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Type | Label | Description |
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Statement | ||
Theorem | 00sr 7601 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) |
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Theorem | ltasrg 7602 | Ordering property of addition. (Contributed by NM, 10-May-1996.) |
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Theorem | pn0sr 7603 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) |
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Theorem | negexsr 7604* | Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) |
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Theorem | recexgt0sr 7605* | The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) |
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Theorem | recexsrlem 7606* | The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) |
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Theorem | addgt0sr 7607 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) |
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Theorem | ltadd1sr 7608 | Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.) |
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Theorem | ltm1sr 7609 | Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.) |
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Theorem | mulgt0sr 7610 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) |
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Theorem | aptisr 7611 | Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.) |
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Theorem | mulextsr1lem 7612 | Lemma for mulextsr1 7613. (Contributed by Jim Kingdon, 17-Feb-2020.) |
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Theorem | mulextsr1 7613 | Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.) |
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Theorem | archsr 7614* |
For any signed real, there is an integer that is greater than it. This
is also known as the "archimedean property". The expression
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Theorem | srpospr 7615* | Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
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Theorem | prsrcl 7616 | Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
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Theorem | prsrpos 7617 | Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.) |
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Theorem | prsradd 7618 | Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.) |
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Theorem | prsrlt 7619 | Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.) |
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Theorem | prsrriota 7620* | Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
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Theorem | caucvgsrlemcl 7621* | Lemma for caucvgsr 7634. Terms of the sequence from caucvgsrlemgt1 7627 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.) |
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Theorem | caucvgsrlemasr 7622* | Lemma for caucvgsr 7634. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.) |
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Theorem | caucvgsrlemfv 7623* | Lemma for caucvgsr 7634. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
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Theorem | caucvgsrlemf 7624* | Lemma for caucvgsr 7634. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
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Theorem | caucvgsrlemcau 7625* | Lemma for caucvgsr 7634. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
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Theorem | caucvgsrlembound 7626* | Lemma for caucvgsr 7634. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.) |
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Theorem | caucvgsrlemgt1 7627* | Lemma for caucvgsr 7634. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.) |
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Theorem | caucvgsrlemoffval 7628* | Lemma for caucvgsr 7634. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
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Theorem | caucvgsrlemofff 7629* | Lemma for caucvgsr 7634. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
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Theorem | caucvgsrlemoffcau 7630* | Lemma for caucvgsr 7634. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
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Theorem | caucvgsrlemoffgt1 7631* | Lemma for caucvgsr 7634. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
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Theorem | caucvgsrlemoffres 7632* | Lemma for caucvgsr 7634. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
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Theorem | caucvgsrlembnd 7633* | Lemma for caucvgsr 7634. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.) |
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Theorem | caucvgsr 7634* |
A Cauchy sequence of signed reals with a modulus of convergence
converges to a signed real. This is basically Corollary 11.2.13 of
[HoTT], p. (varies). The HoTT book
theorem has a modulus of
convergence (that is, a rate of convergence) specified by (11.2.9) in
HoTT whereas this theorem fixes the rate of convergence to say that
all terms after the nth term must be within ![]() ![]() ![]() This is similar to caucvgprpr 7544 but is for signed reals rather than positive reals. Here is an outline of how we prove it: 1. Choose a lower bound for the sequence (see caucvgsrlembnd 7633). 2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 7629).
