P. 78 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	m1r 7701	The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.)
⊢ -1R ∈ R
Theorem	addclsr 7702	Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.)
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ R)
Theorem	mulclsr 7703	Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.)
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R)
Theorem	addcomsrg 7704	Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴))
Theorem	addasssrg 7705	Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 +R 𝐵) +R 𝐶) = (𝐴 +R (𝐵 +R 𝐶)))
Theorem	mulcomsrg 7706	Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴))
Theorem	mulasssrg 7707	Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶)))
Theorem	distrsrg 7708	Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
Theorem	m1p1sr 7709	Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)
⊢ (-1R +R 1R) = 0R
Theorem	m1m1sr 7710	Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)
⊢ (-1R ·R -1R) = 1R
Theorem	lttrsr 7711*	Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)
⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → ((𝑓 <R 𝑔 ∧ 𝑔 <R ℎ) → 𝑓 <R ℎ))
Theorem	ltposr 7712	Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.)
⊢ <R Po R
Theorem	ltsosr 7713	Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
⊢ <R Or R
Theorem	0lt1sr 7714	0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.)
⊢ 0R <R 1R
Theorem	1ne0sr 7715	1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.)
⊢ ¬ 1R = 0R
Theorem	0idsr 7716	The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.)
⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴)
Theorem	1idsr 7717	1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴)
Theorem	00sr 7718	A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R)
Theorem	ltasrg 7719	Ordering property of addition. (Contributed by NM, 10-May-1996.)
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
Theorem	pn0sr 7720	A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.)
⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R)
Theorem	negexsr 7721*	Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.)
⊢ (𝐴 ∈ R → ∃𝑥 ∈ R (𝐴 +R 𝑥) = 0R)
Theorem	recexgt0sr 7722*	The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.)
⊢ (0R <R 𝐴 → ∃𝑥 ∈ R (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R))
Theorem	recexsrlem 7723*	The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.)
⊢ (0R <R 𝐴 → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R)
Theorem	addgt0sr 7724	The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.)
⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 +R 𝐵))
Theorem	ltadd1sr 7725	Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)
⊢ (𝐴 ∈ R → 𝐴 <R (𝐴 +R 1R))
Theorem	ltm1sr 7726	Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.)
⊢ (𝐴 ∈ R → (𝐴 +R -1R) <R 𝐴)
Theorem	mulgt0sr 7727	The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.)
⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))
Theorem	aptisr 7728	Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ ¬ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)) → 𝐴 = 𝐵)
Theorem	mulextsr1lem 7729	Lemma for mulextsr1 7730. (Contributed by Jim Kingdon, 17-Feb-2020.)
⊢ (((𝑋 ∈ P ∧ 𝑌 ∈ P) ∧ (𝑍 ∈ P ∧ 𝑊 ∈ P) ∧ (𝑈 ∈ P ∧ 𝑉 ∈ P)) → ((((𝑋 ·P 𝑈) +P (𝑌 ·P 𝑉)) +P ((𝑍 ·P 𝑉) +P (𝑊 ·P 𝑈)))<P (((𝑋 ·P 𝑉) +P (𝑌 ·P 𝑈)) +P ((𝑍 ·P 𝑈) +P (𝑊 ·P 𝑉))) → ((𝑋 +P 𝑊)<P (𝑌 +P 𝑍) ∨ (𝑍 +P 𝑌)<P (𝑊 +P 𝑋))))
Theorem	mulextsr1 7730	Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 ·R 𝐶) <R (𝐵 ·R 𝐶) → (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)))
Theorem	archsr 7731*	For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R is the embedding of the positive integer 𝑥 into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
⊢ (𝐴 ∈ R → ∃𝑥 ∈ N 𝐴 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
Theorem	srpospr 7732*	Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)
⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
Theorem	prsrcl 7733	Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.)
⊢ (𝐴 ∈ P → [⟨(𝐴 +P 1P), 1P⟩] ~R ∈ R)
Theorem	prsrpos 7734	Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.)
