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	axicn 11101	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 11125. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
⊢ i ∈ ℂ
Theorem	axaddcl 11102	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 11126 be used later. Instead, in most cases use addcl 11148. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
Theorem	axaddrcl 11103	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 11127 be used later. Instead, in most cases use readdcl 11149. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
Theorem	axmulcl 11104	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 11128 be used later. Instead, in most cases use mulcl 11150. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Theorem	axmulrcl 11105	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 11129 be used later. Instead, in most cases use remulcl 11151. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Theorem	axmulcom 11106	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 11130 be used later. Instead, use mulcom 11152. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	axaddass 11107	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 11131 be used later. Instead, use addass 11153. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Theorem	axmulass 11108	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 11132. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Theorem	axdistr 11109	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 11133 be used later. Instead, use adddi 11155. (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 11110	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 11134. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
⊢ ((i · i) + 1) = 0
Theorem	ax1ne0 11111	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 11135. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
⊢ 1 ≠ 0
Theorem	ax1rid 11112	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 11172, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 11136. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
Theorem	axrnegex 11113*	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 11137. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Theorem	axrrecex 11114*	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 11138. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
Theorem	axcnre 11115*	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 11139. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Theorem	axpre-lttri 11116	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 11247. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 11140. (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 11117	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 11248. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 11141. (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 11118	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 11249. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 11142. (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 11119	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 11250. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 11143. (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 11120*	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 11251. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 11144. (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 11121	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 11122	The complex numbers form a set. This axiom is redundant - see cnexALT 12980- but we provide this axiom because the justification theorem axcnex 11098 does not use ax-rep 5224 even though the redundancy proof does. Proofs should normally use cnex 11147 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
⊢ ℂ ∈ V
Axiom	ax-resscn 11123	The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by Theorem axresscn 11099. (Contributed by NM, 1-Mar-1995.)
⊢ ℝ ⊆ ℂ
Axiom	ax-1cn 11124	1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by Theorem ax1cn 11100. (Contributed by NM, 1-Mar-1995.)
⊢ 1 ∈ ℂ
Axiom	ax-icn 11125	i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by Theorem axicn 11101. (Contributed by NM, 1-Mar-1995.)
⊢ i ∈ ℂ
Axiom	ax-addcl 11126	Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by Theorem axaddcl 11102. Proofs should normally use addcl 11148 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 11127	Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by Theorem axaddrcl 11103. Proofs should normally use readdcl 11149 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
Axiom	ax-mulcl 11128	Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by Theorem axmulcl 11104. Proofs should normally use mulcl 11150 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Axiom	ax-mulrcl 11129	Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by Theorem axmulrcl 11105. Proofs should normally use remulcl 11151 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Axiom	ax-mulcom 11130	Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by Theorem axmulcom 11106. Proofs should normally use mulcom 11152 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Axiom	ax-addass 11131	Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by Theorem axaddass 11107. Proofs should normally use addass 11153 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 11132	Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11108. Proofs should normally use mulass 11154 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 11133	Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by Theorem axdistr 11109. Proofs should normally use adddi 11155 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 11134	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 11110. (Contributed by NM, 29-Jan-1995.)
⊢ ((i · i) + 1) = 0
Axiom	ax-1ne0 11135	1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by Theorem ax1ne0 11111. (Contributed by NM, 29-Jan-1995.)
⊢ 1 ≠ 0
Axiom	ax-1rid 11136	1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by Theorem ax1rid 11112. Weakened from the original axiom in the form of statement in mulrid 11172, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
Axiom	ax-rnegex 11137*	Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by Theorem axrnegex 11113. (Contributed by Eric Schmidt, 21-May-2007.)
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Axiom	ax-rrecex 11138*	Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by Theorem axrrecex 11114. (Contributed by Eric Schmidt, 11-Apr-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
Axiom	ax-cnre 11139*	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 11115. For naming consistency, use cnre 11171 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 11140	Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by Theorem axpre-lttri 11116. Note: The more general version for extended reals is axlttri 11247. Normally new proofs would use xrlttri 13134. (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 11141	Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by Theorem axpre-lttrn 11117. Note: The more general version for extended reals is axlttrn 11248. Normally new proofs would use lttr 11252. (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 11142	Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by Theorem axpre-ltadd 11118. Normally new proofs would use axltadd 11249. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
Axiom	ax-pre-mulgt0 11143	The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by Theorem axpre-mulgt0 11119. Normally new proofs would use axmulgt0 11250. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))
