 Intuitionistic Logic Explorer Most Recent Proofs Mirrors  >  Home  >  ILE Home  >  Th. List  >  Recent MPE Most Recent             Other  >  MM 100

Most recent proofs    These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the iset.mm database for the Intuitionistic Logic Explorer. The iset.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from the commit given on the MPE Most Recent Proofs page. The database from that commit is also available here: iset.mm.

See the MPE Most Recent Proofs page for news and some useful links.

 Color key: Intuitionistic Logic Explorer User Mathboxes

Last updated on 22-Oct-2021 at 12:40 PM ET.
Recent Additions to the Intuitionistic Logic Explorer
DateLabelDescription
Theorem

20-Oct-2021onunsnss 6355 Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵𝐴)

19-Oct-2021snon0 6356 An ordinal which is a singleton is {∅}. (Contributed by Jim Kingdon, 19-Oct-2021.)
((𝐴𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)

18-Oct-2021qdencn 10123 The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 9798 (and also would hold for ℝ × ℝ with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.)
𝑄 = {𝑧 ∈ ℂ ∣ ((ℜ‘𝑧) ∈ ℚ ∧ (ℑ‘𝑧) ∈ ℚ)}       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+) → ∃𝑥𝑄 (abs‘(𝑥𝐴)) < 𝐵)

18-Oct-2021modqmulnn 9184 Move a positive integer in and out of a floor in the first argument of a modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · (⌊‘𝐴)) mod (𝑁 · 𝑀)) ≤ ((⌊‘(𝑁 · 𝐴)) mod (𝑁 · 𝑀)))

18-Oct-2021intqfrac 9181 Break a number into its integer part and its fractional part. (Contributed by Jim Kingdon, 18-Oct-2021.)
(𝐴 ∈ ℚ → 𝐴 = ((⌊‘𝐴) + (𝐴 mod 1)))

18-Oct-2021flqmod 9180 The floor function expressed in terms of the modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) = (𝐴 − (𝐴 mod 1)))

18-Oct-2021modqfrac 9179 The fractional part of a number is the number modulo 1. (Contributed by Jim Kingdon, 18-Oct-2021.)
(𝐴 ∈ ℚ → (𝐴 mod 1) = (𝐴 − (⌊‘𝐴)))

18-Oct-2021modqdifz 9178 The modulo operation differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℤ)

18-Oct-2021modqdiffl 9177 The modulo operation differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵)))

18-Oct-2021modqelico 9176 Modular reduction produces a half-open interval. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ (0[,)𝐵))

18-Oct-2021modqlt 9175 The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) < 𝐵)

18-Oct-2021modqge0 9174 The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵))

18-Oct-2021negqmod0 9173 𝐴 is divisible by 𝐵 iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 0 ↔ (-𝐴 mod 𝐵) = 0))

18-Oct-2021mulqmod0 9172 The product of an integer and a positive rational number is 0 modulo the positive real number. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 · 𝑀) mod 𝑀) = 0)

18-Oct-2021flqdiv 9163 Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ) → (⌊‘((⌊‘𝐴) / 𝑁)) = (⌊‘(𝐴 / 𝑁)))

18-Oct-2021intqfrac2 9161 Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.)
𝑍 = (⌊‘𝐴)    &   𝐹 = (𝐴𝑍)       (𝐴 ∈ ℚ → (0 ≤ 𝐹𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))

17-Oct-2021modq0 9171 𝐴 mod 𝐵 is zero iff 𝐴 is evenly divisible by 𝐵. (Contributed by Jim Kingdon, 17-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 0 ↔ (𝐴 / 𝐵) ∈ ℤ))

16-Oct-2021modqcld 9170 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑 → 0 < 𝐵)       (𝜑 → (𝐴 mod 𝐵) ∈ ℚ)

16-Oct-2021flqpmodeq 9169 Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (((⌊‘(𝐴 / 𝐵)) · 𝐵) + (𝐴 mod 𝐵)) = 𝐴)

16-Oct-2021modqcl 9168 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ)

16-Oct-2021modqvalr 9167 The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − ((⌊‘(𝐴 / 𝐵)) · 𝐵)))

16-Oct-2021modqval 9166 The value of the modulo operation. The modulo congruence notation of number theory, 𝐽𝐾 (modulo 𝑁), can be expressed in our notation as (𝐽 mod 𝑁) = (𝐾 mod 𝑁). Definition 1 in Knuth, The Art of Computer Programming, Vol. I (1972), p. 38. Knuth uses "mod" for the operation and "modulo" for the congruence. Unlike Knuth, we restrict the second argument to positive numbers to simplify certain theorems. (This also gives us future flexibility to extend it to any one of several different conventions for a zero or negative second argument, should there be an advantage in doing so.) As with flqcl 9117 we only prove this for rationals although other particular kinds of real numbers may be possible. (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))

15-Oct-2021qdenre 9798 The rational numbers are dense in : any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 9111. (Contributed by BJ, 15-Oct-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℚ (abs‘(𝑥𝐴)) < 𝐵)

14-Oct-2021qbtwnrelemcalc 9110 Lemma for qbtwnre 9111. Calculations involved in showing the constructed rational number is less than 𝐵. (Contributed by Jim Kingdon, 14-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 < (𝐴 · (2 · 𝑁)))    &   (𝜑 → (1 / 𝑁) < (𝐵𝐴))       (𝜑 → ((𝑀 + 2) / (2 · 𝑁)) < 𝐵)

13-Oct-2021rebtwn2z 9109 A real number can be bounded by integers above and below which are two apart.