3. Since a signed real (element of 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7627). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7632). (Contributed by Jim Kingdon, 20-Jun-2021.) |
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Theorem | ltpsrprg 7635 | Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) |
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Theorem | mappsrprg 7636 | Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) |
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Theorem | map2psrprg 7637* | Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | suplocsrlemb 7638* |
Lemma for suplocsr 7641. The set ![]() |
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Theorem | suplocsrlempr 7639* |
Lemma for suplocsr 7641. The set ![]() |
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Theorem | suplocsrlem 7640* |
Lemma for suplocsr 7641. The set ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | suplocsr 7641* | An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Syntax | cc 7642 | Class of complex numbers. |
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Syntax | cr 7643 | Class of real numbers. |
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Syntax | cc0 7644 | Extend class notation to include the complex number 0. |
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Syntax | c1 7645 | Extend class notation to include the complex number 1. |
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Syntax | ci 7646 | Extend class notation to include the complex number i. |
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Syntax | caddc 7647 | Addition on complex numbers. |
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Syntax | cltrr 7648 | 'Less than' predicate (defined over real subset of complex numbers). |
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Syntax | cmul 7649 |
Multiplication on complex numbers. The token ![]() |
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Definition | df-c 7650 | Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.) |
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Definition | df-0 7651 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) |
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Definition | df-1 7652 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) |
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Definition | df-i 7653 |
Define the complex number ![]() |
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Definition | df-r 7654 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) |
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Definition | df-add 7655* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) |
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Definition | df-mul 7656* | Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Definition | df-lt 7657* | Define 'less than' on the real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) |
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Theorem | opelcn 7658 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | opelreal 7659 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | elreal 7660* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) |
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Theorem | elrealeu 7661* | The real number mapping in elreal 7660 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | elreal2 7662 | Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) |
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Theorem | 0ncn 7663 | The empty set is not a complex number. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. See also cnm 7664 which is a related property. (Contributed by NM, 2-May-1996.) |
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Theorem | cnm 7664* | A complex number is an inhabited set. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by Jim Kingdon, 23-Oct-2023.) (New usage is discouraged.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | ltrelre 7665 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) |
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Theorem | addcnsr 7666 | Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) |
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Theorem | mulcnsr 7667 | Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | eqresr 7668 | Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | addresr 7669 | Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | mulresr 7670 | Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | ltresr 7671 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | ltresr2 7672 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dfcnqs 7673 |
Technical trick to permit reuse of previous lemmas to prove arithmetic
operation laws in ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | addcnsrec 7674 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 7673 and mulcnsrec 7675. (Contributed by NM, 13-Aug-1995.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | mulcnsrec 7675 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6501, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 7673. (Contributed by NM, 13-Aug-1995.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | addvalex 7676 |
Existence of a sum. This is dependent on how we define ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | pitonnlem1 7677* | Lemma for pitonn 7680. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | pitonnlem1p1 7678 | Lemma for pitonn 7680. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | pitonnlem2 7679* | Lemma for pitonn 7680. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | pitonn 7680* |
Mapping from ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | pitoregt0 7681* |
Embedding from ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | pitore 7682* |
Embedding from ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | recnnre 7683* |
Embedding the reciprocal of a natural number into ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | peano1nnnn 7684* |
One is an element of ![]() |
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Theorem | peano2nnnn 7685* | A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 8756 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7732). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
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Theorem | ltrennb 7686* |
Ordering of natural numbers with ![]() ![]() |
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Theorem | ltrenn 7687* |
Ordering of natural numbers with ![]() ![]() |
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Theorem | recidpipr 7688* | Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.) |
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Theorem | recidpirqlemcalc 7689 | Lemma for recidpirq 7690. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.) |
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Theorem | recidpirq 7690* |
A real number times its reciprocal is one, where reciprocal is expressed
with ![]() |
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Theorem | axcnex 7691 | The complex numbers form a set. Use cnex 7768 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
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Theorem | axresscn 7692 | The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn 7736. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
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Theorem | ax1cn 7693 | 1 is a complex number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1cn 7737. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.) |
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Theorem | ax1re 7694 |
1 is a real number. Axiom for real and complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly; instead, use ax-1re 7738.
In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 7737 and the other axioms. It is not known whether we can do so here, but the Metamath Proof Explorer proof (accessed 13-Jan-2020) uses excluded middle. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.) |
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Theorem | axicn 7695 |
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Theorem | axaddcl 7696 | Closure law for addition of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcl 7740 be used later. Instead, in most cases use addcl 7769. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
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Theorem | axaddrcl 7697 | Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addrcl 7741 be used later. Instead, in most cases use readdcl 7770. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
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Theorem | axmulcl 7698 | Closure law for multiplication of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcl 7742 be used later. Instead, in most cases use mulcl 7771. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
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Theorem | axmulrcl 7699 | Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulrcl 7743 be used later. Instead, in most cases use remulcl 7772. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
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Theorem | axaddf 7700 | Addition is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axaddcl 7696. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 7766. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
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