P. 112 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 11101-11200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	m1p1sr 11101	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 11102	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 11103	Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
⊢ <R Or R
Theorem	0lt1sr 11104	0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
⊢ 0R <R 1R
Theorem	1ne0sr 11105	1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
⊢ ¬ 1R = 0R
Theorem	0idsr 11106	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 11107	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 11108	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 11109	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 11110	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 11111*	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 11112*	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 11113	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 11114	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 11115	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 11116*	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 11117	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 11118	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 11119*	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 11120*	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 11121*	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 11122	Class of complex numbers.
class ℂ
Syntax	cr 11123	Class of real numbers.
class ℝ
Syntax	cc0 11124	Extend class notation to include the complex number 0.
class 0
Syntax	c1 11125	Extend class notation to include the complex number 1.
class 1
Syntax	ci 11126	Extend class notation to include the complex number i.
class i
Syntax	caddc 11127	Addition on complex numbers.
class +
Syntax	cltrr 11128	'Less than' predicate (defined over real subset of complex numbers).
class <ℝ
Syntax	cmul 11129	Multiplication on complex numbers. The token · is a center dot.
class ·
Definition	df-c 11130	Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 11157. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
⊢ ℂ = (R × R)
Definition	df-0 11131	Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
⊢ 0 = ⟨0R, 0R⟩
Definition	df-1 11132	Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
⊢ 1 = ⟨1R, 0R⟩
Definition	df-i 11133	Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
⊢ i = ⟨0R, 1R⟩
Definition	df-r 11134	Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
⊢ ℝ = (R × {0R})
Definition	df-add 11135*	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 11136*	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 11137*	Define 'less than' on the real subset of complex numbers. Proofs should typically use < instead; see df-ltxr 11272. (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 11138	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 11139	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 11140*	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 11141	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 11142	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 11143	'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
⊢ <ℝ ⊆ (ℝ × ℝ)
Theorem	addcnsr 11144	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 11145	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 11146	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 11147	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 11148	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 11149	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 11150	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 11151	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 8781, 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 11130), and allows 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 11152	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 11151 and mulcnsrec 11153. (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 11153	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 8780, which shows that the coset of the converse membership relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 11151. 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 10853. (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 )
5.1.2  Final derivation of real and complex number postulates
Theorem	axaddf 11154	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 11160. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 11203. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
⊢ + :(ℂ × ℂ)⟶ℂ
Theorem	axmulf 11155	Multiplication is an operation on the complex numbers. This is the construction-dependent version of ax-mulf 11204 and it should not be referenced outside the construction. We generally prefer to develop our theory using the less specific mulcl 11208. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
⊢ · :(ℂ × ℂ)⟶ℂ
Theorem	axcnex 11156	The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 13036), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 5232 in later theorems by invoking Axiom ax-cnex 11180 instead of cnexALT 13036. Use cnex 11205 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
⊢ ℂ ∈ V
Theorem	axresscn 11157	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 11181. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
⊢ ℝ ⊆ ℂ
Theorem	ax1cn 11158	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 11182. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
⊢ 1 ∈ ℂ
Theorem	axicn 11159	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 11183. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
⊢ i ∈ ℂ
Theorem	axaddcl 11160	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 11184 be used later. Instead, in most cases use addcl 11206. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
Theorem	axaddrcl 11161	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 11185 be used later. Instead, in most cases use readdcl 11207. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
Theorem	axmulcl 11162	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 11186 be used later. Instead, in most cases use mulcl 11208. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Theorem	axmulrcl 11163	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 11187 be used later. Instead, in most cases use remulcl 11209. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Theorem	axmulcom 11164	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 11188 be used later. Instead, use mulcom 11210. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	axaddass 11165	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 11189 be used later. Instead, use addass 11211. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Theorem	axmulass 11166	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 11190. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Theorem	axdistr 11167	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 11191 be used later. Instead, use adddi 11213. (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 11168	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 11192. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
⊢ ((i · i) + 1) = 0
Theorem	ax1ne0 11169	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 11193. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
⊢ 1 ≠ 0
Theorem	ax1rid 11170	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 mulrid 11230, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 11194. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
Theorem	axrnegex 11171*	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 11195. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Theorem	axrrecex 11172*	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 11196. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
Theorem	axcnre 11173*	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 11197. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Theorem	axpre-lttri 11174	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 11305. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 11198. (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 11175	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 11306. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 11199. (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 11176	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 11307. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 11200. (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 11177	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 11308. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 11201. (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 11178*	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 11309. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 11202. (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 11179	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)
5.1.3  Real and complex number postulates restated as axioms
Axiom	ax-cnex 11180	The complex numbers form a set. This axiom is redundant - see cnexALT 13036- but we provide this axiom because the justification theorem axcnex 11156 does not use ax-rep 5232 even though the redundancy proof does. Proofs should normally use cnex 11205 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
⊢ ℂ ∈ V
Axiom	ax-resscn 11181	The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by Theorem axresscn 11157. (Contributed by NM, 1-Mar-1995.)
⊢ ℝ ⊆ ℂ
Axiom	ax-1cn 11182	1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by Theorem ax1cn 11158. (Contributed by NM, 1-Mar-1995.)
⊢ 1 ∈ ℂ
Axiom	ax-icn 11183	i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by Theorem axicn 11159. (Contributed by NM, 1-Mar-1995.)
⊢ i ∈ ℂ
Axiom	ax-addcl 11184	Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by Theorem axaddcl 11160. Proofs should normally use addcl 11206 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 11185	Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by Theorem axaddrcl 11161. Proofs should normally use readdcl 11207 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
Axiom	ax-mulcl 11186	Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by Theorem axmulcl 11162. Proofs should normally use mulcl 11208 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Axiom	ax-mulrcl 11187	Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by Theorem axmulrcl 11163. Proofs should normally use remulcl 11209 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Axiom	ax-mulcom 11188	Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by Theorem axmulcom 11164. Proofs should normally use mulcom 11210 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Axiom	ax-addass 11189	Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by Theorem axaddass 11165. Proofs should normally use addass 11211 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Axiom	ax-mulass 11190	Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11166. Proofs should normally use mulass 11212 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Axiom	ax-distr 11191	Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by Theorem axdistr 11167. Proofs should normally use adddi 11213 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Axiom	ax-i2m1 11192	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 11168. (Contributed by NM, 29-Jan-1995.)
⊢ ((i · i) + 1) = 0
Axiom	ax-1ne0 11193	1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by Theorem ax1ne0 11169. (Contributed by NM, 29-Jan-1995.)
⊢ 1 ≠ 0
Axiom	ax-1rid 11194	1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by Theorem ax1rid 11170. Weakened from the original axiom in the form of statement in mulrid 11230, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
Axiom	ax-rnegex 11195*	Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by Theorem axrnegex 11171. (Contributed by Eric Schmidt, 21-May-2007.)
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Axiom	ax-rrecex 11196*	Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by Theorem axrrecex 11172. (Contributed by Eric Schmidt, 11-Apr-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
Axiom	ax-cnre 11197*	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 11173. For naming consistency, use cnre 11229 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Axiom	ax-pre-lttri 11198	Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by Theorem axpre-lttri 11174. Note: The more general version for extended reals is axlttri 11305. Normally new proofs would use xrlttri 13190. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))
Axiom	ax-pre-lttrn 11199	Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by Theorem axpre-lttrn 11175. Note: The more general version for extended reals is axlttrn 11306. Normally new proofs would use lttr 11310. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))
Axiom	ax-pre-ltadd 11200	Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by Theorem axpre-ltadd 11176. Normally new proofs would use axltadd 11307. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))