The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)

(𝐴 ∈ ℝ → ∃𝑥 ∈ ℤ (𝑥 < 𝐴𝐴 < (𝑥 + 2)))

13-Oct-2021rebtwn2zlemshrink 9108 Lemma for rebtwn2z 9109. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.)
((𝐴 ∈ ℝ ∧ 𝐽 ∈ (ℤ‘2) ∧ ∃𝑚 ∈ ℤ (𝑚 < 𝐴𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴𝐴 < (𝑥 + 2)))

13-Oct-2021rebtwn2zlemstep 9107 Lemma for rebtwn2z 9109. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.)
((𝐾 ∈ (ℤ‘2) ∧ 𝐴 ∈ ℝ ∧ ∃𝑚 ∈ ℤ (𝑚 < 𝐴𝐴 < (𝑚 + (𝐾 + 1)))) → ∃𝑚 ∈ ℤ (𝑚 < 𝐴𝐴 < (𝑚 + 𝐾)))

11-Oct-2021flqeqceilz 9160 A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌊‘𝐴) = (⌈‘𝐴)))

11-Oct-2021flqleceil 9159 The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) ≤ (⌈‘𝐴))

11-Oct-2021ceilqidz 9158 A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))

11-Oct-2021ceilqle 9156 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴𝐵) → (⌈‘𝐴) ≤ 𝐵)

11-Oct-2021ceiqle 9155 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴𝐵) → -(⌊‘-𝐴) ≤ 𝐵)

11-Oct-2021ceilqm1lt 9154 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → ((⌈‘𝐴) − 1) < 𝐴)

11-Oct-2021ceiqm1l 9153 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (-(⌊‘-𝐴) − 1) < 𝐴)

11-Oct-2021ceilqge 9152 The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → 𝐴 ≤ (⌈‘𝐴))

11-Oct-2021ceiqge 9151 The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → 𝐴 ≤ -(⌊‘-𝐴))

11-Oct-2021ceilqcl 9150 Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)

11-Oct-2021ceiqcl 9149 The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℤ)

11-Oct-2021qdceq 9102 Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 = 𝐵)

11-Oct-2021qltlen 8575 Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7621 which is a similar result for real numbers. (Contributed by Jim Kingdon, 11-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ (𝐴𝐵𝐵𝐴)))

10-Oct-2021ceilqval 9148 The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
(𝐴 ∈ ℚ → (⌈‘𝐴) = -(⌊‘-𝐴))

10-Oct-2021flqmulnn0 9141 Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)
((𝑁 ∈ ℕ0𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))

10-Oct-2021flqzadd 9140 An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℚ) → (⌊‘(𝑁 + 𝐴)) = (𝑁 + (⌊‘𝐴)))

10-Oct-2021flqaddz 9139 An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁))

10-Oct-2021flqge1nn 9136 The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
((𝐴 ∈ ℚ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ)

10-Oct-2021flqge0nn0 9135 The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
((𝐴 ∈ ℚ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0)

10-Oct-20214ap0 8015 The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
4 # 0

10-Oct-20213ap0 8012 The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
3 # 0

9-Oct-2021flqbi2 9133 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℚ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹𝐹 < 1)))

9-Oct-2021flqbi 9132 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵𝐴𝐴 < (𝐵 + 1))))

9-Oct-2021flqword2 9131 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴𝐵) → (⌊‘𝐵) ∈ (ℤ‘(⌊‘𝐴)))

9-Oct-2021flqwordi 9130 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵))

9-Oct-2021flqltnz 9129 If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) < 𝐴)

9-Oct-2021flqidz 9128 A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.)
(𝐴 ∈ ℚ → ((⌊‘𝐴) = 𝐴𝐴 ∈ ℤ))

8-Oct-2021flqidm 9127 The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴))

8-Oct-2021flqlt 9125 The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (⌊‘𝐴) < 𝐵))

8-Oct-2021flqge 9124 The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵𝐴𝐵 ≤ (⌊‘𝐴)))

8-Oct-2021qfracge0 9123 The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → 0 ≤ (𝐴 − (⌊‘𝐴)))

8-Oct-2021qfraclt1 9122 The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (𝐴 − (⌊‘𝐴)) < 1)

8-Oct-2021flqltp1 9121 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → 𝐴 < ((⌊‘𝐴) + 1))

8-Oct-2021flqle 9120 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)

8-Oct-2021flqcld 9119 The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → (⌊‘𝐴) ∈ ℤ)

8-Oct-2021flqlelt 9118 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → ((⌊‘𝐴) ≤ 𝐴𝐴 < ((⌊‘𝐴) + 1)))

8-Oct-2021flqcl 9117 The floor (greatest integer) function yields an integer when applied to a rational (closure law). It would presumably be possible to prove a similar result for some real numbers (for example, those apart from any integer), but not real numbers in general. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)

8-Oct-2021qbtwnz 9106 There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → ∃!𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))

8-Oct-2021qbtwnzlemex 9105 Lemma for qbtwnz 9106. Existence of the integer.

The proof starts by finding two integers which are less than and greater than the given rational number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on rational number trichotomy, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)

(𝐴 ∈ ℚ → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))

8-Oct-2021qbtwnzlemshrink 9104 Lemma for qbtwnz 9106. Shrinking the range around the given rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐽 ∈ ℕ ∧ ∃𝑚 ∈ ℤ (𝑚𝐴𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))

8-Oct-2021qbtwnzlemstep 9103 Lemma for qbtwnz 9106. Induction step. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐾 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ ∃𝑚 ∈ ℤ (𝑚𝐴𝐴 < (𝑚 + (𝐾 + 1)))) → ∃𝑚 ∈ ℤ (𝑚𝐴𝐴 < (𝑚 + 𝐾)))

8-Oct-2021qltnle 9101 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))

7-Oct-2021qlelttric 9100 Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴𝐵𝐵 < 𝐴))

6-Oct-2021qletric 9099 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴𝐵𝐵𝐴))

6-Oct-2021qtri3or 9098 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
((𝑀 ∈ ℚ ∧ 𝑁 ∈ ℚ) → (𝑀 < 𝑁𝑀 = 𝑁𝑁 < 𝑀))

3-Oct-2021reg3exmid 4304 If any inhabited set satisfying df-wetr 4071 for E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
(( E We 𝑧 ∧ ∃𝑤 𝑤𝑧) → ∃𝑥𝑧𝑦𝑧 𝑥𝑦)       (𝜑 ∨ ¬ 𝜑)

3-Oct-2021reg3exmidlemwe 4303 Lemma for reg3exmid 4304. Our counterexample 𝐴 satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}        E We 𝐴

2-Oct-2021reg2exmid 4261 If any inhabited set has a minimal element (when expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
𝑧(∃𝑤 𝑤𝑧 → ∃𝑥𝑧𝑦𝑧 𝑥𝑦)       (𝜑 ∨ ¬ 𝜑)

2-Oct-2021reg2exmidlema 4259 Lemma for reg2exmid 4261. If 𝐴 has a minimal element (expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}       (∃𝑢𝐴𝑣𝐴 𝑢𝑣 → (𝜑 ∨ ¬ 𝜑))

30-Sep-2021fin0or 6343 A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
(𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑥 𝑥𝐴))

30-Sep-2021wessep 4302 A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)
(( E We 𝐴𝐵𝐴) → E We 𝐵)

