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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | prarloclemcalc 7401 | Some calculations for prarloc 7402. (Contributed by Jim Kingdon, 26-Oct-2019.) |
+_{Q0} ~_{Q0} ·_{Q0} | ||
Theorem | prarloc 7402* |
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 7403 instead. (Contributed by Jim Kingdon, 22-Oct-2019.) |
Theorem | prarloc2 7403* | A Dedekind cut is arithmetically located. This is a variation of prarloc 7402 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 7404 | Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) |
Theorem | ltdfpr 7405* | More convenient form of df-iltp 7369. (Contributed by Jim Kingdon, 15-Dec-2019.) |
Theorem | genpdflem 7406* | Simplification of upper or lower cut expression. Lemma for genpdf 7407. (Contributed by Jim Kingdon, 30-Sep-2019.) |
Theorem | genpdf 7407* | Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.) |
Theorem | genipv 7408* | Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.) |
Theorem | genplt2i 7409* | Operating on both sides of two inequalities, when the operation is consistent with . (Contributed by Jim Kingdon, 6-Oct-2019.) |
Theorem | genpelxp 7410* | Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.) |
Theorem | genpelvl 7411* | Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.) |
Theorem | genpelvu 7412* | Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.) |
Theorem | genpprecll 7413* | Pre-closure law for general operation on lower cuts. (Contributed by Jim Kingdon, 2-Oct-2019.) |
Theorem | genppreclu 7414* | Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-2019.) |
Theorem | genipdm 7415* | Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.) |
Theorem | genpml 7416* | The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.) |
Theorem | genpmu 7417* | The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.) |
Theorem | genpcdl 7418* | Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.) |
Theorem | genpcuu 7419* | Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.) |
Theorem | genprndl 7420* | The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.) |
Theorem | genprndu 7421* | The upper cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.) |
Theorem | genpdisj 7422* | The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (Contributed by Jim Kingdon, 15-Oct-2019.) |
Theorem | genpassl 7423* | Associativity of lower cuts. Lemma for genpassg 7425. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | genpassu 7424* | Associativity of upper cuts. Lemma for genpassg 7425. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | genpassg 7425* | Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | addnqprllem 7426 | Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.) |
Theorem | addnqprulem 7427 | Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.) |
Theorem | addnqprl 7428 | Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) |
Theorem | addnqpru 7429 | Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) |
Theorem | addlocprlemlt 7430 | Lemma for addlocpr 7435. The case. (Contributed by Jim Kingdon, 6-Dec-2019.) |
Theorem | addlocprlemeqgt 7431 | Lemma for addlocpr 7435. This is a step used in both the and cases. (Contributed by Jim Kingdon, 7-Dec-2019.) |
Theorem | addlocprlemeq 7432 | Lemma for addlocpr 7435. The case. (Contributed by Jim Kingdon, 6-Dec-2019.) |
Theorem | addlocprlemgt 7433 | Lemma for addlocpr 7435. The case. (Contributed by Jim Kingdon, 6-Dec-2019.) |
Theorem | addlocprlem 7434 | Lemma for addlocpr 7435. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.) |
Theorem | addlocpr 7435* | 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 7402 to both and , and uses nqtri3or 7295 rather than prloc 7390 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 7436 | 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 7437* | Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | mpvlu 7438* | Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | dmplp 7439 | Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) |
Theorem | dmmp 7440 | Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) |
Theorem | nqprm 7441* | A cut produced from a rational is inhabited. Lemma for nqprlu 7446. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | nqprrnd 7442* | A cut produced from a rational is rounded. Lemma for nqprlu 7446. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | nqprdisj 7443* | A cut produced from a rational is disjoint. Lemma for nqprlu 7446. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | nqprloc 7444* | A cut produced from a rational is located. Lemma for nqprlu 7446. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | nqprxx 7445* | 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 7446* | The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.) |
Theorem | recnnpr 7447* | The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.) |
Theorem | ltnqex 7448 | The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.) |
