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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elprnql 7301 | An element of a positive real's lower cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.) |
Theorem | elprnqu 7302 | An element of a positive real's upper cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.) |
Theorem | 0npr 7303 | The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) |
Theorem | prcdnql 7304 | A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.) |
Theorem | prcunqu 7305 | An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.) |
Theorem | prubl 7306 | A positive fraction not in a lower cut is an upper bound. (Contributed by Jim Kingdon, 29-Sep-2019.) |
Theorem | prltlu 7307 | An element of a lower cut is less than an element of the corresponding upper cut. (Contributed by Jim Kingdon, 15-Oct-2019.) |
Theorem | prnmaxl 7308* | A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.) |
Theorem | prnminu 7309* | An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.) |
Theorem | prnmaddl 7310* | A lower cut has no largest member. Addition version. (Contributed by Jim Kingdon, 29-Sep-2019.) |
Theorem | prloc 7311 | A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.) |
Theorem | prdisj 7312 | A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.) |
Theorem | prarloclemlt 7313 | Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7323. (Contributed by Jim Kingdon, 10-Nov-2019.) |
Theorem | prarloclemlo 7314* | Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 7323. (Contributed by Jim Kingdon, 10-Nov-2019.) |
+_{Q0} ~_{Q0} ·_{Q0} +_{Q0} ~_{Q0} ·_{Q0} | ||
Theorem | prarloclemup 7315 | Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 7323. (Contributed by Jim Kingdon, 10-Nov-2019.) |
+_{Q0} ~_{Q0} ·_{Q0} +_{Q0} ~_{Q0} ·_{Q0} | ||
Theorem | prarloclem3step 7316* | Induction step for prarloclem3 7317. (Contributed by Jim Kingdon, 9-Nov-2019.) |
+_{Q0} ~_{Q0} ·_{Q0} +_{Q0} ~_{Q0} ·_{Q0} | ||
Theorem | prarloclem3 7317* | Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7323. (Contributed by Jim Kingdon, 27-Oct-2019.) |
+_{Q0} ~_{Q0} ·_{Q0} +_{Q0} ~_{Q0} ·_{Q0} | ||
Theorem | prarloclem4 7318* | A slight rearrangement of prarloclem3 7317. Lemma for prarloc 7323. (Contributed by Jim Kingdon, 4-Nov-2019.) |
+_{Q0} ~_{Q0} ·_{Q0} +_{Q0} ~_{Q0} ·_{Q0} | ||
Theorem | prarloclemn 7319* | Subtracting two from a positive integer. Lemma for prarloc 7323. (Contributed by Jim Kingdon, 5-Nov-2019.) |
Theorem | prarloclem5 7320* | A substitution of zero for and minus two for . Lemma for prarloc 7323. (Contributed by Jim Kingdon, 4-Nov-2019.) |
+_{Q0} ~_{Q0} ·_{Q0} | ||
Theorem | prarloclem 7321* | A special case of Lemma 6.16 from [BauerTaylor], p. 32. Given evenly spaced rational numbers from to (which are in the lower and upper cuts, respectively, of a real number), there are a pair of numbers, two positions apart in the even spacing, which straddle the cut. (Contributed by Jim Kingdon, 22-Oct-2019.) |
+_{Q0} ~_{Q0} ·_{Q0} | ||
Theorem | prarloclemcalc 7322 | Some calculations for prarloc 7323. (Contributed by Jim Kingdon, 26-Oct-2019.) |
+_{Q0} ~_{Q0} ·_{Q0} | ||
Theorem | prarloc 7323* |
A Dedekind cut is arithmetically located. Part of Proposition 11.15 of
[BauerTaylor], p. 52, slightly
modified. It states that given a
tolerance ,
there are elements of the lower and upper cut which
are within that tolerance of each other.
Usually, proofs will be shorter if they use prarloc2 7324 instead. (Contributed by Jim Kingdon, 22-Oct-2019.) |
Theorem | prarloc2 7324* | A Dedekind cut is arithmetically located. This is a variation of prarloc 7323 which only constructs one (named) point and is therefore often easier to work with. It states that given a tolerance , there are elements of the lower and upper cut which are exactly that tolerance from each other. (Contributed by Jim Kingdon, 26-Dec-2019.) |
Theorem | ltrelpr 7325 | Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) |
Theorem | ltdfpr 7326* | More convenient form of df-iltp 7290. (Contributed by Jim Kingdon, 15-Dec-2019.) |
Theorem | genpdflem 7327* | Simplification of upper or lower cut expression. Lemma for genpdf 7328. (Contributed by Jim Kingdon, 30-Sep-2019.) |
Theorem | genpdf 7328* | Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.) |
Theorem | genipv 7329* | Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.) |
Theorem | genplt2i 7330* | Operating on both sides of two inequalities, when the operation is consistent with . (Contributed by Jim Kingdon, 6-Oct-2019.) |
Theorem | genpelxp 7331* | Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.) |
Theorem | genpelvl 7332* | Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.) |
Theorem | genpelvu 7333* | Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.) |
Theorem | genpprecll 7334* | Pre-closure law for general operation on lower cuts. (Contributed by Jim Kingdon, 2-Oct-2019.) |
Theorem | genppreclu 7335* | Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-2019.) |
Theorem | genipdm 7336* | Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.) |
Theorem | genpml 7337* | The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.) |
Theorem | genpmu 7338* | The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.) |
Theorem | genpcdl 7339* | Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.) |
Theorem | genpcuu 7340* | Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.) |
Theorem | genprndl 7341* | The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.) |
Theorem | genprndu 7342* | The upper cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.) |
Theorem | genpdisj 7343* | The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (Contributed by Jim Kingdon, 15-Oct-2019.) |
Theorem | genpassl 7344* | Associativity of lower cuts. Lemma for genpassg 7346. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | genpassu 7345* | Associativity of upper cuts. Lemma for genpassg 7346. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | genpassg 7346* | Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | addnqprllem 7347 | Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.) |
Theorem | addnqprulem 7348 | Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.) |