28-Sep-2021frind 4089 Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   ((𝜒𝑥𝐴) → (∀𝑦𝐴 (𝑦𝑅𝑥𝜓) → 𝜑))    &   (𝜒𝑅 Fr 𝐴)    &   (𝜒𝐴𝑉)       ((𝜒𝑥𝐴) → 𝜑)

26-Sep-2021wetriext 4301 A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∀𝑧𝐴 (𝑧𝑅𝐵𝑧𝑅𝐶))       (𝜑𝐵 = 𝐶)

25-Sep-2021nnwetri 6354 A natural number is well-ordered by E. More specifically, this order both satisfies We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
(𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))

23-Sep-2021wepo 4096 A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
((𝑅 We 𝐴𝐴𝑉) → 𝑅 Po 𝐴)

23-Sep-2021df-wetr 4071 Define the well-ordering predicate. It is unusual to define "well-ordering" in the absence of excluded middle, but we mean an ordering which is like the ordering which we have for ordinals (for example, it does not entail trichotomy because ordinals don't have that as seen at ordtriexmid 4247). Given excluded middle, well-ordering is usually defined to require trichotomy (and the defintion of Fr is typically also different). (Contributed by Mario Carneiro and Jim Kingdon, 23-Sep-2021.)
(𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))

22-Sep-2021frforeq3 4084 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
(𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))

22-Sep-2021frforeq2 4082 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
(𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))

22-Sep-2021frforeq1 4080 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
(𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))

22-Sep-2021df-frfor 4068 Define the well-founded relation predicate where 𝐴 might be a proper class. By passing in 𝑆 we allow it potentially to be a proper class rather than a set. (Contributed by Jim Kingdon and Mario Carneiro, 22-Sep-2021.)
( FrFor 𝑅𝐴𝑆 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑆) → 𝑥𝑆) → 𝐴𝑆))

21-Sep-2021frirrg 4087 A well-founded relation is irreflexive. This is the case where 𝐴 exists. (Contributed by Jim Kingdon, 21-Sep-2021.)
((𝑅 Fr 𝐴𝐴𝑉𝐵𝐴) → ¬ 𝐵𝑅𝐵)

21-Sep-2021df-frind 4069 Define the well-founded relation predicate. In the presence of excluded middle, there are a variety of equivalent ways to define this. In our case, this definition, in terms of an inductive principle, works better than one along the lines of "there is an element which is minimal when A is ordered by R". Because 𝑠 is constrained to be a set (not a proper class) here, sometimes it may be necessary to use FrFor directly rather than via Fr. (Contributed by Jim Kingdon and Mario Carneiro, 21-Sep-2021.)
(𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)

17-Sep-2021ontr2exmid 4250 An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
𝑥 ∈ On ∀𝑦𝑧 ∈ On ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)       (𝜑 ∨ ¬ 𝜑)

16-Sep-2021nnsseleq 6079 For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

15-Sep-2021fientri3 6353 Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵𝐵𝐴))

15-Sep-2021nntri2or2 6076 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐵𝐴))

14-Sep-2021findcard2sd 6349 Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.)
(𝑥 = ∅ → (𝜓𝜒))    &   (𝑥 = 𝑦 → (𝜓𝜃))    &   (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓𝜏))    &   (𝑥 = 𝐴 → (𝜓𝜂))    &   (𝜑𝜒)    &   (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (𝜃𝜏))    &   (𝜑𝐴 ∈ Fin)       (𝜑𝜂)

13-Sep-2021php5fin 6339 A finite set is not equinumerous to a set which adds one element. (Contributed by Jim Kingdon, 13-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) → ¬ 𝐴 ≈ (𝐴 ∪ {𝐵}))

13-Sep-2021fiunsnnn 6338 Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.)
(((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴𝑁)) → (𝐴 ∪ {𝐵}) ≈ suc 𝑁)

12-Sep-2021fisbth 6340 Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
(((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵𝐵𝐴)) → 𝐴𝐵)

11-Sep-2021diffisn 6350 Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → (𝐴 ∖ {𝐵}) ∈ Fin)

10-Sep-2021fin0 6342 A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)
(𝐴 ∈ Fin → (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴))

9-Sep-2021fidifsnid 6332 If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3510 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

9-Sep-2021fidifsnen 6331 All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)
((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))

8-Sep-2021diffifi 6351 Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵𝐴) → (𝐴𝐵) ∈ Fin)

8-Sep-2021diffitest 6344 If subtracting any set from a finite set gives a finite set, any proposition of the form ¬ 𝜑 is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove 𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)
𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin       𝜑 ∨ ¬ ¬ 𝜑)

6-Sep-2021phpelm 6328 Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → ¬ 𝐴𝐵)

5-Sep-2021fidceq 6330 Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that {𝐵, 𝐶} is finite would require showing it is equinumerous to 1𝑜 or to 2𝑜 but to show that you'd need to know 𝐵 = 𝐶 or ¬ 𝐵 = 𝐶, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) → DECID 𝐵 = 𝐶)

5-Sep-2021phplem4on 6329 Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵𝐴𝐵))

4-Sep-2021ex-ceil 9896 Example for df-ceil 9115. (Contributed by AV, 4-Sep-2021.)
((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)

3-Sep-20210elsucexmid 4289 If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)
𝑥 ∈ On ∅ ∈ suc 𝑥       (𝜑 ∨ ¬ 𝜑)

1-Sep-2021php5dom 6325 A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
(𝐴 ∈ ω → ¬ suc 𝐴𝐴)

1-Sep-2021phplem4dom 6324 Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≼ suc 𝐵𝐴𝐵))

1-Sep-2021snnen2oprc 6323 A singleton {𝐴} is never equinumerous with the ordinal number 2. If 𝐴 is a set, see snnen2og 6322. (Contributed by Jim Kingdon, 1-Sep-2021.)
𝐴 ∈ V → ¬ {𝐴} ≈ 2𝑜)

1-Sep-2021snnen2og 6322 A singleton {𝐴} is never equinumerous with the ordinal number 2. If 𝐴 is a proper class, see snnen2oprc 6323. (Contributed by Jim Kingdon, 1-Sep-2021.)
(𝐴𝑉 → ¬ {𝐴} ≈ 2𝑜)

1-Sep-2021phplem3g 6319 A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6317 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))

31-Aug-2021nndifsnid 6080 If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3510 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

30-Aug-2021carden2bex 6369 If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → (card‘𝐴) = (card‘𝐵))

