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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ltsrpr 10501 | Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.) |
⊢ ([〈𝐴, 𝐵〉] ~R <R [〈𝐶, 𝐷〉] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶)) | ||
Theorem | gt0srpr 10502 | Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ 𝐵<P 𝐴) | ||
Theorem | 0nsr 10503 | 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 10504 | The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ 0R ∈ R | ||
Theorem | 1sr 10505 | The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ 1R ∈ R | ||
Theorem | m1r 10506 | The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ -1R ∈ R | ||
Theorem | addclsr 10507 | 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 10508 | 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 10509 | Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
⊢ dom +R = (R × R) | ||
Theorem | dmmulsr 10510 | Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
⊢ dom ·R = (R × R) | ||
Theorem | addcomsr 10511 | 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 10512 | 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 10513 | 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 10514 | 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 10515 | 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 10516 | 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 10517 | 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 10518 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.) |
⊢ <R Or R | ||
Theorem | 0lt1sr 10519 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
⊢ 0R <R 1R | ||
Theorem | 1ne0sr 10520 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
⊢ ¬ 1R = 0R | ||
Theorem | 0idsr 10521 | 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 10522 | 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 10523 | 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 10524 | 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 10525 | 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 10526* | 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 10527* | 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 10528 | 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 10529 | 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 10530 | 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 10531* | 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 10532 | 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 10533 | 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 10534* | 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 10535* | 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 10536* | 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 10537 | Class of complex numbers. |
class ℂ | ||
Syntax | cr 10538 | Class of real numbers. |
class ℝ | ||
Syntax | cc0 10539 | Extend class notation to include the complex number 0. |
class 0 | ||
Syntax | c1 10540 | Extend class notation to include the complex number 1. |
class 1 | ||
Syntax | ci 10541 | Extend class notation to include the complex number i. |
class i | ||
Syntax | caddc 10542 | Addition on complex numbers. |
class + | ||
Syntax | cltrr 10543 | 'Less than' predicate (defined over real subset of complex numbers). |
class <ℝ | ||
Syntax | cmul 10544 | Multiplication on complex numbers. The token · is a center dot. |
class · | ||
Definition | df-c 10545 | Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 10572. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ ℂ = (R × R) | ||
Definition | df-0 10546 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ 0 = 〈0R, 0R〉 | ||
Definition | df-1 10547 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ 1 = 〈1R, 0R〉 | ||
Definition | df-i 10548 | Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ i = 〈0R, 1R〉 | ||
Definition | df-r 10549 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ ℝ = (R × {0R}) | ||
Definition | df-add 10550* | 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 10551* | 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 10552* | Define 'less than' on the real subset of complex numbers. Proofs should typically use < instead; see df-ltxr 10682. (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 10553 | 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 10554 | 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 10555* | 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 10556 | 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 10557 | 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 10558 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ <ℝ ⊆ (ℝ × ℝ) | ||
Theorem | addcnsr 10559 | 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 10560 | 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 10561 | 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 10562 | 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 10563 | 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 10564 | 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 10565 | 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 10566 | Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 8365, 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 10545), 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 10567 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 10566 and mulcnsrec 10568. (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 10568 |
Technical trick to permit re-use of some equivalence class lemmas for
operation laws. The trick involves ecid 8364,
which shows that the coset of
the converse membership relation (which is not an equivalence relation)
leaves a set unchanged. See also dfcnqs 10566.
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 10268. (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 10569 | 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 10575. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 10618. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
⊢ + :(ℂ × ℂ)⟶ℂ | ||
Theorem | axmulf 10570 | 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 10577. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 10619. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
⊢ · :(ℂ × ℂ)⟶ℂ | ||
Theorem | axcnex 10571 | The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 12388), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 5192 in later theorems by invoking the axiom ax-cnex 10595 instead of cnexALT 12388. Use cnex 10620 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
⊢ ℂ ∈ V | ||
Theorem | axresscn 10572 | 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 10596. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
⊢ ℝ ⊆ ℂ | ||
Theorem | ax1cn 10573 | 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 10597. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.) |
⊢ 1 ∈ ℂ | ||
Theorem | axicn 10574 | 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 10598. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
⊢ i ∈ ℂ | ||
Theorem | axaddcl 10575 | 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 10599 be used later. Instead, in most cases use addcl 10621. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) | ||
Theorem | axaddrcl 10576 | 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 10600 be used later. Instead, in most cases use readdcl 10622. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | ||
Theorem | axmulcl 10577 | 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 10601 be used later. Instead, in most cases use mulcl 10623. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) | ||
Theorem | axmulrcl 10578 | 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 10602 be used later. Instead, in most cases use remulcl 10624. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | ||
Theorem | axmulcom 10579 | 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 10603 be used later. Instead, use mulcom 10625. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | axaddass 10580 | 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 10604 be used later. Instead, use addass 10626. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | ||
Theorem | axmulass 10581 | 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 10605. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | ||
Theorem | axdistr 10582 | 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 10606 be used later. Instead, use adddi 10628. (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 10583 | 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 10607. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
⊢ ((i · i) + 1) = 0 | ||
Theorem | ax1ne0 10584 | 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 10608. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.) |
⊢ 1 ≠ 0 | ||
Theorem | ax1rid 10585 | 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 10641, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 10609. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | ||
Theorem | axrnegex 10586* | 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 10610. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | ||
Theorem | axrrecex 10587* | 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 10611. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) | ||
Theorem | axcnre 10588* | 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 10612. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | ||
Theorem | axpre-lttri 10589 | 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 10714. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 10613. (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 10590 | 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 10715. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 10614. (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 10591 | 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 10716. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 10615. (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 10592 | 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 10717. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 10616. (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 10593* | 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 10718. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 10617. (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 10594 | 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 10595 | The complex numbers form a set. This axiom is redundant - see cnexALT 12388- but we provide this axiom because the justification theorem axcnex 10571 does not use ax-rep 5192 even though the redundancy proof does. Proofs should normally use cnex 10620 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
⊢ ℂ ∈ V | ||
Axiom | ax-resscn 10596 | The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 10572. (Contributed by NM, 1-Mar-1995.) |
⊢ ℝ ⊆ ℂ | ||
Axiom | ax-1cn 10597 | 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 10573. (Contributed by NM, 1-Mar-1995.) |
⊢ 1 ∈ ℂ | ||
Axiom | ax-icn 10598 | i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 10574. (Contributed by NM, 1-Mar-1995.) |
⊢ i ∈ ℂ | ||
Axiom | ax-addcl 10599 | Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 10575. Proofs should normally use addcl 10621 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 10600 | Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 10576. Proofs should normally use readdcl 10622 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) |
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