Axiom	ax-pre-sup 11144*	A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by Theorem axpre-sup 11120. Note: Normally new proofs would use axsup 11251. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
⊢ ((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)))
Axiom	ax-addf 11145	Addition is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first-order or second-order statement (see https://us.metamath.org/downloads/schmidt-cnaxioms.pdf). It may be deleted in the future and should be avoided for new theorems. Instead, the less specific addcl 11148 should be used. Note that uses of ax-addf 11145 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) in place of +, from which this axiom (with the defined operation in place of +) follows as a theorem. This axiom is justified by Theorem axaddf 11096. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)
⊢ + :(ℂ × ℂ)⟶ℂ
Axiom	ax-mulf 11146	Multiplication is an operation on the complex numbers. This axiom tells us that · is defined only on complex numbers which is analogous to the way that other operations are defined, for example see subf 11425 or eff 16101. However, while Metamath can handle this axiom, if we wish to work with weaker complex number axioms, we can avoid it by using the less specific mulcl 11150. Note that uses of ax-mulf 11146 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11161. This axiom is justified by Theorem axmulf 11097. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)
⊢ · :(ℂ × ℂ)⟶ℂ
5.2  Derive the basic properties from the field axioms
5.2.1  Some deductions from the field axioms for complex numbers
Theorem	cnex 11147	Alias for ax-cnex 11122. See also cnexALT 12980. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ ℂ ∈ V
Theorem	addcl 11148	Alias for ax-addcl 11126, for naming consistency with addcli 11181. Use this theorem instead of ax-addcl 11126 or axaddcl 11102. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
Theorem	readdcl 11149	Alias for ax-addrcl 11127, for naming consistency with readdcli 11190. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
Theorem	mulcl 11150	Alias for ax-mulcl 11128, for naming consistency with mulcli 11182. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Theorem	remulcl 11151	Alias for ax-mulrcl 11129, for naming consistency with remulcli 11191. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Theorem	mulcom 11152	Alias for ax-mulcom 11130, for naming consistency with mulcomi 11183. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	addass 11153	Alias for ax-addass 11131, for naming consistency with addassi 11185. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Theorem	mulass 11154	Alias for ax-mulass 11132, for naming consistency with mulassi 11186. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Theorem	adddi 11155	Alias for ax-distr 11133, for naming consistency with adddii 11187. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Theorem	recn 11156	A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
Theorem	reex 11157	The real numbers form a set. See also reexALT 12978. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ ℝ ∈ V
Theorem	reelprrecn 11158	Reals are a subset of the pair of real and complex numbers. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ ℝ ∈ {ℝ, ℂ}
Theorem	cnelprrecn 11159	Complex numbers are a subset of the pair of real and complex numbers . (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ ℂ ∈ {ℝ, ℂ}
Theorem	mpoaddf 11160*	Addition is an operation on complex numbers. Version of ax-addf 11145 using maps-to notation, proved from the axioms of set theory and ax-addcl 11126. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)):(ℂ × ℂ)⟶ℂ
Theorem	mpomulf 11161*	Multiplication is an operation on complex numbers. Version of ax-mulf 11146 using maps-to notation, proved from the axioms of set theory and ax-mulcl 11128. (Contributed by GG, 16-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
Theorem	elimne0 11162	Hypothesis for weak deduction theorem to eliminate 𝐴 ≠ 0. (Contributed by NM, 15-May-1999.)
⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0
Theorem	adddir 11163	Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
Theorem	0cn 11164	Zero is a complex number. See also 0cnALT 11411. (Contributed by NM, 19-Feb-2005.)
⊢ 0 ∈ ℂ
Theorem	0cnd 11165	Zero is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℂ)
Theorem	c0ex 11166	Zero is a set. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ∈ V
Theorem	0elpr01 11167	0 is an element of {0, 1}. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 ∈ {0, 1}
Theorem	1cnd 11168	One is a complex number, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 1 ∈ ℂ)
Theorem	1ex 11169	One is a set. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 1 ∈ V
Theorem	1elpr01 11170	1 is an element of {0, 1}. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 1 ∈ {0, 1}
Theorem	cnre 11171*	Alias for ax-cnre 11139, for naming consistency. (Contributed by NM, 3-Jan-2013.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Theorem	mulrid 11172	The number 1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
Theorem	mullid 11173	Identity law for multiplication. See mulrid 11172 for commuted version. (Contributed by NM, 8-Oct-1999.)
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
Theorem	1re 11174	The number 1 is real. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax-1cn 11124, by exploiting properties of the imaginary unit i. (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
⊢ 1 ∈ ℝ
Theorem	1red 11175	The number 1 is real, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 1 ∈ ℝ)
Theorem	0re 11176	The number 0 is real. Remark: the first step could also be ax-icn 11125. See also 0reALT 11521. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 11-Oct-2022.)
⊢ 0 ∈ ℝ
Theorem	0red 11177	The number 0 is real, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℝ)
Theorem	pr01ssre 11178	The pair {0, 1} is a subset of ℝ. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ {0, 1} ⊆ ℝ
Theorem	mulridi 11179	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 1) = 𝐴
Theorem	mullidi 11180	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (1 · 𝐴) = 𝐴
Theorem	addcli 11181	Closure law for addition. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 + 𝐵) ∈ ℂ
Theorem	mulcli 11182	Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℂ
Theorem	mulcomi 11183	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)
Theorem	mulcomli 11184	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 · 𝐵) = 𝐶    ⇒   ⊢ (𝐵 · 𝐴) = 𝐶
Theorem	addassi 11185	Associative law for addition. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
Theorem	mulassi 11186	Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
Theorem	adddii 11187	Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
Theorem	adddiri 11188	Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))
Theorem	recni 11189	A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	readdcli 11190	Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 + 𝐵) ∈ ℝ
Theorem	remulcli 11191	Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℝ
Theorem	mulridd 11192	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 1) = 𝐴)
Theorem	mullidd 11193	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (1 · 𝐴) = 𝐴)
Theorem	addcld 11194	Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
Theorem	mulcld 11195	Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ)
Theorem	mulcomd 11196	Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	addassd 11197	Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Theorem	mulassd 11198	Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Theorem	adddid 11199	Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Theorem	adddird 11200	Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))