30-Aug-2021cardval3ex 6365 The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.)
(∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})

30-Aug-2021cardcl 6361 The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
(∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) ∈ On)

30-Aug-2021onintrab2im 4244 An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.)
(∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)

30-Aug-2021onintonm 4243 The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)
((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ On)

29-Aug-2021onintexmid 4297 If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) → 𝑦𝑦)       (𝜑 ∨ ¬ 𝜑)

29-Aug-2021ordtri2or2exmid 4296 Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)       (𝜑 ∨ ¬ 𝜑)

29-Aug-2021ordtri2or2exmidlem 4251 A set which is 2𝑜 if 𝜑 or if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
{𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On

29-Aug-20212ordpr 4249 Version of 2on 6009 with the definition of 2𝑜 expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
Ord {∅, {∅}}

25-Aug-2021nnap0d 7959 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 # 0)

24-Aug-2021climcaucn 9870 A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 9866 but adds the part that (𝐹𝑘) is complex. (Contributed by Jim Kingdon, 24-Aug-2021.)
𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥))

24-Aug-2021climcvg1nlem 9868 Lemma for climcvg1n 9869. We construct sequences of the real and imaginary parts of each term of 𝐹, show those converge, and use that to show that 𝐹 converges. (Contributed by Jim Kingdon, 24-Aug-2021.)
(𝜑𝐹:ℕ⟶ℂ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)(abs‘((𝐹𝑘) − (𝐹𝑛))) < (𝐶 / 𝑛))    &   𝐺 = (𝑥 ∈ ℕ ↦ (ℜ‘(𝐹𝑥)))    &   𝐻 = (𝑥 ∈ ℕ ↦ (ℑ‘(𝐹𝑥)))    &   𝐽 = (𝑥 ∈ ℕ ↦ (i · (𝐻𝑥)))       (𝜑𝐹 ∈ dom ⇝ )

23-Aug-2021climcvg1n 9869 A Cauchy sequence of complex numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within 𝐶 / 𝑛 of the nth term, where 𝐶 is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
(𝜑𝐹:ℕ⟶ℂ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)(abs‘((𝐹𝑘) − (𝐹𝑛))) < (𝐶 / 𝑛))       (𝜑𝐹 ∈ dom ⇝ )

23-Aug-2021climrecvg1n 9867 A Cauchy sequence of real numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within 𝐶 / 𝑛 of the nth term, where 𝐶 is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)(abs‘((𝐹𝑘) − (𝐹𝑛))) < (𝐶 / 𝑛))       (𝜑𝐹 ∈ dom ⇝ )

22-Aug-2021climserile 9865 The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by Jim Kingdon, 22-Aug-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → 0 ≤ (𝐹𝑘))       (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘𝑁) ≤ 𝐴)

22-Aug-2021iserige0 9863 The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → 0 ≤ (𝐹𝑘))       (𝜑 → 0 ≤ 𝐴)

22-Aug-2021iserile 9862 Comparison of the limits of two infinite series. (Contributed by Jim Kingdon, 22-Aug-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴)    &   (𝜑 → seq𝑀( + , 𝐺, ℂ) ⇝ 𝐵)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ (𝐺𝑘))       (𝜑𝐴𝐵)

22-Aug-2021serile 9253 Comparison of partial sums of two infinite series of reals. (Contributed by Jim Kingdon, 22-Aug-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ≤ (𝐺𝑘))       (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘𝑁) ≤ (seq𝑀( + , 𝐺, ℂ)‘𝑁))

22-Aug-2021serige0 9252 A finite sum of nonnegative terms is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → 0 ≤ (𝐹𝑘))       (𝜑 → 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁))

21-Aug-2021clim2iser2 9858 The limit of an infinite series with an initial segment added. (Contributed by Jim Kingdon, 21-Aug-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   (𝜑 → seq(𝑁 + 1)( + , 𝐹, ℂ) ⇝ 𝐴)       (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ (𝐴 + (seq𝑀( + , 𝐹, ℂ)‘𝑁)))

21-Aug-2021iseqdistr 9249 The distributive property for series. (Contributed by Jim Kingdon, 21-Aug-2021.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) = (𝐶𝑇(𝐺𝑥)))    &   (𝜑𝑆𝑉)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))

21-Aug-2021iseqhomo 9248 Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 21-Aug-2021.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   (𝜑𝑆𝑉)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)       (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁))

20-Aug-2021clim2iser 9857 The limit of an infinite series with an initial segment removed. (Contributed by Jim Kingdon, 20-Aug-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴)       (𝜑 → seq(𝑁 + 1)( + , 𝐹, ℂ) ⇝ (𝐴 − (seq𝑀( + , 𝐹, ℂ)‘𝑁)))

20-Aug-2021iseqex 9213 Existence of the sequence builder operation. (Contributed by Jim Kingdon, 20-Aug-2021.)
seq𝑀( + , 𝐹, 𝑆) ∈ V

20-Aug-2021frecex 5981 Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
frec(𝐹, 𝐴) ∈ V

20-Aug-2021tfrfun 5955 Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
Fun recs(𝐹)

19-Aug-2021iserclim0 9826 The zero series converges to zero. (Contributed by Jim Kingdon, 19-Aug-2021.)
(𝑀 ∈ ℤ → seq𝑀( + , ((ℤ𝑀) × {0}), ℂ) ⇝ 0)

19-Aug-2021shftval4g 9438 Value of a sequence shifted by -𝐴. (Contributed by Jim Kingdon, 19-Aug-2021.)
((𝐹𝑉𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift -𝐴)‘𝐵) = (𝐹‘(𝐴 + 𝐵)))

19-Aug-2021iser0f 9251 A zero-valued infinite series is equal to the constant zero function. (Contributed by Jim Kingdon, 19-Aug-2021.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0}), ℂ) = (𝑍 × {0}))

19-Aug-2021iser0 9250 The value of the partial sums in a zero-valued infinite series. (Contributed by Jim Kingdon, 19-Aug-2021.)
𝑍 = (ℤ𝑀)       (𝑁𝑍 → (seq𝑀( + , (𝑍 × {0}), ℂ)‘𝑁) = 0)

19-Aug-2021ovexg 5539 Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.)
((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐴𝐹𝐵) ∈ V)

18-Aug-2021iseqid3s 9246 A sequence that consists of zeroes up to 𝑁 sums to zero at 𝑁. In this case by "zero" we mean whatever the identity 𝑍 is for the operation +). (Contributed by Jim Kingdon, 18-Aug-2021.)
(𝜑 → (𝑍 + 𝑍) = 𝑍)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = 𝑍)    &   (𝜑𝑍𝑆)    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)

