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Type | Label | Description |
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Statement | ||
Theorem | axmulrcl 7501 | 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 7541 be used later. Instead, in most cases use remulcl 7567. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
Theorem | axaddcom 7502 |
Addition commutes. 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-addcom 7542 be used later.
Instead, use addcom 7716.
In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on real number trichotomy and it is not known whether it is possible to prove this from the other axioms without it. (Contributed by Jim Kingdon, 17-Jan-2020.) (New usage is discouraged.) |
Theorem | axmulcom 7503 | Multiplication of complex numbers is commutative. 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-mulcom 7543 be used later. Instead, use mulcom 7568. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
Theorem | axaddass 7504 | Addition of complex numbers is associative. This theorem transfers the associative laws for the real and imaginary signed real components of complex number pairs, to complex number addition itself. 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-addass 7544 be used later. Instead, use addass 7569. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axmulass 7505 | Multiplication of complex numbers is associative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 7545. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
Theorem | axdistr 7506 | Distributive law for complex numbers (left-distributivity). 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-distr 7546 be used later. Instead, use adddi 7571. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axi2m1 7507 | i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 7547. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Theorem | ax0lt1 7508 |
0 is less than 1. Axiom for real and complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly; instead, use ax-0lt1 7548.
The version of this axiom in the Metamath Proof Explorer reads ; here we change it to . The proof of from in the Metamath Proof Explorer (accessed 12-Jan-2020) relies on real number trichotomy. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) |
Theorem | ax1rid 7509 | is an identity element for real multiplication. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 7549. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
Theorem | ax0id 7510 |
is an identity element
for real addition. Axiom for real and
complex numbers, derived from set theory. This construction-dependent
theorem should not be referenced directly; instead, use ax-0id 7550.
In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on excluded middle and it is not known whether it is possible to prove this from the other axioms without excluded middle. (Contributed by Jim Kingdon, 16-Jan-2020.) (New usage is discouraged.) |
Theorem | axrnegex 7511* | Existence of negative of real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 7551. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | axprecex 7512* |
Existence of positive reciprocal of positive real number. Axiom for
real and complex numbers, derived from set theory. This
construction-dependent theorem should not be referenced directly;
instead, use ax-precex 7552.
In treatments which assume excluded middle, the condition is generally replaced by , and it may not be necessary to state that the reciproacal is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) (New usage is discouraged.) |
Theorem | axcnre 7513* | A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 7553. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-ltirr 7514 | Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltirr 7554. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) |
Theorem | axpre-ltwlin 7515 | Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltwlin 7555. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) |
Theorem | axpre-lttrn 7516 | Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 7556. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | axpre-apti 7517 |
Apartness of reals is tight. Axiom for real and complex numbers,
derived from set theory. This construction-dependent theorem should not
be referenced directly; instead, use ax-pre-apti 7557.
(Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.) |
Theorem | axpre-ltadd 7518 | Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 7558. (Contributed by NM, 11-May-1996.) (New usage is discouraged.) |
Theorem | axpre-mulgt0 7519 | The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 7559. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-mulext 7520 |
Strong extensionality of multiplication (expressed in terms of
).
Axiom for real and complex numbers, derived from set theory.
This construction-dependent theorem should not be referenced directly;
instead, use ax-pre-mulext 7560.
(Contributed by Jim Kingdon, 18-Feb-2020.) (New usage is discouraged.) |
Theorem | rereceu 7521* | The reciprocal from axprecex 7512 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.) |
Theorem | recriota 7522* | Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.) |
Theorem | axarch 7523* |
Archimedean axiom. The Archimedean property is more naturally stated
once we have defined . Unless we find another way to state it,
we'll just use the right hand side of dfnn2 8522 in stating what we mean by
"natural number" in the context of this axiom.
