554* | Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.) |
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Theorem | addnqprllem 7555 | Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.) |
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Theorem | addnqprulem 7556 | Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.) |
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Theorem | addnqprl 7557 | Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) |
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Theorem | addnqpru 7558 | Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) |
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Theorem | addlocprlemlt 7559 |
Lemma for addlocpr 7564. The ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | addlocprlemeqgt 7560 |
Lemma for addlocpr 7564. This is a step used in both the
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Theorem | addlocprlemeq 7561 |
Lemma for addlocpr 7564. The ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | addlocprlemgt 7562 |
Lemma for addlocpr 7564. The ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | addlocprlem 7563 | Lemma for addlocpr 7564. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | addlocpr 7564* |
Locatedness of addition on positive reals. Lemma 11.16 in
[BauerTaylor], p. 53. The proof in
BauerTaylor relies on signed
rationals, so we replace it with another proof which applies prarloc 7531
to both ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | addclpr 7565 | Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. Combination of Lemma 11.13 and Lemma 11.16 in [BauerTaylor], p. 53. (Contributed by NM, 13-Mar-1996.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | plpvlu 7566* | Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | mpvlu 7567* | Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dmplp 7568 | Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) |
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Theorem | dmmp 7569 | Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) |
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Theorem | nqprm 7570* | A cut produced from a rational is inhabited. Lemma for nqprlu 7575. (Contributed by Jim Kingdon, 8-Dec-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | nqprrnd 7571* | A cut produced from a rational is rounded. Lemma for nqprlu 7575. (Contributed by Jim Kingdon, 8-Dec-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | nqprdisj 7572* | A cut produced from a rational is disjoint. Lemma for nqprlu 7575. (Contributed by Jim Kingdon, 8-Dec-2019.) |
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Theorem | nqprloc 7573* | A cut produced from a rational is located. Lemma for nqprlu 7575. (Contributed by Jim Kingdon, 8-Dec-2019.) |
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Theorem | nqprxx 7574* | The canonical embedding of the rationals into the reals, expressed with the same variable for the lower and upper cuts. (Contributed by Jim Kingdon, 8-Dec-2019.) |
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Theorem | nqprlu 7575* | The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.) |
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Theorem | recnnpr 7576* | The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.) |
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Theorem | ltnqex 7577 | The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.) |
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Theorem | gtnqex 7578 | The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.) |
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Theorem | nqprl 7579* |
Comparing a fraction to a real can be done by whether it is an element
of the lower cut, or by ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | nqpru 7580* |
Comparing a fraction to a real can be done by whether it is an element
of the upper cut, or by ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | nnprlu 7581* | The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.) |
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Theorem | 1pr 7582 | The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) |
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Theorem | 1prl 7583 | The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) |
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Theorem | 1pru 7584 | The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) |
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Theorem | addnqprlemrl 7585* | Lemma for addnqpr 7589. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
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Theorem | addnqprlemru 7586* | Lemma for addnqpr 7589. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
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Theorem | addnqprlemfl 7587* | Lemma for addnqpr 7589. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
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Theorem | addnqprlemfu 7588* | Lemma for addnqpr 7589. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
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Theorem | addnqpr 7589* | Addition of fractions embedded into positive reals. One can either add the fractions as fractions, or embed them into positive reals and add them as positive reals, and get the same result. (Contributed by Jim Kingdon, 19-Aug-2020.) |
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Theorem | addnqpr1 7590* | Addition of one to a fraction embedded into a positive real. One can either add the fraction one to the fraction, or the positive real one to the positive real, and get the same result. Special case of addnqpr 7589. (Contributed by Jim Kingdon, 26-Apr-2020.) |
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Theorem | appdivnq 7591* |
Approximate division for positive rationals. Proposition 12.7 of
[BauerTaylor], p. 55 (a special case
where ![]() ![]() ![]() |
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Theorem | appdiv0nq 7592* |
Approximate division for positive rationals. This can be thought of as
a variation of appdivnq 7591 in which ![]() |
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Theorem | prmuloclemcalc 7593 | Calculations for prmuloc 7594. (Contributed by Jim Kingdon, 9-Dec-2019.) |
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Theorem | prmuloc 7594* | Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | prmuloc2 7595* |
Positive reals are multiplicatively located. This is a variation of
prmuloc 7594 which only constructs one (named) point and
is therefore often
easier to work with. It states that given a ratio ![]() |
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Theorem | mulnqprl 7596 | Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | mulnqpru 7597 | Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.) |
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Theorem | mullocprlem 7598 | Calculations for mullocpr 7599. (Contributed by Jim Kingdon, 10-Dec-2019.) |
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Theorem | mullocpr 7599* |
Locatedness of multiplication on positive reals. Lemma 12.9 in
[BauerTaylor], p. 56 (but where both
![]() ![]() ![]() |
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Theorem | mulclpr 7600 | Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.) |
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