18-Aug-2021iseqid3 9245 A sequence that consists entirely of zeroes (or whatever the identity 𝑍 is for operation +) sums to zero. (Contributed by Jim Kingdon, 18-Aug-2021.)
(𝜑 → (𝑍 + 𝑍) = 𝑍)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) = 𝑍)    &   (𝜑𝑍𝑉)       (𝜑 → (seq𝑀( + , 𝐹, {𝑍})‘𝑁) = 𝑍)

18-Aug-2021iseqss 9226 Specifying a larger universe for seq. As long as 𝐹 and + are closed over 𝑆, then any set which contains 𝑆 can be used as the last argument to seq. This theorem does not allow 𝑇 to be a proper class, however. It also currently requires that + be closed over 𝑇 (as well as 𝑆). (Contributed by Jim Kingdon, 18-Aug-2021.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑇𝑉)    &   (𝜑𝑆𝑇)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑇𝑦𝑇)) → (𝑥 + 𝑦) ∈ 𝑇)       (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))

17-Aug-2021iseqcaopr 9242 The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Jim Kingdon, 17-Aug-2021.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘) + (𝐺𝑘)))    &   (𝜑𝑆𝑉)       (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (seq𝑀( + , 𝐺, 𝑆)‘𝑁)))

16-Aug-2021iseqcaopr3 9240 Lemma for iseqcaopr2 . (Contributed by Jim Kingdon, 16-Aug-2021.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))    &   ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1)))))    &   (𝜑𝑆𝑉)       (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))

16-Aug-2021iseq1p 9239 Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   (𝜑𝑆𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = ((𝐹𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))

16-Aug-2021iseqsplit 9238 Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   (𝜑𝑆𝑉)    &   (𝜑𝑀 ∈ (ℤ𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)       (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))

15-Aug-2021shftfibg 9421 Value of a fiber of the relation 𝐹. (Contributed by Jim Kingdon, 15-Aug-2021.)
((𝐹𝑉𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) “ {𝐵}) = (𝐹 “ {(𝐵𝐴)}))

15-Aug-2021ovshftex 9420 Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)
((𝐹𝑉𝐴 ∈ ℂ) → (𝐹 shift 𝐴) ∈ V)

15-Aug-2021isermono 9237 The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑𝑁 ∈ (ℤ𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ ℝ)    &   ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (𝐹𝑥))       (𝜑 → (seq𝑀( + , 𝐹, ℝ)‘𝐾) ≤ (seq𝑀( + , 𝐹, ℝ)‘𝑁))

15-Aug-2021iserf 9233 An infinite series of complex terms is a function from to . (Contributed by Jim Kingdon, 15-Aug-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)       (𝜑 → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℂ)

15-Aug-2021iseqshft2 9232 Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐾 ∈ ℤ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))

15-Aug-2021iseqfeq 9231 Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = (𝐺𝑘))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))

13-Aug-2021absdivapd 9791 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))

13-Aug-2021absrpclapd 9784 The absolute value of a complex number apart from zero is a positive real. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (abs‘𝐴) ∈ ℝ+)

13-Aug-2021absdivapzi 9750 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))

13-Aug-2021absgt0ap 9695 The absolute value of a number apart from zero is positive. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝐴 ∈ ℂ → (𝐴 # 0 ↔ 0 < (abs‘𝐴)))

13-Aug-2021recvalap 9693 Reciprocal expressed with a real denominator. (Contributed by Jim Kingdon, 13-Aug-2021.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) = ((∗‘𝐴) / ((abs‘𝐴)↑2)))

13-Aug-2021sqap0 9320 A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝐴 ∈ ℂ → ((𝐴↑2) # 0 ↔ 𝐴 # 0))

13-Aug-2021leltapd 7628 '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 < 𝐵𝐵 # 𝐴))

13-Aug-2021leltap 7615 '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (𝐴 < 𝐵𝐵 # 𝐴))

12-Aug-2021abssubap0 9686 If the absolute value of a complex number is less than a real, its difference from the real is apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ (abs‘𝐴) < 𝐵) → (𝐵𝐴) # 0)

12-Aug-2021ltabs 9683 A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.)
((𝐴 ∈ ℝ ∧ 𝐴 < (abs‘𝐴)) → 𝐴 < 0)

12-Aug-2021absext 9661 Strong extensionality for absolute value. (Contributed by Jim Kingdon, 12-Aug-2021.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) # (abs‘𝐵) → 𝐴 # 𝐵))

12-Aug-2021sq11ap 9414 Analogue to sq11 9326 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) # (𝐵↑2) ↔ 𝐴 # 𝐵))

12-Aug-2021subap0d 7631 Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 𝐵)       (𝜑 → (𝐴𝐵) # 0)

12-Aug-2021ltapd 7627 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴 # 𝐵)

12-Aug-2021gtapd 7626 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵 # 𝐴)

12-Aug-2021ltapi 7625 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵𝐵 # 𝐴)

12-Aug-2021ltapii 7624 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴 < 𝐵       𝐴 # 𝐵

12-Aug-2021gtapii 7623 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴 < 𝐵       𝐵 # 𝐴

12-Aug-2021ltap 7622 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 # 𝐴)

11-Aug-2021absexpzap 9676 Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴𝑁)) = ((abs‘𝐴)↑𝑁))

11-Aug-2021absdivap 9668 Absolute value distributes over division. (Contributed by Jim Kingdon, 11-Aug-2021.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))

11-Aug-2021abs00ap 9660 The absolute value of a number is apart from zero iff the number is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
(𝐴 ∈ ℂ → ((abs‘𝐴) # 0 ↔ 𝐴 # 0))

11-Aug-2021absrpclap 9659 The absolute value of a number apart from zero is a positive real. (Contributed by Jim Kingdon, 11-Aug-2021.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (abs‘𝐴) ∈ ℝ+)

11-Aug-2021sqrt11ap 9636 Analogue to sqrt11 9637 but for apartness. (Contributed by Jim Kingdon, 11-Aug-2021.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((√‘𝐴) # (√‘𝐵) ↔ 𝐴 # 𝐵))

11-Aug-2021cjap0d 9548 A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (∗‘𝐴) # 0)

11-Aug-2021ap0gt0d 7630 A nonzero nonnegative number is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 # 0)       (𝜑 → 0 < 𝐴)