This construction-dependent theorem should not be referenced directly; instead, use ax-arch 7561. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.) |
Theorem | peano5nnnn 7524* | Peano's inductive postulate. This is a counterpart to peano5nni 8523 designed for real number axioms which involve natural numbers (notably, axcaucvg 7532). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
Theorem | nnindnn 7525* | Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 8536 designed for real number axioms which involve natural numbers (notably, axcaucvg 7532). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
Theorem | nntopi 7526* | Mapping from to . (Contributed by Jim Kingdon, 13-Jul-2021.) |
Theorem | axcaucvglemcl 7527* | Lemma for axcaucvg 7532. Mapping to and . (Contributed by Jim Kingdon, 10-Jul-2021.) |
Theorem | axcaucvglemf 7528* | Lemma for axcaucvg 7532. Mapping to and yields a sequence. (Contributed by Jim Kingdon, 9-Jul-2021.) |
Theorem | axcaucvglemval 7529* | Lemma for axcaucvg 7532. Value of sequence when mapping to and . (Contributed by Jim Kingdon, 10-Jul-2021.) |
Theorem | axcaucvglemcau 7530* | Lemma for axcaucvg 7532. The result of mapping to and satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.) |
Theorem | axcaucvglemres 7531* | Lemma for axcaucvg 7532. Mapping the limit from and . (Contributed by Jim Kingdon, 10-Jul-2021.) |
Theorem | axcaucvg 7532* |
Real number completeness axiom. A Cauchy sequence with a modulus of
convergence converges. 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 of the nth term (it should
later
be able to prove versions of this theorem with a different fixed rate
or a modulus of convergence supplied as a hypothesis).
Because we are stating this axiom before we have introduced notations for or division, we use for the natural numbers and express a reciprocal in terms of . This construction-dependent theorem should not be referenced directly; instead, use ax-caucvg 7562. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.) |
Axiom | ax-cnex 7533 | The complex numbers form a set. Proofs should normally use cnex 7563 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-resscn 7534 | The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by theorem axresscn 7494. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-1cn 7535 | 1 is a complex number. Axiom for real and complex numbers, justified by theorem ax1cn 7495. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-1re 7536 | 1 is a real number. Axiom for real and complex numbers, justified by theorem ax1re 7496. Proofs should use 1re 7584 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.) |
Axiom | ax-icn 7537 | is a complex number. Axiom for real and complex numbers, justified by theorem axicn 7497. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-addcl 7538 | Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by theorem axaddcl 7498. Proofs should normally use addcl 7564 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-addrcl 7539 | Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axaddrcl 7499. Proofs should normally use readdcl 7565 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulcl 7540 | Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by theorem axmulcl 7500. Proofs should normally use mulcl 7566 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulrcl 7541 | Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 7501. Proofs should normally use remulcl 7567 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-addcom 7542 | Addition commutes. Axiom for real and complex numbers, justified by theorem axaddcom 7502. Proofs should normally use addcom 7716 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.) |
Axiom | ax-mulcom 7543 | Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by theorem axmulcom 7503. Proofs should normally use mulcom 7568 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-addass 7544 | Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 7504. Proofs should normally use addass 7569 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulass 7545 | Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7505. Proofs should normally use mulass 7570 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-distr 7546 | Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by theorem axdistr 7506. Proofs should normally use adddi 7571 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-i2m1 7547 | i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by theorem axi2m1 7507. (Contributed by NM, 29-Jan-1995.) |
Axiom | ax-0lt1 7548 | 0 is less than 1. Axiom for real and complex numbers, justified by theorem ax0lt1 7508. Proofs should normally use 0lt1 7707 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.) |
Axiom | ax-1rid 7549 | is an identity element for real multiplication. Axiom for real and complex numbers, justified by theorem ax1rid 7509. (Contributed by NM, 29-Jan-1995.) |
Axiom | ax-0id 7550 |
is an identity element
for real addition. Axiom for real and
complex numbers, justified by theorem ax0id 7510.
Proofs should normally use addid1 7717 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.) |
Axiom | ax-rnegex 7551* | Existence of negative of real number. Axiom for real and complex numbers, justified by theorem axrnegex 7511. (Contributed by Eric Schmidt, 21-May-2007.) |
Axiom | ax-precex 7552* | Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 7512. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Axiom | ax-cnre 7553* | A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, justified by theorem axcnre 7513. For naming consistency, use cnre 7581 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.) |
Axiom | ax-pre-ltirr 7554 | Real number less-than is irreflexive. Axiom for real and complex numbers, justified by theorem ax-pre-ltirr 7554. (Contributed by Jim Kingdon, 12-Jan-2020.) |
Axiom | ax-pre-ltwlin 7555 | Real number less-than is weakly linear. Axiom for real and complex numbers, justified by theorem axpre-ltwlin 7515. (Contributed by Jim Kingdon, 12-Jan-2020.) |
Axiom | ax-pre-lttrn 7556 | Ordering on reals is transitive. Axiom for real and complex numbers, justified by theorem axpre-lttrn 7516. (Contributed by NM, 13-Oct-2005.) |
Axiom | ax-pre-apti 7557 | Apartness of reals is tight. Axiom for real and complex numbers, justified by theorem axpre-apti 7517. (Contributed by Jim Kingdon, 29-Jan-2020.) |
Axiom | ax-pre-ltadd 7558 | Ordering property of addition on reals. Axiom for real and complex numbers, justified by theorem axpre-ltadd 7518. (Contributed by NM, 13-Oct-2005.) |
Axiom | ax-pre-mulgt0 7559 | The product of two positive reals is positive. Axiom for real and complex numbers, justified by theorem axpre-mulgt0 7519. (Contributed by NM, 13-Oct-2005.) |
Axiom | ax-pre-mulext 7560 |
Strong extensionality of multiplication (expressed in terms of ).
