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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 0nsr 10501 | The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.) |
⊢ ¬ ∅ ∈ R | ||
Theorem | 0r 10502 | The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ 0R ∈ R | ||
Theorem | 1sr 10503 | The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ 1R ∈ R | ||
Theorem | m1r 10504 | The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ -1R ∈ R | ||
Theorem | addclsr 10505 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ R) | ||
Theorem | mulclsr 10506 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) | ||
Theorem | dmaddsr 10507 | Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
⊢ dom +R = (R × R) | ||
Theorem | dmmulsr 10508 | Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
⊢ dom ·R = (R × R) | ||
Theorem | addcomsr 10509 | 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) | ||
Theorem | addasssr 10510 | 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)) | ||
Theorem | mulcomsr 10511 | 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) | ||
Theorem | mulasssr 10512 | 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)) | ||
Theorem | distrsr 10513 | 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)) | ||
Theorem | m1p1sr 10514 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
⊢ (-1R +R 1R) = 0R | ||
Theorem | m1m1sr 10515 | 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 | ||
Theorem | ltsosr 10516 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.) |
⊢ <R Or R | ||
Theorem | 0lt1sr 10517 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
⊢ 0R <R 1R | ||
Theorem | 1ne0sr 10518 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
⊢ ¬ 1R = 0R | ||
Theorem | 0idsr 10519 | The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) | ||
Theorem | 1idsr 10520 | 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴) | ||
Theorem | 00sr 10521 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | ||
Theorem | ltasr 10522 | Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
⊢ (𝐶 ∈ R → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵))) | ||
Theorem | pn0sr 10523 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) | ||
Theorem | negexsr 10524* | 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.) |
⊢ (𝐴 ∈ R → ∃𝑥 ∈ R (𝐴 +R 𝑥) = 0R) | ||
Theorem | recexsrlem 10525* | 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 → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R) | ||
Theorem | addgt0sr 10526 | 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)) | ||
Theorem | mulgt0sr 10527 | 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)) | ||
Theorem | sqgt0sr 10528 | The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ R ∧ 𝐴 ≠ 0R) → 0R <R (𝐴 ·R 𝐴)) | ||
Theorem | recexsr 10529* | 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.) |
⊢ ((𝐴 ∈ R ∧ 𝐴 ≠ 0R) → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R) | ||
Theorem | mappsrpr 10530 | 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.) |
⊢ 𝐶 ∈ R ⇒ ⊢ ((𝐶 +R -1R) <R (𝐶 +R [〈𝐴, 1P〉] ~R ) ↔ 𝐴 ∈ P) | ||
Theorem | ltpsrpr 10531 | 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.) |
⊢ 𝐶 ∈ R ⇒ ⊢ ((𝐶 +R [〈𝐴, 1P〉] ~R ) <R (𝐶 +R [〈𝐵, 1P〉] ~R ) ↔ 𝐴<P 𝐵) | ||
Theorem | map2psrpr 10532* | Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
⊢ 𝐶 ∈ R ⇒ ⊢ ((𝐶 +R -1R) <R 𝐴 ↔ ∃𝑥 ∈ P (𝐶 +R [〈𝑥, 1P〉] ~R ) = 𝐴) | ||
Theorem | supsrlem 10533* | 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} & ⊢ 𝐶 ∈ R ⇒ ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) | ||
Theorem | supsr 10534* | 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 ≠ ∅ ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) | ||
Syntax | cc 10535 | Class of complex numbers. |
class ℂ | ||
Syntax | cr 10536 | Class of real numbers. |
class ℝ | ||
Syntax | cc0 10537 | Extend class notation to include the complex number 0. |
class 0 | ||
Syntax | c1 10538 | Extend class notation to include the complex number 1. |
class 1 | ||
Syntax | ci 10539 | Extend class notation to include the complex number i. |
class i | ||
Syntax | caddc 10540 | Addition on complex numbers. |
class + | ||
Syntax | cltrr 10541 | 'Less than' predicate (defined over real subset of complex numbers). |
class <ℝ | ||
Syntax | cmul 10542 | Multiplication on complex numbers. The token · is a center dot. |
class · | ||
Definition | df-c 10543 | Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 10570. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ ℂ = (R × R) | ||
Definition | df-0 10544 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ 0 = 〈0R, 0R〉 | ||
Definition | df-1 10545 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ 1 = 〈1R, 0R〉 | ||
Definition | df-i 10546 | Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ i = 〈0R, 1R〉 | ||
Definition | df-r 10547 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ ℝ = (R × {0R}) | ||
Definition | df-add 10548* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
⊢ + = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))} | ||
Definition | df-mul 10549* | Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ · = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))〉))} | ||
Definition | df-lt 10550* | Define 'less than' on the real subset of complex numbers. Proofs should typically use < instead; see df-ltxr 10680. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ <ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} | ||
Theorem | opelcn 10551 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
⊢ (〈𝐴, 𝐵〉 ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R)) | ||
Theorem | opelreal 10552 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 𝐴 ∈ R) | ||
Theorem | elreal 10553* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | ||
Theorem | elreal2 10554 | Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) | ||
Theorem | 0ncn 10555 | 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 | ||
Theorem | ltrelre 10556 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ <ℝ ⊆ (ℝ × ℝ) | ||
Theorem | addcnsr 10557 | Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉) = 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉) | ||
Theorem | mulcnsr 10558 | Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (〈𝐴, 𝐵〉 · 〈𝐶, 𝐷〉) = 〈((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))〉) | ||
Theorem | eqresr 10559 | 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) | ||
Theorem | addresr 10560 | Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 + 〈𝐵, 0R〉) = 〈(𝐴 +R 𝐵), 0R〉) | ||
Theorem | mulresr 10561 | Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 · 〈𝐵, 0R〉) = 〈(𝐴 ·R 𝐵), 0R〉) | ||
Theorem | ltresr 10562 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ (〈𝐴, 0R〉 <ℝ 〈𝐵, 0R〉 ↔ 𝐴 <R 𝐵) | ||
Theorem | ltresr2 10563 | 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 <ℝ 𝐵 ↔ (1st ‘𝐴) <R (1st ‘𝐵))) | ||
Theorem | dfcnqs 10564 | Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 8363, 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 ℂ is a quotient set, even though it is not (compare df-c 10543), 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 ) | ||
Theorem | addcnsrec 10565 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 10564 and mulcnsrec 10566. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([〈𝐴, 𝐵〉]◡ E + [〈𝐶, 𝐷〉]◡ E ) = [〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉]◡ E ) | ||
Theorem | mulcnsrec 10566 |
Technical trick to permit re-use of some equivalence class lemmas for
operation laws. The trick involves ecid 8362,
which shows that the coset of
the converse membership relation (which is not an equivalence relation)
leaves a set unchanged. See also dfcnqs 10564.