11-Aug-2021ap0gt0 7629 A nonnegative number is apart from zero if and only if it is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 # 0 ↔ 0 < 𝐴))

10-Aug-2021rersqrtthlem 9628 Lemma for resqrtth 9629. (Contributed by Jim Kingdon, 10-Aug-2021.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (√‘𝐴)))

10-Aug-2021rersqreu 9626 Existence and uniqueness for the real square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃!𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))

10-Aug-2021rsqrmo 9625 Uniqueness for the square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃*𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))

9-Aug-2021resqrexlemoverl 9619 Lemma for resqrex 9624. Every term in the sequence is an overestimate compared with the limit 𝐿. Although this theorem is stated in terms of a particular sequence the proof could be adapted for any decreasing convergent sequence. (Contributed by Jim Kingdon, 9-Aug-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐿 ∈ ℝ)    &   (𝜑 → ∀𝑒 ∈ ℝ+𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ𝑗)((𝐹𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹𝑖) + 𝑒)))    &   (𝜑𝐾 ∈ ℕ)       (𝜑𝐿 ≤ (𝐹𝐾))

8-Aug-2021resqrexlemga 9621 Lemma for resqrex 9624. The sequence formed by squaring each term of 𝐹 converges to 𝐴. (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐿 ∈ ℝ)    &   (𝜑 → ∀𝑒 ∈ ℝ+𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ𝑗)((𝐹𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹𝑖) + 𝑒)))    &   𝐺 = (𝑥 ∈ ℕ ↦ ((𝐹𝑥)↑2))       (𝜑 → ∀𝑒 ∈ ℝ+𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)((𝐺𝑘) < (𝐴 + 𝑒) ∧ 𝐴 < ((𝐺𝑘) + 𝑒)))

8-Aug-2021resqrexlemglsq 9620 Lemma for resqrex 9624. The sequence formed by squaring each term of 𝐹 converges to (𝐿↑2). (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐿 ∈ ℝ)    &   (𝜑 → ∀𝑒 ∈ ℝ+𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ𝑗)((𝐹𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹𝑖) + 𝑒)))    &   𝐺 = (𝑥 ∈ ℕ ↦ ((𝐹𝑥)↑2))       (𝜑 → ∀𝑒 ∈ ℝ+𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)((𝐺𝑘) < ((𝐿↑2) + 𝑒) ∧ (𝐿↑2) < ((𝐺𝑘) + 𝑒)))

8-Aug-2021recvguniqlem 9592 Lemma for recvguniq 9593. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝐴 < ((𝐹𝐾) + ((𝐴𝐵) / 2)))    &   (𝜑 → (𝐹𝐾) < (𝐵 + ((𝐴𝐵) / 2)))       (𝜑 → ⊥)

8-Aug-2021ifcldcd 3358 Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)    &   (𝜑DECID 𝜓)       (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)

8-Aug-2021ifbothdc 3357 A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))       ((𝜓𝜒DECID 𝜑) → 𝜃)

7-Aug-2021resqrexlemsqa 9622 Lemma for resqrex 9624. The square of a limit is 𝐴. (Contributed by Jim Kingdon, 7-Aug-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐿 ∈ ℝ)    &   (𝜑 → ∀𝑒 ∈ ℝ+𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ𝑗)((𝐹𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹𝑖) + 𝑒)))       (𝜑 → (𝐿↑2) = 𝐴)

7-Aug-2021resqrexlemgt0 9618 Lemma for resqrex 9624. A limit is nonnegative. (Contributed by Jim Kingdon, 7-Aug-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐿 ∈ ℝ)    &   (𝜑 → ∀𝑒 ∈ ℝ+𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ𝑗)((𝐹𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹𝑖) + 𝑒)))       (𝜑 → 0 ≤ 𝐿)

7-Aug-2021recvguniq 9593 Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐿 ∈ ℝ)    &   (𝜑 → ∀𝑥 ∈ ℝ+𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) < (𝐿 + 𝑥) ∧ 𝐿 < ((𝐹𝑘) + 𝑥)))    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → ∀𝑥 ∈ ℝ+𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) < (𝑀 + 𝑥) ∧ 𝑀 < ((𝐹𝑘) + 𝑥)))       (𝜑𝐿 = 𝑀)

6-Aug-2021resqrexlemcvg 9617 Lemma for resqrex 9624. The sequence has a limit. (Contributed by Jim Kingdon, 6-Aug-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℝ+𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ𝑗)((𝐹𝑖) < (𝑟 + 𝑥) ∧ 𝑟 < ((𝐹𝑖) + 𝑥)))

6-Aug-2021cvg1nlemcxze 9581 Lemma for cvg1n 9585. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)
(𝜑𝐶 ∈ ℝ+)    &   (𝜑𝑋 ∈ ℝ+)    &   (𝜑𝑍 ∈ ℕ)    &   (𝜑𝐸 ∈ ℕ)    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑 → ((((𝐶 · 2) / 𝑋) / 𝑍) + 𝐴) < 𝐸)       (𝜑 → (𝐶 / (𝐸 · 𝑍)) < (𝑋 / 2))

1-Aug-2021cvg1n 9585 Convergence of real sequences.

This is a version of caucvgre 9580 with a constant multiplier 𝐶 on the rate of convergence. That is, all terms after the nth term must be within 𝐶 / 𝑛 of the nth term.

(Contributed by Jim Kingdon, 1-Aug-2021.)

(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((𝐹𝑛) < ((𝐹𝑘) + (𝐶 / 𝑛)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝐶 / 𝑛))))       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ+𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ𝑗)((𝐹𝑖) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹𝑖) + 𝑥)))

1-Aug-2021cvg1nlemres 9584 Lemma for cvg1n 9585. The original sequence 𝐹 has a limit (turns out it is the same as the limit of the modified sequence 𝐺). (Contributed by Jim Kingdon, 1-Aug-2021.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((𝐹𝑛) < ((𝐹𝑘) + (𝐶 / 𝑛)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝐶 / 𝑛))))    &   𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍)))    &   (𝜑𝑍 ∈ ℕ)    &   (𝜑𝐶 < 𝑍)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ+𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ𝑗)((𝐹𝑖) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹𝑖) + 𝑥)))

