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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-imp 7301* |
Define multiplication on positive reals. Here we use a simple
definition which is similar to df-iplp 7300 or the definition of
multiplication on positive reals in Metamath Proof Explorer. This is as
opposed to the more complicated definition of multiplication given in
Section 11.2.1 of [HoTT], p. (varies),
which appears to be motivated by
handling negative numbers or handling modified Dedekind cuts in which
locatedness is omitted.
This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.) |
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Definition | df-iltp 7302* |
Define ordering on positive reals. We define ![]() ![]() ![]() ![]() ![]() ![]() This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.) |
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Theorem | npsspw 7303 | Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.) |
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Theorem | preqlu 7304 | Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.) |
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Theorem | npex 7305 | The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) |
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Theorem | elinp 7306* | Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.) |
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Theorem | prop 7307 | 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 7308* |
Membership in positive reals, using ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | prml 7309* | A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
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Theorem | prmu 7310* | A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
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Theorem | prssnql 7311 | 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 7312 | 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 7313 | 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 7314 | 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 7315 | The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) |
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Theorem | prcdnql 7316 | A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.) |
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Theorem | prcunqu 7317 | An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.) |
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Theorem | prubl 7318 | 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 7319 | 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 7320* | A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.) |
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Theorem | prnminu 7321* | An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.) |
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Theorem | prnmaddl 7322* | A lower cut has no largest member. Addition version. (Contributed by Jim Kingdon, 29-Sep-2019.) |
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Theorem | prloc 7323 | A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.) |
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Theorem | prdisj 7324 | A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.) |
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Theorem | prarloclemlt 7325 | Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7335. (Contributed by Jim Kingdon, 10-Nov-2019.) |
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Theorem | prarloclemlo 7326* | Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 7335. (Contributed by Jim Kingdon, 10-Nov-2019.) |
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Theorem | prarloclemup 7327 | Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 7335. (Contributed by Jim Kingdon, 10-Nov-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | prarloclem3step 7328* | Induction step for prarloclem3 7329. (Contributed by Jim Kingdon, 9-Nov-2019.) |
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Theorem | prarloclem3 7329* | Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7335. (Contributed by Jim Kingdon, 27-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | prarloclem4 7330* | A slight rearrangement of prarloclem3 7329. Lemma for prarloc 7335. (Contributed by Jim Kingdon, 4-Nov-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | prarloclemn 7331* | Subtracting two from a positive integer. Lemma for prarloc 7335. (Contributed by Jim Kingdon, 5-Nov-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | prarloclem5 7332* |
A substitution of zero for ![]() ![]() ![]() |
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Theorem | prarloclem 7333* |
A special case of Lemma 6.16 from [BauerTaylor], p. 32. Given evenly
spaced rational numbers from ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | prarloclemcalc 7334 | Some calculations for prarloc 7335. (Contributed by Jim Kingdon, 26-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | prarloc 7335* |
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 7336 instead. (Contributed by Jim Kingdon, 22-Oct-2019.) |
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Theorem | prarloc2 7336* |
A Dedekind cut is arithmetically located. This is a variation of
prarloc 7335 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 7337 | Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) |
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Theorem | ltdfpr 7338* | More convenient form of df-iltp 7302. (Contributed by Jim Kingdon, 15-Dec-2019.) |
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Theorem | genpdflem 7339* | Simplification of upper or lower cut expression. Lemma for genpdf 7340. (Contributed by Jim Kingdon, 30-Sep-2019.) |
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Theorem | genpdf 7340* | Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | genipv 7341* | Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.) |
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Theorem | genplt2i 7342* |
Operating on both sides of two inequalities, when the operation is
consistent with ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | genpelxp 7343* | Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.) |
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Theorem | genpelvl 7344* | 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 7345* | Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.) |
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Theorem | genpprecll 7346* | Pre-closure law for general operation on lower cuts. (Contributed by Jim Kingdon, 2-Oct-2019.) |
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Theorem | genppreclu 7347* | Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-2019.) |
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Theorem | genipdm 7348* | Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.) |
