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Theorem List for Metamath Proof Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulasssr 10501 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 10502 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 10503 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
(-1R +R 1R) = 0R
 
Theoremm1m1sr 10504 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 10505 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
<R Or R
 
Theorem0lt1sr 10506 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
0R <R 1R
 
Theorem1ne0sr 10507 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
¬ 1R = 0R
 
Theorem0idsr 10508 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 10509 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
(𝐴R → (𝐴 ·R 1R) = 𝐴)
 
Theorem00sr 10510 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
(𝐴R → (𝐴 ·R 0R) = 0R)
 
Theoremltasr 10511 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 10512 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 10513* 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 10514* 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 10515 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 10516 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 10517 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 10518* 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 10519 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 10520 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 10521* 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 10522* 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 10523* 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 10524 Class of complex numbers.
class
 
Syntaxcr 10525 Class of real numbers.
class
 
Syntaxcc0 10526 Extend class notation to include the complex number 0.
class 0
 
Syntaxc1 10527 Extend class notation to include the complex number 1.
class 1
 
Syntaxci 10528 Extend class notation to include the complex number i.
class i
 
Syntaxcaddc 10529 Addition on complex numbers.
class +
 
Syntaxcltrr 10530 'Less than' predicate (defined over real subset of complex numbers).
class <
 
Syntaxcmul 10531 Multiplication on complex numbers. The token · is a center dot.
class ·
 
Definitiondf-c 10532 Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 10559. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
ℂ = (R × R)
 
Definitiondf-0 10533 Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
0 = ⟨0R, 0R
 
Definitiondf-1 10534 Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
1 = ⟨1R, 0R
 
Definitiondf-i 10535 Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
i = ⟨0R, 1R
 
Definitiondf-r 10536 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
ℝ = (R × {0R})
 
Definitiondf-add 10537* 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 10538* 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 10539* Define 'less than' on the real subset of complex numbers. Proofs should typically use < instead; see df-ltxr 10669. (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 10540 Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
(⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴R𝐵R))
 
Theoremopelreal 10541 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 10542* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
(𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
 
Theoremelreal2 10543 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 10544 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 10545 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
< ⊆ (ℝ × ℝ)
 
Theoremaddcnsr 10546 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 10547 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 10548 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 10549 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 10550 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 10551 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 10552 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 10553 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in R. The trick involves qsid 8353, 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 10532), 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 10554 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 10553 and mulcnsrec 10555. (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 10555 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 8352, which shows that the coset of the converse membership relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 10553.

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 10255. (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 10556 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 10562. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 10605. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
+ :(ℂ × ℂ)⟶ℂ
 
Theoremaxmulf 10557 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 10564. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 10606. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
· :(ℂ × ℂ)⟶ℂ
 
Theoremaxcnex 10558 The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 12375), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 5182 in later theorems by invoking the axiom ax-cnex 10582 instead of cnexALT 12375. Use cnex 10607 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
ℂ ∈ V
 
Theoremaxresscn 10559 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 10583. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
ℝ ⊆ ℂ
 
Theoremax1cn 10560 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 10584. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
1 ∈ ℂ
 
Theoremaxicn 10561 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 10585. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
i ∈ ℂ
 
Theoremaxaddcl 10562 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 10586 be used later. Instead, in most cases use addcl 10608. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
 
Theoremaxaddrcl 10563 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 10587 be used later. Instead, in most cases use readdcl 10609. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
 
Theoremaxmulcl 10564 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 10588 be used later. Instead, in most cases use mulcl 10610. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
 
Theoremaxmulrcl 10565 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 10589 be used later. Instead, in most cases use remulcl 10611. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
 
Theoremaxmulcom 10566 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 10590 be used later. Instead, use mulcom 10612. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremaxaddass 10567 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 10591 be used later. Instead, use addass 10613. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
 
Theoremaxmulass 10568 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 10592. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
 
Theoremaxdistr 10569 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 10593 be used later. Instead, use adddi 10615. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 
Theoremaxi2m1 10570 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 10594. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
((i · i) + 1) = 0
 
Theoremax1ne0 10571 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 10595. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
1 ≠ 0
 
Theoremax1rid 10572 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 10628, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 10596. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
(𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
 
Theoremaxrnegex 10573* 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 10597. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
(𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
 
Theoremaxrrecex 10574* 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 10598. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
 
Theoremaxcnre 10575* 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 10599. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
 
Theoremaxpre-lttri 10576 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 10701. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 10600. (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 10577 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 10702. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 10601. (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 10578 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 10703. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 10602. (Contributed by NM, 11-May-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))
 
Theoremaxpre-mulgt0 10579 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 10704. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 10603. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))
 
Theoremaxpre-sup 10580* 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 10705. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 10604. (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 10581 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 10582 The complex numbers form a set. This axiom is redundant - see cnexALT 12375- but we provide this axiom because the justification theorem axcnex 10558 does not use ax-rep 5182 even though the redundancy proof does. Proofs should normally use cnex 10607 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
ℂ ∈ V
 
Axiomax-resscn 10583 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 10559. (Contributed by NM, 1-Mar-1995.)
ℝ ⊆ ℂ
 
Axiomax-1cn 10584 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 10560. (Contributed by NM, 1-Mar-1995.)
1 ∈ ℂ
 
Axiomax-icn 10585 i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 10561. (Contributed by NM, 1-Mar-1995.)
i ∈ ℂ
 
Axiomax-addcl 10586 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 10562. Proofs should normally use addcl 10608 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 10587 Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 10563. Proofs should normally use readdcl 10609 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
 
Axiomax-mulcl 10588 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 10564. Proofs should normally use mulcl 10610 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
 
Axiomax-mulrcl 10589 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 10565. Proofs should normally use remulcl 10611 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
 
Axiomax-mulcom 10590 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 10566. Proofs should normally use mulcom 10612 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Axiomax-addass 10591 Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 10567. Proofs should normally use addass 10613 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
 
Axiomax-mulass 10592 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 10568. Proofs should normally use mulass 10614 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
 
Axiomax-distr 10593 Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 10569. Proofs should normally use adddi 10615 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 
Axiomax-i2m1 10594 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, justified by theorem axi2m1 10570. (Contributed by NM, 29-Jan-1995.)
((i · i) + 1) = 0
 
Axiomax-1ne0 10595 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 10571. (Contributed by NM, 29-Jan-1995.)
1 ≠ 0
 
Axiomax-1rid 10596 1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 10572. Weakened from the original axiom in the form of statement in mulid1 10628, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)
(𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
 
Axiomax-rnegex 10597* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 10573. (Contributed by Eric Schmidt, 21-May-2007.)
(𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
 
Axiomax-rrecex 10598* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 10574. (Contributed by Eric Schmidt, 11-Apr-2007.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
 
Axiomax-cnre 10599* 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, justified by theorem axcnre 10575. For naming consistency, use cnre 10627 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
 
Axiomax-pre-lttri 10600 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 10576. Note: The more general version for extended reals is axlttri 10701. Normally new proofs would use xrlttri 12522. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
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