⊢ (𝐴 ∈ P → 0R <R [⟨(𝐴 +P 1P), 1P⟩] ~R )
Theorem	prsradd 7735	Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R = ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ))
Theorem	prsrlt 7736	Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ [⟨(𝐴 +P 1P), 1P⟩] ~R <R [⟨(𝐵 +P 1P), 1P⟩] ~R ))
Theorem	prsrriota 7737*	Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → [⟨((℩𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩] ~R = 𝐴)
Theorem	caucvgsrlemcl 7738*	Lemma for caucvgsr 7751. Terms of the sequence from caucvgsrlemgt1 7744 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)
Theorem	caucvgsrlemasr 7739*	Lemma for caucvgsr 7751. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)
⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    ⇒   ⊢ (𝜑 → 𝐴 ∈ R)
Theorem	caucvgsrlemfv 7740*	Lemma for caucvgsr 7751. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))    &   ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ N) → [⟨((𝐺‘𝐴) +P 1P), 1P⟩] ~R = (𝐹‘𝐴))
Theorem	caucvgsrlemf 7741*	Lemma for caucvgsr 7751. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))    &   ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))    ⇒   ⊢ (𝜑 → 𝐺:N⟶P)
Theorem	caucvgsrlemcau 7742*	Lemma for caucvgsr 7751. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))    &   ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))    ⇒   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛)<P ((𝐺‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐺‘𝑘)<P ((𝐺‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
Theorem	caucvgsrlembound 7743*	Lemma for caucvgsr 7751. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))    &   ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))    ⇒   ⊢ (𝜑 → ∀𝑚 ∈ N 1P<P (𝐺‘𝑚))
Theorem	caucvgsrlemgt1 7744*	Lemma for caucvgsr 7751. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑖 ∈ N (𝑗 <N 𝑖 → ((𝐹‘𝑖) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑖) +R 𝑥)))))
Theorem	caucvgsrlemoffval 7745*	Lemma for caucvgsr 7751. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    &   ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))    ⇒   ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ((𝐺‘𝐽) +R 𝐴) = ((𝐹‘𝐽) +R 1R))
Theorem	caucvgsrlemofff 7746*	Lemma for caucvgsr 7751. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    &   ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))    ⇒   ⊢ (𝜑 → 𝐺:N⟶R)
Theorem	caucvgsrlemoffcau 7747*	Lemma for caucvgsr 7751. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    &   ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))    ⇒   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
Theorem	caucvgsrlemoffgt1 7748*	Lemma for caucvgsr 7751. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    &   ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))    ⇒   ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐺‘𝑚))
Theorem	caucvgsrlemoffres 7749*	Lemma for caucvgsr 7751. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    &   ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥)))))
Theorem	caucvgsrlembnd 7750*	Lemma for caucvgsr 7751. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥)))))
Theorem	caucvgsr 7751*	A Cauchy sequence of signed reals with a modulus of convergence converges to a signed real. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). This is similar to caucvgprpr 7661 but is for signed reals rather than positive reals. Here is an outline of how we prove it: 1. Choose a lower bound for the sequence (see caucvgsrlembnd 7750). 2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 7746). 3. Since a signed real (element of R) which is greater than zero can be mapped to a positive real (element of P), perform that mapping on each element of the sequence and invoke caucvgprpr 7661 to get a limit (see caucvgsrlemgt1 7744). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7744). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7749). (Contributed by Jim Kingdon, 20-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R)    &   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥)))))
Theorem	ltpsrprg 7752	Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → ((𝐶 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐶 +R [⟨𝐵, 1P⟩] ~R ) ↔ 𝐴<P 𝐵))
Theorem	mappsrprg 7753	Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.)
⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ R) → (𝐶 +R -1R) <R (𝐶 +R [⟨𝐴, 1P⟩] ~R ))
Theorem	map2psrprg 7754*	Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.)
⊢ (𝐶 ∈ R → ((𝐶 +R -1R) <R 𝐴 ↔ ∃𝑥 ∈ P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴))
Theorem	suplocsrlemb 7755*	Lemma for suplocsr 7758. The set 𝐵 is located. (Contributed by Jim Kingdon, 18-Jan-2024.)
⊢ 𝐵 = {𝑤 ∈ P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   ⊢ (𝜑 → 𝐴 ⊆ R)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)    &   ⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦)))    ⇒   ⊢ (𝜑 → ∀𝑢 ∈ P ∀𝑣 ∈ P (𝑢<P 𝑣 → (∃𝑞 ∈ 𝐵 𝑢<P 𝑞 ∨ ∀𝑞 ∈ 𝐵 𝑞<P 𝑣)))
Theorem	suplocsrlempr 7756*	Lemma for suplocsr 7758. The set 𝐵 has a least upper bound. (Contributed by Jim Kingdon, 19-Jan-2024.)
⊢ 𝐵 = {𝑤 ∈ P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   ⊢ (𝜑 → 𝐴 ⊆ R)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)    &   ⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦)))    ⇒   ⊢ (𝜑 → ∃𝑣 ∈ P (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)))
Theorem	suplocsrlem 7757*	Lemma for suplocsr 7758. The set 𝐴 has a least upper bound. (Contributed by Jim Kingdon, 16-Jan-2024.)
⊢ 𝐵 = {𝑤 ∈ P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   ⊢ (𝜑 → 𝐴 ⊆ R)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)    &   ⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦)))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
Theorem	suplocsr 7758*	An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.)
⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)    &   ⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦)))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
Syntax	cc 7759	Class of complex numbers.
class ℂ
Syntax	cr 7760	Class of real numbers.
class ℝ
Syntax	cc0 7761	Extend class notation to include the complex number 0.
class 0
Syntax	c1 7762	Extend class notation to include the complex number 1.
class 1
Syntax	ci 7763	Extend class notation to include the complex number i.
class i
Syntax	caddc 7764	Addition on complex numbers.
class +
Syntax	cltrr 7765	'Less than' predicate (defined over real subset of complex numbers).
class <ℝ
Syntax	cmul 7766	Multiplication on complex numbers. The token · is a center dot.
class ·
Definition	df-c 7767	Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.)