Theorem | gtnqex 7449 | The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.) |
Theorem | nqprl 7450* | 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 7451* | 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 7452* | The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.) |
Theorem | 1pr 7453 | The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) |
Theorem | 1prl 7454 | The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) |
Theorem | 1pru 7455 | The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) |
Theorem | addnqprlemrl 7456* | Lemma for addnqpr 7460. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
Theorem | addnqprlemru 7457* | Lemma for addnqpr 7460. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
Theorem | addnqprlemfl 7458* | Lemma for addnqpr 7460. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
Theorem | addnqprlemfu 7459* | Lemma for addnqpr 7460. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
Theorem | addnqpr 7460* | 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 7461* | 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 7460. (Contributed by Jim Kingdon, 26-Apr-2020.) |
Theorem | appdivnq 7462* | 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 7463* | Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7462 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 7464 | Calculations for prmuloc 7465. (Contributed by Jim Kingdon, 9-Dec-2019.) |
Theorem | prmuloc 7465* | Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Theorem | prmuloc2 7466* | Positive reals are multiplicatively located. This is a variation of prmuloc 7465 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 7467 | Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.) |
Theorem | mulnqpru 7468 | Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.) |
Theorem | mullocprlem 7469 | Calculations for mullocpr 7470. (Contributed by Jim Kingdon, 10-Dec-2019.) |
Theorem | mullocpr 7470* | 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 7471 | 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 7472* | Lemma for mulnqpr 7476. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.) |
Theorem | mulnqprlemru 7473* | Lemma for mulnqpr 7476. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.) |
Theorem | mulnqprlemfl 7474* | Lemma for mulnqpr 7476. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.) |
Theorem | mulnqprlemfu 7475* | Lemma for mulnqpr 7476. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.) |
Theorem | mulnqpr 7476* | 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 7477 | Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | addassprg 7478 | Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | mulcomprg 7479 | Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | mulassprg 7480 | Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | distrlem1prl 7481 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Theorem | distrlem1pru 7482 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Theorem | distrlem4prl 7483* | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Theorem | distrlem4pru 7484* | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Theorem | distrlem5prl 7485 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Theorem | distrlem5pru 7486 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Theorem | distrprg 7487 | Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Theorem | ltprordil 7488 | If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.) |
Theorem | 1idprl 7489 | Lemma for 1idpr 7491. (Contributed by Jim Kingdon, 13-Dec-2019.) |
Theorem | 1idpru 7490 | Lemma for 1idpr 7491. (Contributed by Jim Kingdon, 13-Dec-2019.) |
Theorem | 1idpr 7491 | 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.) |
Theorem | ltnqpr 7492* | We can order fractions via or . (Contributed by Jim Kingdon, 19-Jun-2021.) |
Theorem | ltnqpri 7493* | We can order fractions via or . (Contributed by Jim Kingdon, 8-Jan-2021.) |
Theorem | ltpopr 7494 | Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 7495. (Contributed by Jim Kingdon, 15-Dec-2019.) |
Theorem | ltsopr 7495 | Positive real 'less than' is a weak linear order (in the sense of df-iso 4252). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.) |
Theorem | ltaddpr 7496 | The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) |
Theorem | ltexprlemell 7497* | Element in lower cut of the constructed difference. Lemma for ltexpri 7512. (Contributed by Jim Kingdon, 21-Dec-2019.) |
Theorem | ltexprlemelu 7498* | Element in upper cut of the constructed difference. Lemma for ltexpri 7512. (Contributed by Jim Kingdon, 21-Dec-2019.) |
Theorem | ltexprlemm 7499* | Our constructed difference is inhabited. Lemma for ltexpri 7512. (Contributed by Jim Kingdon, 17-Dec-2019.) |
Theorem | ltexprlemopl 7500* | The lower cut of our constructed difference is open. Lemma for ltexpri 7512. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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