1-Aug-2021cvg1nlemcau 9583 Lemma for cvg1n 9585. By selecting spaced out terms for the modified sequence 𝐺, the terms are within 1 / 𝑛 (without the constant 𝐶). (Contributed by Jim Kingdon, 1-Aug-2021.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((𝐹𝑛) < ((𝐹𝑘) + (𝐶 / 𝑛)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝐶 / 𝑛))))    &   𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍)))    &   (𝜑𝑍 ∈ ℕ)    &   (𝜑𝐶 < 𝑍)       (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((𝐺𝑛) < ((𝐺𝑘) + (1 / 𝑛)) ∧ (𝐺𝑘) < ((𝐺𝑛) + (1 / 𝑛))))

1-Aug-2021cvg1nlemf 9582 Lemma for cvg1n 9585. The modified sequence 𝐺 is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((𝐹𝑛) < ((𝐹𝑘) + (𝐶 / 𝑛)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝐶 / 𝑛))))    &   𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍)))    &   (𝜑𝑍 ∈ ℕ)    &   (𝜑𝐶 < 𝑍)       (𝜑𝐺:ℕ⟶ℝ)

31-Jul-2021resqrexlemnm 9616 Lemma for resqrex 9624. The difference between two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 31-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁𝑀)       (𝜑 → ((𝐹𝑁) − (𝐹𝑀)) < ((((𝐹‘1)↑2) · 2) / (2↑(𝑁 − 1))))

31-Jul-2021resqrexlemdecn 9610 Lemma for resqrex 9624. The sequence is decreasing. (Contributed by Jim Kingdon, 31-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 < 𝑀)       (𝜑 → (𝐹𝑀) < (𝐹𝑁))

30-Jul-2021resqrexlemnmsq 9615 Lemma for resqrex 9624. The difference between the squares of two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 30-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁𝑀)       (𝜑 → (((𝐹𝑁)↑2) − ((𝐹𝑀)↑2)) < (((𝐹‘1)↑2) / (4↑(𝑁 − 1))))

30-Jul-2021divmuldivapd 7806 Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐷 # 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷)))

29-Jul-2021resqrexlemcalc3 9614 Lemma for resqrex 9624. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       ((𝜑𝑁 ∈ ℕ) → (((𝐹𝑁)↑2) − 𝐴) ≤ (((𝐹‘1)↑2) / (4↑(𝑁 − 1))))

29-Jul-2021resqrexlemcalc2 9613 Lemma for resqrex 9624. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       ((𝜑𝑁 ∈ ℕ) → (((𝐹‘(𝑁 + 1))↑2) − 𝐴) ≤ ((((𝐹𝑁)↑2) − 𝐴) / 4))

29-Jul-2021resqrexlemcalc1 9612 Lemma for resqrex 9624. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       ((𝜑𝑁 ∈ ℕ) → (((𝐹‘(𝑁 + 1))↑2) − 𝐴) = (((((𝐹𝑁)↑2) − 𝐴)↑2) / (4 · ((𝐹𝑁)↑2))))

29-Jul-2021resqrexlemlo 9611 Lemma for resqrex 9624. A (variable) lower bound for each term of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       ((𝜑𝑁 ∈ ℕ) → (1 / (2↑𝑁)) < (𝐹𝑁))

29-Jul-2021resqrexlemdec 9609 Lemma for resqrex 9624. The sequence is decreasing. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       ((𝜑𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) < (𝐹𝑁))

28-Jul-2021resqrexlemp1rp 9604 Lemma for resqrex 9624. Applying the recursion rule yields a positive real (expressed in a way that will help apply iseqf 9224 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       ((𝜑 ∧ (𝐵 ∈ ℝ+𝐶 ∈ ℝ+)) → (𝐵(𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2))𝐶) ∈ ℝ+)

28-Jul-2021resqrexlem1arp 9603 Lemma for resqrex 9624. 1 + 𝐴 is a positive real (expressed in a way that will help apply iseqf 9224 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       ((𝜑𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) ∈ ℝ+)

28-Jul-2021rpap0d 8628 A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 # 0)

27-Jul-2021resqrexlemex 9623 Lemma for resqrex 9624. Existence of square root given a sequence which converges to the square root. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))

27-Jul-2021resqrexlemover 9608 Lemma for resqrex 9624. Each element of the sequence is an overestimate. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       ((𝜑𝑁 ∈ ℕ) → 𝐴 < ((𝐹𝑁)↑2))

27-Jul-2021resqrexlemfp1 9607 Lemma for resqrex 9624. Recursion rule. This sequence is the ancient method for computing square roots, often known as the babylonian method, although known to many ancient cultures. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       ((𝜑𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) = (((𝐹𝑁) + (𝐴 / (𝐹𝑁))) / 2))

27-Jul-2021resqrexlemf1 9606 Lemma for resqrex 9624. Initial value. Although this sequence converges to the square root with any positive initial value, this choice makes various steps in the proof of convergence easier. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (𝐹‘1) = (1 + 𝐴))

27-Jul-2021resqrexlemf 9605 Lemma for resqrex 9624. The sequence is a function. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑𝐹:ℕ⟶ℝ+)

23-Jul-2021iseqf 9224 Range of the recursive sequence builder. (Contributed by Jim Kingdon, 23-Jul-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑆𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥𝑍) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹, 𝑆):𝑍𝑆)

22-Jul-2021ialgrlemconst 9882 Lemma for ialgr0 9883. Closure of a constant function, in a form suitable for theorems such as iseq1 9222 or iseqfn 9221. (Contributed by Jim Kingdon, 22-Jul-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑆)       ((𝜑𝑥 ∈ (ℤ𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)

22-Jul-2021ialgrlem1st 9881 Lemma for ialgr0 9883. Expressing algrflemg 5851 in a form suitable for theorems such as iseq1 9222 or iseqfn 9221. (Contributed by Jim Kingdon, 22-Jul-2021.)
(𝜑𝐹:𝑆𝑆)       ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)

22-Jul-2021algrflemg 5851 Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.)
((𝐵𝑉𝐶𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵))

21-Jul-2021zmod1congr 9183 Two arbitrary integers are congruent modulo 1, see example 4 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 mod 1) = (𝐵 mod 1))

20-Jul-2021caucvgrelemcau 9579 Lemma for caucvgre 9580. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((𝐹𝑛) < ((𝐹𝑘) + (1 / 𝑛)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (1 / 𝑛))))       (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ ℕ (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))

20-Jul-2021caucvgrelemrec 9578 Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (𝑟 ∈ ℝ (𝐴 · 𝑟) = 1) = (1 / 𝐴))

19-Jul-2021caucvgre 9580 Convergence of real sequences.

A Cauchy sequence (as defined here, which has a rate of convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / 𝑛 of the nth term.