⊢ ℂ = (R × R)
Definition	df-0 7768	Define the complex number 0. (Contributed by NM, 22-Feb-1996.)
⊢ 0 = ⟨0R, 0R⟩
Definition	df-1 7769	Define the complex number 1. (Contributed by NM, 22-Feb-1996.)
⊢ 1 = ⟨1R, 0R⟩
Definition	df-i 7770	Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.)
⊢ i = ⟨0R, 1R⟩
Definition	df-r 7771	Define the set of real numbers. (Contributed by NM, 22-Feb-1996.)
⊢ ℝ = (R × {0R})
Definition	df-add 7772*	Define addition over complex numbers. (Contributed by NM, 28-May-1995.)
⊢ + = {⟨⟨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 7773*	Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.)
⊢ · = {⟨⟨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 7774*	Define 'less than' on the real subset of complex numbers. (Contributed by NM, 22-Feb-1996.)
⊢ <ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
Theorem	opelcn 7775	Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.)
⊢ (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R))
Theorem	opelreal 7776	Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.)
⊢ (⟨𝐴, 0R⟩ ∈ ℝ ↔ 𝐴 ∈ R)
Theorem	elreal 7777*	Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.)
⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
Theorem	elrealeu 7778*	The real number mapping in elreal 7777 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
⊢ (𝐴 ∈ ℝ ↔ ∃!𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
Theorem	elreal2 7779	Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.)
⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩))
Theorem	0ncn 7780	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. See also cnm 7781 which is a related property. (Contributed by NM, 2-May-1996.)
⊢ ¬ ∅ ∈ ℂ
Theorem	cnm 7781*	A complex number is an inhabited set. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by Jim Kingdon, 23-Oct-2023.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 𝑥 ∈ 𝐴)
Theorem	ltrelre 7782	'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
⊢ <ℝ ⊆ (ℝ × ℝ)
Theorem	addcnsr 7783	Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.)
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
Theorem	mulcnsr 7784	Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.)
⊢ (((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 7785	Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (⟨𝐴, 0R⟩ = ⟨𝐵, 0R⟩ ↔ 𝐴 = 𝐵)
Theorem	addresr 7786	Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (⟨𝐴, 0R⟩ + ⟨𝐵, 0R⟩) = ⟨(𝐴 +R 𝐵), 0R⟩)
Theorem	mulresr 7787	Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (⟨𝐴, 0R⟩ · ⟨𝐵, 0R⟩) = ⟨(𝐴 ·R 𝐵), 0R⟩)
Theorem	ltresr 7788	Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.)
⊢ (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ ↔ 𝐴 <R 𝐵)
Theorem	ltresr2 7789	Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ (1st ‘𝐴) <R (1st ‘𝐵)))
Theorem	dfcnqs 7790	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 6574, which shows that the coset of the converse epsilon 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 7767), and allows us to reuse some of the equivalence class lemmas we developed for the transition from positive reals to signed reals, etc. (Contributed by NM, 13-Aug-1995.)
⊢ ℂ = ((R × R) / ◡ E )
Theorem	addcnsrec 7791	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 7790 and mulcnsrec 7792. (Contributed by NM, 13-Aug-1995.)
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([⟨𝐴, 𝐵⟩]◡ E + [⟨𝐶, 𝐷⟩]◡ E ) = [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩]◡ E )
Theorem	mulcnsrec 7792	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6573, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 7790. (Contributed by NM, 13-Aug-1995.)
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([⟨𝐴, 𝐵⟩]◡ E · [⟨𝐶, 𝐷⟩]◡ E ) = [⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩]◡ E )
Theorem	addvalex 7793	Existence of a sum. This is dependent on how we define + so once we proceed to real number axioms we will replace it with theorems such as addcl 7886. (Contributed by Jim Kingdon, 14-Jul-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 + 𝐵) ∈ V)
Theorem	pitonnlem1 7794*	Lemma for pitonn 7797. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.)
⊢ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
Theorem	pitonnlem1p1 7795	Lemma for pitonn 7797. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (𝐴 ∈ P → [⟨(𝐴 +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨(𝐴 +P 1P), 1P⟩] ~R )
Theorem	pitonnlem2 7796*	Lemma for pitonn 7797. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.)
⊢ (𝐾 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
Theorem	pitonn 7797*	Mapping from N to ℕ. (Contributed by Jim Kingdon, 22-Apr-2020.)
⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
Theorem	pitoregt0 7798*	Embedding from N to ℝ yields a number greater than zero. (Contributed by Jim Kingdon, 15-Jul-2021.)
⊢ (𝑁 ∈ N → 0 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
Theorem	pitore 7799*	Embedding from N to ℝ. Similar to pitonn 7797 but separate in the sense that we have not proved nnssre 8869 yet. (Contributed by Jim Kingdon, 15-Jul-2021.)
⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)
Theorem	recnnre 7800*	Embedding the reciprocal of a natural number into ℝ. (Contributed by Jim Kingdon, 15-Jul-2021.)
⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)