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 10266. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([〈𝐴, 𝐵〉]◡ E · [〈𝐶, 𝐷〉]◡ E ) = [〈((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))〉]◡ E ) | ||
Theorem | axaddf 10567 | 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 10573. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 10616. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
⊢ + :(ℂ × ℂ)⟶ℂ | ||
Theorem | axmulf 10568 | 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 10575. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 10617. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
⊢ · :(ℂ × ℂ)⟶ℂ | ||
Theorem | axcnex 10569 | The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 12386), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 5190 in later theorems by invoking the axiom ax-cnex 10593 instead of cnexALT 12386. Use cnex 10618 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
⊢ ℂ ∈ V | ||
Theorem | axresscn 10570 | 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 10594. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
⊢ ℝ ⊆ ℂ | ||
Theorem | ax1cn 10571 | 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 10595. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.) |
⊢ 1 ∈ ℂ | ||
Theorem | axicn 10572 | 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 10596. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
⊢ i ∈ ℂ | ||
Theorem | axaddcl 10573 | 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 10597 be used later. Instead, in most cases use addcl 10619. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) | ||
Theorem | axaddrcl 10574 | 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 10598 be used later. Instead, in most cases use readdcl 10620. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | ||
Theorem | axmulcl 10575 | 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 10599 be used later. Instead, in most cases use mulcl 10621. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) | ||
Theorem | axmulrcl 10576 | 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 10600 be used later. Instead, in most cases use remulcl 10622. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | ||
Theorem | axmulcom 10577 | 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 10601 be used later. Instead, use mulcom 10623. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | axaddass 10578 | 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 10602 be used later. Instead, use addass 10624. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | ||
Theorem | axmulass 10579 | 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 10603. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | ||
Theorem | axdistr 10580 | 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 10604 be used later. Instead, use adddi 10626. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) | ||
Theorem | axi2m1 10581 | 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 10605. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
⊢ ((i · i) + 1) = 0 | ||
Theorem | ax1ne0 10582 | 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 10606. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.) |
⊢ 1 ≠ 0 | ||
Theorem | ax1rid 10583 | 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 10639, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 10607. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | ||
Theorem | axrnegex 10584* | 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 10608. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | ||
Theorem | axrrecex 10585* | 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 10609. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) | ||
Theorem | axcnre 10586* | 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 10610. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | ||
Theorem | axpre-lttri 10587 | 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 10712. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 10611. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) | ||
Theorem | axpre-lttrn 10588 | 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 10713. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 10612. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶)) | ||
Theorem | axpre-ltadd 10589 | 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 10714. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 10613. (Contributed by NM, 11-May-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))) | ||
Theorem | axpre-mulgt0 10590 | 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 10715. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 10614. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵))) | ||
Theorem | axpre-sup 10591* | 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 10716. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 10615. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))) | ||
Theorem | wuncn 10592 | 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.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) ⇒ ⊢ (𝜑 → ℂ ∈ 𝑈) | ||
Axiom | ax-cnex 10593 | The complex numbers form a set. This axiom is redundant - see cnexALT 12386- but we provide this axiom because the justification theorem axcnex 10569 does not use ax-rep 5190 even though the redundancy proof does. Proofs should normally use cnex 10618 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
⊢ ℂ ∈ V | ||
Axiom | ax-resscn 10594 | The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 10570. (Contributed by NM, 1-Mar-1995.) |
⊢ ℝ ⊆ ℂ | ||
Axiom | ax-1cn 10595 | 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 10571. (Contributed by NM, 1-Mar-1995.) |
⊢ 1 ∈ ℂ | ||
Axiom | ax-icn 10596 | i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 10572. (Contributed by NM, 1-Mar-1995.) |
⊢ i ∈ ℂ | ||
Axiom | ax-addcl 10597 | Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 10573. Proofs should normally use addcl 10619 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) | ||
Axiom | ax-addrcl 10598 | Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 10574. Proofs should normally use readdcl 10620 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | ||
Axiom | ax-mulcl 10599 | Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 10575. Proofs should normally use mulcl 10621 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) | ||
Axiom | ax-mulrcl 10600 | Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 10576. Proofs should normally use remulcl 10622 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) |
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