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Theorem List for Metamath Proof Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1sr 10501 The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
1RR
 
Theoremm1r 10502 The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
-1RR
 
Theoremaddclsr 10503 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
((𝐴R𝐵R) → (𝐴 +R 𝐵) ∈ R)
 
Theoremmulclsr 10504 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
((𝐴R𝐵R) → (𝐴 ·R 𝐵) ∈ R)
 
Theoremdmaddsr 10505 Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
dom +R = (R × R)
 
Theoremdmmulsr 10506 Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
dom ·R = (R × R)
 
Theoremaddcomsr 10507 Addition of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
(𝐴 +R 𝐵) = (𝐵 +R 𝐴)
 
Theoremaddasssr 10508 Addition of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
((𝐴 +R 𝐵) +R 𝐶) = (𝐴 +R (𝐵 +R 𝐶))
 
Theoremmulcomsr 10509 Multiplication of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
(𝐴 ·R 𝐵) = (𝐵 ·R 𝐴)
 
Theoremmulasssr 10510 Multiplication of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶))
 
Theoremdistrsr 10511 Multiplication of signed reals is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
(𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶))
 
Theoremm1p1sr 10512 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
(-1R +R 1R) = 0R
 
Theoremm1m1sr 10513 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
(-1R ·R -1R) = 1R
 
Theoremltsosr 10514 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
<R Or R
 
Theorem0lt1sr 10515 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
0R <R 1R
 
Theorem1ne0sr 10516 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
¬ 1R = 0R
 
Theorem0idsr 10517 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
(𝐴R → (𝐴 +R 0R) = 𝐴)
 
Theorem1idsr 10518 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
(𝐴R → (𝐴 ·R 1R) = 𝐴)
 
Theorem00sr 10519 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
(𝐴R → (𝐴 ·R 0R) = 0R)
 
Theoremltasr 10520 Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
(𝐶R → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
 
Theorempn0sr 10521 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
(𝐴R → (𝐴 +R (𝐴 ·R -1R)) = 0R)
 
Theoremnegexsr 10522* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
(𝐴R → ∃𝑥R (𝐴 +R 𝑥) = 0R)
 
Theoremrecexsrlem 10523* The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
(0R <R 𝐴 → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
 
Theoremaddgt0sr 10524 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 +R 𝐵))
 
Theoremmulgt0sr 10525 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))
 
Theoremsqgt0sr 10526 The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
((𝐴R𝐴 ≠ 0R) → 0R <R (𝐴 ·R 𝐴))
 
Theoremrecexsr 10527* The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
((𝐴R𝐴 ≠ 0R) → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
 
Theoremmappsrpr 10528 Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
𝐶R       ((𝐶 +R -1R) <R (𝐶 +R [⟨𝐴, 1P⟩] ~R ) ↔ 𝐴P)
 
Theoremltpsrpr 10529 Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
𝐶R       ((𝐶 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐶 +R [⟨𝐵, 1P⟩] ~R ) ↔ 𝐴<P 𝐵)
 
Theoremmap2psrpr 10530* Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
𝐶R       ((𝐶 +R -1R) <R 𝐴 ↔ ∃𝑥P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴)
 
Theoremsupsrlem 10531* Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
𝐵 = {𝑤 ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   𝐶R       ((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
 
Theoremsupsr 10532* A nonempty, bounded set of signed reals has a supremum. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
((𝐴 ≠ ∅ ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
 
Syntaxcc 10533 Class of complex numbers.
class
 
Syntaxcr 10534 Class of real numbers.
class
 
Syntaxcc0 10535 Extend class notation to include the complex number 0.
class 0
 
Syntaxc1 10536 Extend class notation to include the complex number 1.
class 1
 
Syntaxci 10537 Extend class notation to include the complex number i.
class i
 
Syntaxcaddc 10538 Addition on complex numbers.
class +
 
Syntaxcltrr 10539 'Less than' predicate (defined over real subset of complex numbers).
class <
 
Syntaxcmul 10540 Multiplication on complex numbers. The token · is a center dot.
class ·
 
Definitiondf-c 10541 Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 10568. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
ℂ = (R × R)
 
Definitiondf-0 10542 Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
0 = ⟨0R, 0R
 
Definitiondf-1 10543 Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
1 = ⟨1R, 0R
 
Definitiondf-i 10544 Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
i = ⟨0R, 1R
 
Definitiondf-r 10545 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
ℝ = (R × {0R})
 
Definitiondf-add 10546* Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
+ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
 
Definitiondf-mul 10547* Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
· = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩))}
 
Definitiondf-lt 10548* Define 'less than' on the real subset of complex numbers. Proofs should typically use < instead; see df-ltxr 10678. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
< = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
 
Theoremopelcn 10549 Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
(⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴R𝐵R))
 
Theoremopelreal 10550 Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
(⟨𝐴, 0R⟩ ∈ ℝ ↔ 𝐴R)
 
Theoremelreal 10551* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
(𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
 
Theoremelreal2 10552 Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
(𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
 
Theorem0ncn 10553 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. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
¬ ∅ ∈ ℂ
 
Theoremltrelre 10554 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
< ⊆ (ℝ × ℝ)
 
Theoremaddcnsr 10555 Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
 
Theoremmulcnsr 10556 Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨𝐴, 𝐵⟩ · ⟨𝐶, 𝐷⟩) = ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩)
 
Theoremeqresr 10557 Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
𝐴 ∈ V       (⟨𝐴, 0R⟩ = ⟨𝐵, 0R⟩ ↔ 𝐴 = 𝐵)
 
Theoremaddresr 10558 Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
((𝐴R𝐵R) → (⟨𝐴, 0R⟩ + ⟨𝐵, 0R⟩) = ⟨(𝐴 +R 𝐵), 0R⟩)
 
