P. 110 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10901-11000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elnpi 10901*	Membership in positive reals. (Contributed by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
Theorem	prn0 10902	A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ P → 𝐴 ≠ ∅)
Theorem	prpssnq 10903	A positive real is a subset of the positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
Theorem	elprnq 10904	A positive real is a set of positive fractions. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)
Theorem	0npr 10905	The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
⊢ ¬ ∅ ∈ P
Theorem	prcdnq 10906	A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → (𝐶 <Q 𝐵 → 𝐶 ∈ 𝐴))
Theorem	prub 10907	A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ Q) → (¬ 𝐶 ∈ 𝐴 → 𝐵 <Q 𝐶))
Theorem	prnmax 10908*	A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <Q 𝑥)
Theorem	npomex 10909	A simplifying observation, and an indication of why any attempt to develop a theory of the real numbers without the Axiom of Infinity is doomed to failure: since every member of P is an infinite set, the negation of Infinity implies that P, and hence ℝ, is empty. (Note that this proof, which used the fact that Dedekind cuts have no maximum, could just as well have used that they have no minimum, since they are downward-closed by prcdnq 10906 and nsmallnq 10890). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
⊢ (𝐴 ∈ P → ω ∈ V)
Theorem	prnmadd 10910*	A positive real has no largest member. Addition version. (Contributed by NM, 7-Apr-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥(𝐵 +Q 𝑥) ∈ 𝐴)
Theorem	ltrelpr 10911	Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
⊢ <P ⊆ (P × P)
Theorem	genpv 10912*	Value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)})
Theorem	genpelv 10913*	Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ)))
Theorem	genpprecl 10914*	Pre-closure law for general operation on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶𝐺𝐷) ∈ (𝐴𝐹𝐵)))
Theorem	genpdm 10915*	Domain of general operation on positive reals. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ dom 𝐹 = (P × P)
Theorem	genpn0 10916*	The result of an operation on positive reals is not empty. (Contributed by NM, 28-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∅ ⊊ (𝐴𝐹𝐵))
Theorem	genpss 10917*	The result of an operation on positive reals is a subset of the positive fractions. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ⊆ Q)
Theorem	genpnnp 10918*	The result of an operation on positive reals is different from the set of positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ (𝑧 ∈ Q → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ¬ (𝐴𝐹𝐵) = Q)
Theorem	genpcd 10919*	Downward closure of an operation on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔𝐺ℎ) → 𝑥 ∈ (𝐴𝐹𝐵)))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵))))
Theorem	genpnmax 10920*	An operation on positive reals has no largest member. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ (𝑣 ∈ Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤)))    &   ⊢ (𝑧𝐺𝑤) = (𝑤𝐺𝑧)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
Theorem	genpcl 10921*	Closure of an operation on reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ (ℎ ∈ Q → (𝑓 <Q 𝑔 ↔ (ℎ𝐺𝑓) <Q (ℎ𝐺𝑔)))    &   ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)    &   ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔𝐺ℎ) → 𝑥 ∈ (𝐴𝐹𝐵)))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ P)
Theorem	genpass 10922*	Associativity of an operation on reals. (Contributed by NM, 18-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ dom 𝐹 = (P × P)    &   ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P)    &   ⊢ ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ))    ⇒   ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
Theorem	plpv 10923*	Value of addition on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +Q 𝑧)})
Theorem	mpv 10924*	Value of multiplication on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ·Q 𝑧)})
Theorem	dmplp 10925	Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
⊢ dom +P = (P × P)
Theorem	dmmp 10926	Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
⊢ dom ·P = (P × P)
Theorem	nqpr 10927*	The canonical embedding of the rationals into the reals. (Contributed by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ Q → {𝑥 ∣ 𝑥 <Q 𝐴} ∈ P)
Theorem	1pr 10928	The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 1P ∈ P
Theorem	addclprlem1 10929	Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴))
Theorem	addclprlem2 10930*	Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → 𝑥 ∈ (𝐴 +P 𝐵)))
Theorem	addclpr 10931	Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
Theorem	mulclprlem 10932*	Lemma to prove downward closure in positive real multiplication. Part of proof of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 14-Mar-1996.) (New usage is discouraged.)
⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 ·Q ℎ) → 𝑥 ∈ (𝐴 ·P 𝐵)))
Theorem	mulclpr 10933	Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
Theorem	addcompr 10934	Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
⊢ (𝐴 +P 𝐵) = (𝐵 +P 𝐴)
Theorem	addasspr 10935	Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
⊢ ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶))
Theorem	mulcompr 10936	Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
⊢ (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴)
Theorem	mulasspr 10937	Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
⊢ ((𝐴 ·P 𝐵) ·P 𝐶) = (𝐴 ·P (𝐵 ·P 𝐶))
Theorem	distrlem1pr 10938	Lemma for distributive law for positive reals. (Contributed by NM, 1-May-1996.) (Revised by Mario Carneiro, 13-Jun-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ⊆ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
Theorem	distrlem4pr 10939*	Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑓 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
Theorem	distrlem5pr 10940	Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ⊆ (𝐴 ·P (𝐵 +P 𝐶)))
Theorem	distrpr 10941	Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
⊢ (𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))
Theorem	1idpr 10942	1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ P → (𝐴 ·P 1P) = 𝐴)
Theorem	ltprord 10943	Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ 𝐴 ⊊ 𝐵))
Theorem	psslinpr 10944	Proper subset is a linear ordering on positive reals. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴))
Theorem	ltsopr 10945	Positive real 'less than' is a strict ordering. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
⊢ <P Or P
Theorem	prlem934 10946*	Lemma 9-3.4 of [Gleason] p. 122. (Contributed by NM, 25-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ P → ∃𝑥 ∈ 𝐴 ¬ (𝑥 +Q 𝐵) ∈ 𝐴)
Theorem	ltaddpr 10947	The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐴<P (𝐴 +P 𝐵))
Theorem	ltaddpr2 10948	The sum of two positive reals is greater than one of them. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
⊢ (𝐶 ∈ P → ((𝐴 +P 𝐵) = 𝐶 → 𝐴<P 𝐶))
Theorem	ltexprlem1 10949*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)
⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}    ⇒   ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → 𝐶 ≠ ∅))
Theorem	ltexprlem2 10950*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)
⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}    ⇒   ⊢ (𝐵 ∈ P → 𝐶 ⊊ Q)
Theorem	ltexprlem3 10951*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}    ⇒   ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∀𝑧(𝑧 <Q 𝑥 → 𝑧 ∈ 𝐶)))
Theorem	ltexprlem4 10952*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}    ⇒   ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑥 <Q 𝑧)))
Theorem	ltexprlem5 10953*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}    ⇒   ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ∈ P)
Theorem	ltexprlem6 10954*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}    ⇒   ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵) → (𝐴 +P 𝐶) ⊆ 𝐵)
Theorem	ltexprlem7 10955*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}    ⇒   ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵) → 𝐵 ⊆ (𝐴 +P 𝐶))
Theorem	ltexpri 10956*	Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
⊢ (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵)
Theorem	ltaprlem 10957	Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
⊢ (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
Theorem	ltapr 10958	Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
⊢ (𝐶 ∈ P → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
Theorem	addcanpr 10959	Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by NM, 9-Apr-1996.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))
Theorem	prlem936 10960*	Lemma 9-3.6 of [Gleason] p. 124. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥 ∈ 𝐴 ¬ (𝑥 ·Q 𝐵) ∈ 𝐴)
Theorem	reclem2pr 10961*	Lemma for Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q‘𝑦) ∈ 𝐴)}    ⇒   ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
Theorem	reclem3pr 10962*	Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q‘𝑦) ∈ 𝐴)}    ⇒   ⊢ (𝐴 ∈ P → 1P ⊆ (𝐴 ·P 𝐵))
Theorem	reclem4pr 10963*	Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (New usage is discouraged.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q‘𝑦) ∈ 𝐴)}    ⇒   ⊢ (𝐴 ∈ P → (𝐴 ·P 𝐵) = 1P)
Theorem	recexpr 10964*	The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ P → ∃𝑥 ∈ P (𝐴 ·P 𝑥) = 1P)
Theorem	suplem1pr 10965*	The union of a nonempty, bounded set of positive reals is a positive real. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → ∪ 𝐴 ∈ P)
Theorem	suplem2pr 10966*	The union of a set of positive reals (if a positive real) is its supremum (the least upper bound). Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ (𝐴 ⊆ P → ((𝑦 ∈ 𝐴 → ¬ ∪ 𝐴<P 𝑦) ∧ (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))
Theorem	supexpr 10967*	The union of a nonempty, bounded set of positive reals has a supremum. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (New usage is discouraged.)
⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → ∃𝑥 ∈ P (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))
Definition	df-enr 10968*	Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 11034, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
⊢ ~R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))}
Definition	df-nr 10969	Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 11034, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
⊢ R = ((P × P) / ~R )
Definition	df-plr 10970*	Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 11034, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
⊢ +R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R ))}
Definition	df-mr 10971*	Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 11034, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
⊢ ·R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R ))}
Definition	df-ltr 10972*	Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 11034, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
⊢ <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
Definition	df-0r 10973	Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 11034, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
⊢ 0R = [⟨1P, 1P⟩] ~R
Definition	df-1r 10974	Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 11034, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
Definition	df-m1r 10975	Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 11034, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
Theorem	enrer 10976	The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.)
⊢ ~R Er (P × P)
Theorem	nrex1 10977	The class of signed reals is a set. Note that a shorter proof is possible using qsex 8707 (and not requiring enrer 10976), but it would add a dependency on ax-rep 5221. (Contributed by Mario Carneiro, 17-Nov-2014.) Extract proof from that of axcnex 11060. (Revised by BJ, 4-Feb-2023.) (New usage is discouraged.)
⊢ R ∈ V
Theorem	enrbreq 10978	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 10979	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 10980	The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
⊢ ~R ∈ V
Theorem	ltrelsr 10981	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 10982	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 10983	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 10984	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 10985*	Decomposing signed reals into positive reals. Lemma for addsrpr 10988 and mulsrpr 10989. (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 10986*	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 10987*	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 10988	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 10989	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 10990	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 10991	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 10992	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 10993	The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
⊢ 0R ∈ R
Theorem	1sr 10994	The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
⊢ 1R ∈ R
Theorem	m1r 10995	The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
⊢ -1R ∈ R
Theorem	addclsr 10996	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 10997	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 10998	Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
⊢ dom +R = (R × R)
Theorem	dmmulsr 10999	Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
⊢ dom ·R = (R × R)
Theorem	addcomsr 11000	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)