Theorem | addnqprl 7349 | Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) |
Theorem | addnqpru 7350 | Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) |
Theorem | addlocprlemlt 7351 | Lemma for addlocpr 7356. The case. (Contributed by Jim Kingdon, 6-Dec-2019.) |
Theorem | addlocprlemeqgt 7352 | Lemma for addlocpr 7356. This is a step used in both the and cases. (Contributed by Jim Kingdon, 7-Dec-2019.) |
Theorem | addlocprlemeq 7353 | Lemma for addlocpr 7356. The case. (Contributed by Jim Kingdon, 6-Dec-2019.) |
Theorem | addlocprlemgt 7354 | Lemma for addlocpr 7356. The case. (Contributed by Jim Kingdon, 6-Dec-2019.) |
Theorem | addlocprlem 7355 | Lemma for addlocpr 7356. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.) |
Theorem | addlocpr 7356* | 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 7323 to both and , and uses nqtri3or 7216 rather than prloc 7311 to decide whether is too big to be in the lower cut of (and deduce that if it is, then must be in the upper cut). What the two proofs have in common is that they take the difference between and to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.) |
Theorem | addclpr 7357 | 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 7358* | Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | mpvlu 7359* | Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | dmplp 7360 | Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) |
Theorem | dmmp 7361 | Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) |
Theorem | nqprm 7362* | A cut produced from a rational is inhabited. Lemma for nqprlu 7367. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | nqprrnd 7363* | A cut produced from a rational is rounded. Lemma for nqprlu 7367. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | nqprdisj 7364* | A cut produced from a rational is disjoint. Lemma for nqprlu 7367. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | nqprloc 7365* | A cut produced from a rational is located. Lemma for nqprlu 7367. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | nqprxx 7366* | 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.) |
Theorem | nqprlu 7367* | The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.) |
Theorem | recnnpr 7368* | The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.) |
Theorem | ltnqex 7369 | The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.) |
Theorem | gtnqex 7370 | The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.) |
Theorem | nqprl 7371* | Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by . (Contributed by Jim Kingdon, 8-Jul-2020.) |
Theorem | nqpru 7372* | Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by . (Contributed by Jim Kingdon, 29-Nov-2020.) |
Theorem | nnprlu 7373* | The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.) |
Theorem | 1pr 7374 | The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) |
Theorem | 1prl 7375 | The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) |
Theorem | 1pru 7376 | The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) |
Theorem | addnqprlemrl 7377* | Lemma for addnqpr 7381. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
Theorem | addnqprlemru 7378* | Lemma for addnqpr 7381. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
Theorem | addnqprlemfl 7379* | Lemma for addnqpr 7381. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
Theorem | addnqprlemfu 7380* | Lemma for addnqpr 7381. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
Theorem | addnqpr 7381* | 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.) |
Theorem | addnqpr1 7382* | 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 7381. (Contributed by Jim Kingdon, 26-Apr-2020.) |
Theorem | appdivnq 7383* | Approximate division for positive rationals. Proposition 12.7 of [BauerTaylor], p. 55 (a special case where and are positive, as well as ). Our proof is simpler than the one in BauerTaylor because we have reciprocals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | appdiv0nq 7384* | Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7383 in which is zero, although it can be stated and proved in terms of positive rationals alone, without zero as such. (Contributed by Jim Kingdon, 9-Dec-2019.) |
Theorem | prmuloclemcalc 7385 | Calculations for prmuloc 7386. (Contributed by Jim Kingdon, 9-Dec-2019.) |
Theorem | prmuloc 7386* | Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | prmuloc2 7387* | Positive reals are multiplicatively located. This is a variation of prmuloc 7386 which only constructs one (named) point and is therefore often easier to work with. It states that given a ratio , there are elements of the lower and upper cut which have exactly that ratio between them. (Contributed by Jim Kingdon, 28-Dec-2019.) |
Theorem | mulnqprl 7388 | Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.) |
Theorem | mulnqpru 7389 | Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.) |
Theorem | mullocprlem 7390 | Calculations for mullocpr 7391. (Contributed by Jim Kingdon, 10-Dec-2019.) |
Theorem | mullocpr 7391* | Locatedness of multiplication on positive reals. Lemma 12.9 in [BauerTaylor], p. 56 (but where both and are positive, not just ). (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | mulclpr 7392 | Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.) |
Theorem | mulnqprlemrl 7393* | Lemma for mulnqpr 7397. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.) |
Theorem | mulnqprlemru 7394* | Lemma for mulnqpr 7397. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.) |
Theorem | mulnqprlemfl 7395* | Lemma for mulnqpr 7397. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.) |
Theorem | mulnqprlemfu 7396* | Lemma for mulnqpr 7397. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.) |
Theorem | mulnqpr 7397* | Multiplication of fractions embedded into positive reals. One can either multiply the fractions as fractions, or embed them into positive reals and multiply them as positive reals, and get the same result. (Contributed by Jim Kingdon, 18-Jul-2021.) |
Theorem | addcomprg 7398 | Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | addassprg 7399 | Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | mulcomprg 7400 | Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.) |
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