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	nrex1 11101	The class of signed reals is a set. Note that a shorter proof is possible using qsex 8814 (and not requiring enrer 11100), but it would add a dependency on ax-rep 5284. (Contributed by Mario Carneiro, 17-Nov-2014.) Extract proof from that of axcnex 11184. (Revised by BJ, 4-Feb-2023.) (New usage is discouraged.)
⊢ R ∈ V
Theorem	enrbreq 11102	Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (⟨𝐴, 𝐵⟩ ~R ⟨𝐶, 𝐷⟩ ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)))
Theorem	enreceq 11103	Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) (New usage is discouraged.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)))
Theorem	enrex 11104	The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
⊢ ~R ∈ V
Theorem	ltrelsr 11105	Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
⊢ <R ⊆ (R × R)
Theorem	addcmpblnr 11106	Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) ∧ ((𝐹 ∈ P ∧ 𝐺 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ⟨(𝐴 +P 𝐹), (𝐵 +P 𝐺)⟩ ~R ⟨(𝐶 +P 𝑅), (𝐷 +P 𝑆)⟩))
Theorem	mulcmpblnrlem 11107	Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.)
⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)))))
Theorem	mulcmpblnr 11108	Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.)
⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) ∧ ((𝐹 ∈ P ∧ 𝐺 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ⟨((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)), ((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹))⟩ ~R ⟨((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)), ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅))⟩))
Theorem	prsrlem1 11109*	Decomposing signed reals into positive reals. Lemma for addsrpr 11112 and mulsrpr 11113. (Contributed by Jim Kingdon, 30-Dec-2019.)
⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔))))
Theorem	addsrmo 11110*	There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))
Theorem	mulsrmo 11111*	There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
Theorem	addsrpr 11112	Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )
Theorem	mulsrpr 11113	Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R ·R [⟨𝐶, 𝐷⟩] ~R ) = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R )
Theorem	ltsrpr 11114	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 11115	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 11116	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 11117	The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
⊢ 0R ∈ R
Theorem	1sr 11118	The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
⊢ 1R ∈ R
Theorem	m1r 11119	The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
⊢ -1R ∈ R
Theorem	addclsr 11120	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 11121	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 11122	Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
⊢ dom +R = (R × R)
Theorem	dmmulsr 11123	Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
⊢ dom ·R = (R × R)
Theorem	addcomsr 11124	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 11125	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 11126	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 11127	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 11128	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 11129	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 11130	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 11131	Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
⊢ <R Or R
Theorem	0lt1sr 11132	0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
⊢ 0R <R 1R
Theorem	1ne0sr 11133	1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
⊢ ¬ 1R = 0R
Theorem	0idsr 11134	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 11135	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 11136	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 11137	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 11138	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 11139*	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 11140*	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 11141	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 11142	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 11143	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 11144*	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 11145	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 11146	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 11147*	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 11148*	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 11149*	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 11150	Class of complex numbers.
class ℂ
Syntax	cr 11151	Class of real numbers.
class ℝ
Syntax	cc0 11152	Extend class notation to include the complex number 0.
class 0
Syntax	c1 11153	Extend class notation to include the complex number 1.
class 1
Syntax	ci 11154	Extend class notation to include the complex number i.
class i
Syntax	caddc 11155	Addition on complex numbers.
class +
Syntax	cltrr 11156	'Less than' predicate (defined over real subset of complex numbers).
class <ℝ
Syntax	cmul 11157	Multiplication on complex numbers. The token · is a center dot.
class ·
Definition	df-c 11158	Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 11185. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
⊢ ℂ = (R × R)
Definition	df-0 11159	Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
⊢ 0 = ⟨0R, 0R⟩
Definition	df-1 11160	Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
⊢ 1 = ⟨1R, 0R⟩
Definition	df-i 11161	Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
⊢ i = ⟨0R, 1R⟩
Definition	df-r 11162	Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
⊢ ℝ = (R × {0R})
Definition	df-add 11163*	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 11164*	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 11165*	Define 'less than' on the real subset of complex numbers. Proofs should typically use < instead; see df-ltxr 11297. (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 11166	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 11167	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 11168*	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 11169	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 11170	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 11171	'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
⊢ <ℝ ⊆ (ℝ × ℝ)
Theorem	addcnsr 11172	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 11173	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 11174	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 11175	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 11176	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 11177	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 11178	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 11179	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 8821, 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 11158), 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 11180	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 11179 and mulcnsrec 11181. (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 11181	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 8820, which shows that the coset of the converse membership relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 11179. 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 10881. (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 11182	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 11188. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 11231. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
⊢ + :(ℂ × ℂ)⟶ℂ
Theorem	axmulf 11183	Multiplication is an operation on the complex numbers. This is the construction-dependent version of ax-mulf 11232 and it should not be referenced outside the construction. We generally prefer to develop our theory using the less specific mulcl 11236. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
⊢ · :(ℂ × ℂ)⟶ℂ
Theorem	axcnex 11184	The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 13025), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 5284 in later theorems by invoking Axiom ax-cnex 11208 instead of cnexALT 13025. Use cnex 11233 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
⊢ ℂ ∈ V
Theorem	axresscn 11185	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 11209. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
⊢ ℝ ⊆ ℂ
Theorem	ax1cn 11186	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 11210. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
⊢ 1 ∈ ℂ
Theorem	axicn 11187	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 11211. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
⊢ i ∈ ℂ
Theorem	axaddcl 11188	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 11212 be used later. Instead, in most cases use addcl 11234. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
Theorem	axaddrcl 11189	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 11213 be used later. Instead, in most cases use readdcl 11235. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
Theorem	axmulcl 11190	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 11214 be used later. Instead, in most cases use mulcl 11236. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Theorem	axmulrcl 11191	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 11215 be used later. Instead, in most cases use remulcl 11237. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Theorem	axmulcom 11192	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 11216 be used later. Instead, use mulcom 11238. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	axaddass 11193	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 11217 be used later. Instead, use addass 11239. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Theorem	axmulass 11194	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 11218. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Theorem	axdistr 11195	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 11219 be used later. Instead, use adddi 11241. (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 11196	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 11220. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
⊢ ((i · i) + 1) = 0
Theorem	ax1ne0 11197	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 11221. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
⊢ 1 ≠ 0
Theorem	ax1rid 11198	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 11256, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 11222. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
Theorem	axrnegex 11199*	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 11223. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Theorem	axrrecex 11200*	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 11224. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)