Theoremmulresr 10559 Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
((𝐴R𝐵R) → (⟨𝐴, 0R⟩ · ⟨𝐵, 0R⟩) = ⟨(𝐴 ·R 𝐵), 0R⟩)
 
Theoremltresr 10560 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
(⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ 𝐴 <R 𝐵)
 
Theoremltresr2 10561 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (1st𝐴) <R (1st𝐵)))
 
Theoremdfcnqs 10562 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in R. The trick involves qsid 8359, which shows that the coset of the converse membership relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that is a quotient set, even though it is not (compare df-c 10541), and allows us to reuse some of the equivalence class lemmas we developed for the transition from positive reals to signed reals, etc. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)
ℂ = ((R × R) / E )
 
Theoremaddcnsrec 10563 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 10562 and mulcnsrec 10564. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)
(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → ([⟨𝐴, 𝐵⟩] E + [⟨𝐶, 𝐷⟩] E ) = [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩] E )
 
Theoremmulcnsrec 10564 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 8358, which shows that the coset of the converse membership relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 10562.

Note: This is the last lemma (from which the axioms will be derived) in the construction of real and complex numbers. The construction starts at cnpi 10264. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → ([⟨𝐴, 𝐵⟩] E · [⟨𝐶, 𝐷⟩] E ) = [⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩] E )
 
5.1.2  Final derivation of real and complex number postulates
 
Theoremaxaddf 10565 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 10571. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 10614. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
+ :(ℂ × ℂ)⟶ℂ
 
Theoremaxmulf 10566 Multiplication 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 axmulcl 10573. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 10615. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
· :(ℂ × ℂ)⟶ℂ
 
Theoremaxcnex 10567 The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 12382), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 5176 in later theorems by invoking the axiom ax-cnex 10591 instead of cnexALT 12382. Use cnex 10616 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
ℂ ∈ V
 
Theoremaxresscn 10568 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn 10592. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
ℝ ⊆ ℂ
 
Theoremax1cn 10569 1 is a complex number. Axiom 2 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1cn 10593. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
1 ∈ ℂ
 
Theoremaxicn 10570 i is a complex number. Axiom 3 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-icn 10594. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
i ∈ ℂ
 
Theoremaxaddcl 10571 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcl 10595 be used later. Instead, in most cases use addcl 10617. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
 
Theoremaxaddrcl 10572 Closure law for addition in the real subfield of complex numbers. Axiom 5 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addrcl 10596 be used later. Instead, in most cases use readdcl 10618. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
 
Theoremaxmulcl 10573 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcl 10597 be used later. Instead, in most cases use mulcl 10619. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
 
Theoremaxmulrcl 10574 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulrcl 10598 be used later. Instead, in most cases use remulcl 10620. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
 
Theoremaxmulcom 10575 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcom 10599 be used later. Instead, use mulcom 10621. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremaxaddass 10576 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 9 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 10600 be used later. Instead, use addass 10622. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
 
Theoremaxmulass 10577 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 10601. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
 
Theoremaxdistr 10578 Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-distr 10602 be used later. Instead, use adddi 10624. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 
Theoremaxi2m1 10579 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 10603. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
((i · i) + 1) = 0
 
Theoremax1ne0 10580 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1ne0 10604. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
1 ≠ 0
 
Theoremax1rid 10581 1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, derived from ZF set theory. Weakened from the original axiom in the form of statement in mulid1 10637, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 10605. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
(𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
 
Theoremaxrnegex 10582* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 10606. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
(𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
 
Theoremaxrrecex 10583* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rrecex 10607. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
 
Theoremaxcnre 10584* A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 10608. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
 
Theoremaxpre-lttri 10585 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttri 10710. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 10609. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
 
Theoremaxpre-lttrn 10586 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttrn 10711. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 10610. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
 
Theoremaxpre-ltadd 10587 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axltadd 10712. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 10611. (Contributed by NM, 11-May-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))
 
Theoremaxpre-mulgt0 10588 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axmulgt0 10713. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 10612. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))
 
Theoremaxpre-sup 10589* A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version with ordering on extended reals is axsup 10714. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 10613. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
 
Theoremwuncn 10590 A weak universe containing ω contains the complex number construction. This theorem is construction-dependent in the literal sense, but will also be satisfied by any other reasonable implementation of the complex numbers. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)       (𝜑 → ℂ ∈ 𝑈)
 
5.1.3  Real and complex number postulates restated as axioms
 
Axiomax-cnex 10591 The complex numbers form a set. This axiom is redundant - see cnexALT 12382- but we provide this axiom because the justification theorem axcnex 10567 does not use ax-rep 5176 even though the redundancy proof does. Proofs should normally use cnex 10616 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
ℂ ∈ V
 
Axiomax-resscn 10592 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 10568. (Contributed by NM, 1-Mar-1995.)
ℝ ⊆ ℂ
 
Axiomax-1cn 10593 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 10569. (Contributed by NM, 1-Mar-1995.)
1 ∈ ℂ
 
Axiomax-icn 10594 i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 10570. (Contributed by NM, 1-Mar-1995.)
i ∈ ℂ
 
Axiomax-addcl 10595 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 10571. Proofs should normally use addcl 10617 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
 
Axiomax-addrcl 10596 Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 10572. Proofs should normally use readdcl 10618 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
 
Axiomax-mulcl 10597 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 10573. Proofs should normally use mulcl 10619 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
 
Axiomax-mulrcl 10598 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 10574. Proofs should normally use remulcl 10620 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
 
Axiomax-mulcom 10599 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 10575. Proofs should normally use mulcom 10621 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Axiomax-addass 10600 Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 10576. Proofs should normally use addass 10622 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
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