Axiom for real and complex numbers, justified by theorem axpre-mulext 7520
(Contributed by Jim Kingdon, 18-Feb-2020.) |
Axiom | ax-arch 7561* |
Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for
real and complex numbers, justified by theorem axarch 7523.
This axiom should not be used directly; instead use arch 8768 (which is the same, but stated in terms of and ). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.) |
Axiom | ax-caucvg 7562* |
Completeness. Axiom for real and complex numbers, justified by theorem
axcaucvg 7532.
A Cauchy sequence (as defined here, which has a rate convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within of the nth term. This axiom should not be used directly; instead use caucvgre 10529 (which is the same, but stated in terms of the and notations). (Contributed by Jim Kingdon, 19-Jul-2021.) (New usage is discouraged.) |
Theorem | cnex 7563 | Alias for ax-cnex 7533. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Theorem | addcl 7564 | Alias for ax-addcl 7538, for naming consistency with addcli 7589. Use this theorem instead of ax-addcl 7538 or axaddcl 7498. (Contributed by NM, 10-Mar-2008.) |
Theorem | readdcl 7565 | Alias for ax-addrcl 7539, for naming consistency with readdcli 7598. (Contributed by NM, 10-Mar-2008.) |
Theorem | mulcl 7566 | Alias for ax-mulcl 7540, for naming consistency with mulcli 7590. (Contributed by NM, 10-Mar-2008.) |
Theorem | remulcl 7567 | Alias for ax-mulrcl 7541, for naming consistency with remulcli 7599. (Contributed by NM, 10-Mar-2008.) |
Theorem | mulcom 7568 | Alias for ax-mulcom 7543, for naming consistency with mulcomi 7591. (Contributed by NM, 10-Mar-2008.) |
Theorem | addass 7569 | Alias for ax-addass 7544, for naming consistency with addassi 7593. (Contributed by NM, 10-Mar-2008.) |
Theorem | mulass 7570 | Alias for ax-mulass 7545, for naming consistency with mulassi 7594. (Contributed by NM, 10-Mar-2008.) |
Theorem | adddi 7571 | Alias for ax-distr 7546, for naming consistency with adddii 7595. (Contributed by NM, 10-Mar-2008.) |
Theorem | recn 7572 | A real number is a complex number. (Contributed by NM, 10-Aug-1999.) |
Theorem | reex 7573 | The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Theorem | reelprrecn 7574 | Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | cnelprrecn 7575 | Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | adddir 7576 | Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.) |
Theorem | 0cn 7577 | 0 is a complex number. (Contributed by NM, 19-Feb-2005.) |
Theorem | 0cnd 7578 | 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | c0ex 7579 | 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | 1ex 7580 | 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | cnre 7581* | Alias for ax-cnre 7553, for naming consistency. (Contributed by NM, 3-Jan-2013.) |
Theorem | mulid1 7582 | is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
Theorem | mulid2 7583 | Identity law for multiplication. Note: see mulid1 7582 for commuted version. (Contributed by NM, 8-Oct-1999.) |
Theorem | 1re 7584 | is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.) |
Theorem | 0re 7585 | is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) |
Theorem | 0red 7586 | is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
Theorem | mulid1i 7587 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
Theorem | mulid2i 7588 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
Theorem | addcli 7589 | Closure law for addition. (Contributed by NM, 23-Nov-1994.) |
Theorem | mulcli 7590 | Closure law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Theorem | mulcomi 7591 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Theorem | mulcomli 7592 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Theorem | addassi 7593 | Associative law for addition. (Contributed by NM, 23-Nov-1994.) |
Theorem | mulassi 7594 | Associative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Theorem | adddii 7595 | Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.) |
Theorem | adddiri 7596 | Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.) |
Theorem | recni 7597 | A real number is a complex number. (Contributed by NM, 1-Mar-1995.) |
Theorem | readdcli 7598 | Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.) |
Theorem | remulcli 7599 | Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.) |
Theorem | 1red 7600 | 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
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