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Theorem | genpml 7349* | 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 7350* | 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 7351* | Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.) |
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Theorem | genpcuu 7352* | Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.) |
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Theorem | genprndl 7353* | 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 7354* | 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 7355* | 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 7356* | Associativity of lower cuts. Lemma for genpassg 7358. (Contributed by Jim Kingdon, 11-Dec-2019.) |
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Theorem | genpassu 7357* | Associativity of upper cuts. Lemma for genpassg 7358. (Contributed by Jim Kingdon, 11-Dec-2019.) |
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Theorem | genpassg 7358* | Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.) |
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Theorem | addnqprllem 7359 | Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.) |
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Theorem | addnqprulem 7360 | Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.) |
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Theorem | addnqprl 7361 | Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) |
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Theorem | addnqpru 7362 | Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) |
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Theorem | addlocprlemlt 7363 |
Lemma for addlocpr 7368. The ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | addlocprlemeqgt 7364 |
Lemma for addlocpr 7368. This is a step used in both the
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Theorem | addlocprlemeq 7365 |
Lemma for addlocpr 7368. The ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | addlocprlemgt 7366 |
Lemma for addlocpr 7368. The ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | addlocprlem 7367 | Lemma for addlocpr 7368. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.) |
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Theorem | addlocpr 7368* |
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 7335
to both ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | addclpr 7369 | 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.) |
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Theorem | plpvlu 7370* | Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
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Theorem | mpvlu 7371* | Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
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Theorem | dmplp 7372 | Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) |
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Theorem | dmmp 7373 | Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) |
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Theorem | nqprm 7374* | A cut produced from a rational is inhabited. Lemma for nqprlu 7379. (Contributed by Jim Kingdon, 8-Dec-2019.) |
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Theorem | nqprrnd 7375* | A cut produced from a rational is rounded. Lemma for nqprlu 7379. (Contributed by Jim Kingdon, 8-Dec-2019.) |
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Theorem | nqprdisj 7376* | A cut produced from a rational is disjoint. Lemma for nqprlu 7379. (Contributed by Jim Kingdon, 8-Dec-2019.) |
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Theorem | nqprloc 7377* | A cut produced from a rational is located. Lemma for nqprlu 7379. (Contributed by Jim Kingdon, 8-Dec-2019.) |
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Theorem | nqprxx 7378* | 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 7379* | The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.) |
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Theorem | recnnpr 7380* | The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.) |
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Theorem | ltnqex 7381 | 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 7382 | 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 7383* |
Comparing a fraction to a real can be done by whether it is an element
of the lower cut, or by ![]() |
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Theorem | nqpru 7384* |
Comparing a fraction to a real can be done by whether it is an element
of the upper cut, or by ![]() |
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Theorem | nnprlu 7385* | The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.) |
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Theorem | 1pr 7386 | The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) |
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Theorem | 1prl 7387 | The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) |
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Theorem | 1pru 7388 | The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) |
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Theorem | addnqprlemrl 7389* | Lemma for addnqpr 7393. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
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Theorem | addnqprlemru 7390* | Lemma for addnqpr 7393. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
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Theorem | addnqprlemfl 7391* | Lemma for addnqpr 7393. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
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Theorem | addnqprlemfu 7392* | Lemma for addnqpr 7393. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.) |
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Theorem | addnqpr 7393* | 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 7394* | 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 7393. (Contributed by Jim Kingdon, 26-Apr-2020.) |
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Theorem | appdivnq 7395* |
Approximate division for positive rationals. Proposition 12.7 of
[BauerTaylor], p. 55 (a special case
where ![]() ![]() ![]() |
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Theorem | appdiv0nq 7396* |
Approximate division for positive rationals. This can be thought of as
a variation of appdivnq 7395 in which ![]() |
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Theorem | prmuloclemcalc 7397 | Calculations for prmuloc 7398. (Contributed by Jim Kingdon, 9-Dec-2019.) |
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Theorem | prmuloc 7398* | Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.) |
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Theorem | prmuloc2 7399* |
Positive reals are multiplicatively located. This is a variation of
prmuloc 7398 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 7400 | Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.) |
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