(Contributed by Jim Kingdon, 19-Jul-2021.)

(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((𝐹𝑛) < ((𝐹𝑘) + (1 / 𝑛)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (1 / 𝑛))))       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ+𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ𝑗)((𝐹𝑖) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹𝑖) + 𝑥)))

19-Jul-2021ax-caucvg 7004 Completeness. Axiom for real and complex numbers, justified by theorem axcaucvg 6974.

A Cauchy sequence (as defined here, which has a rate convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / 𝑛 of the nth term.

This axiom should not be used directly; instead use caucvgre 9580 (which is the same, but stated in terms of the and 1 / 𝑛 notations). (Contributed by Jim Kingdon, 19-Jul-2021.) (New usage is discouraged.)

𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 < 𝑥 → ∃𝑗𝑁𝑘𝑁 (𝑗 < 𝑘 → ((𝐹𝑘) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹𝑘) + 𝑥)))))

18-Jul-2021mulnqpr 6675 Multiplication of fractions embedded into positive reals. One can either multiply the fractions as fractions, or embed them into positive reals and multiply them as positive reals, and get the same result. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → ⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))

18-Jul-2021mulnqprlemfu 6674 Lemma for mulnqpr 6675. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))

18-Jul-2021mulnqprlemfl 6673 Lemma for mulnqpr 6675. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))

18-Jul-2021mulnqprlemru 6672 Lemma for mulnqpr 6675. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))

18-Jul-2021mulnqprlemrl 6671 Lemma for mulnqpr 6675. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))

18-Jul-2021lt2mulnq 6503 Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.)
(((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → ((𝐴 <Q 𝐵𝐶 <Q 𝐷) → (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)))

17-Jul-2021recidpirqlemcalc 6933 Lemma for recidpirq 6934. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.)
(𝜑𝐴P)    &   (𝜑𝐵P)    &   (𝜑 → (𝐴 ·P 𝐵) = 1P)       (𝜑 → ((((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) +P 1P) = ((((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) +P (1P +P 1P)))

17-Jul-2021recidpipr 6932 Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.)
(𝑁N → (⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) = 1P)

15-Jul-2021rereceu 6963 The reciprocal from axprecex 6954 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃!𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)

15-Jul-2021recidpirq 6934 A real number times its reciprocal is one, where reciprocal is expressed with *Q. (Contributed by Jim Kingdon, 15-Jul-2021.)
(𝑁N → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = 1)

15-Jul-2021recnnre 6927 Embedding the reciprocal of a natural number into . (Contributed by Jim Kingdon, 15-Jul-2021.)
(𝑁N → ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)

15-Jul-2021pitore 6926 Embedding from N to . Similar to pitonn 6924 but separate in the sense that we have not proved nnssre 7918 yet. (Contributed by Jim Kingdon, 15-Jul-2021.)
(𝑁N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)

15-Jul-2021pitoregt0 6925 Embedding from N to yields a number greater than zero. (Contributed by Jim Kingdon, 15-Jul-2021.)
(𝑁N → 0 < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)

14-Jul-2021adddivflid 9134 The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴))

14-Jul-2021nnindnn 6967 Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 7930 designed for real number axioms which involve natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝑧 = 1 → (𝜑𝜓))    &   (𝑧 = 𝑘 → (𝜑𝜒))    &   (𝑧 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑧 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑘𝑁 → (𝜒𝜃))       (𝐴𝑁𝜏)

14-Jul-2021peano5nnnn 6966 Peano's inductive postulate. This is a counterpart to peano5nni 7917 designed for real number axioms which involve natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}       ((1 ∈ 𝐴 ∧ ∀𝑧𝐴 (𝑧 + 1) ∈ 𝐴) → 𝑁𝐴)

14-Jul-2021peano2nnnn 6929 A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 7926 designed for real number axioms which involve to natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}       (𝐴𝑁 → (𝐴 + 1) ∈ 𝑁)

14-Jul-2021peano1nnnn 6928 One is an element of . This is a counterpart to 1nn 7925 designed for real number axioms which involve natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}       1 ∈ 𝑁

14-Jul-2021addvalex 6920 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 7006. (Contributed by Jim Kingdon, 14-Jul-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴 + 𝐵) ∈ V)

13-Jul-2021nntopi 6968 Mapping from to N. (Contributed by Jim Kingdon, 13-Jul-2021.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}       (𝐴𝑁 → ∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝐴)

13-Jul-2021ltrennb 6930 Ordering of natural numbers with <N or <. (Contributed by Jim Kingdon, 13-Jul-2021.)
((𝐽N𝐾N) → (𝐽 <N 𝐾 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))

12-Jul-2021ltrenn 6931 Ordering of natural numbers with <N or <. (Contributed by Jim Kingdon, 12-Jul-2021.)
(𝐽 <N 𝐾 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)

11-Jul-2021recriota 6964 Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.)
(𝑁N → (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)

11-Jul-2021elrealeu 6906 The real number mapping in elreal 6905 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
(𝐴 ∈ ℝ ↔ ∃!𝑥R𝑥, 0R⟩ = 𝐴)

10-Jul-2021axcaucvglemres 6973 Lemma for axcaucvg 6974. Mapping the limit from N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 < 𝑥 → ∃𝑗𝑁𝑘𝑁 (𝑗 < 𝑘 → ((𝐹𝑘) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹𝑘) + 𝑥)))))

10-Jul-2021axcaucvglemval 6971 Lemma for axcaucvg 6974. Value of sequence when mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))       ((𝜑𝐽N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝐽), 0R⟩)

10-Jul-2021axcaucvglemcl 6969 Lemma for axcaucvg 6974. Mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)       ((𝜑𝐽N) → (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) ∈ R)

9-Jul-2021axcaucvglemcau 6972 Lemma for axcaucvg 6974. The result of mapping to N and R satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))       (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))

9-Jul-2021axcaucvglemf 6970 Lemma for axcaucvg 6974. Mapping to N and R yields a sequence. (Contributed by Jim Kingdon, 9-Jul-2021.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))       (𝜑𝐺:NR)

8-Jul-2021fldiv4p1lem1div2 9147 The floor of an integer equal to 3 or greater than 4, increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
((𝑁 = 3 ∨ 𝑁 ∈ (ℤ‘5)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))

8-Jul-2021divfl0 9138 The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (⌊‘(𝐴 / 𝐵)) = 0))

8-Jul-2021div4p1lem1div2 8177 An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))

8-Jul-2021axcaucvg 6974 Real number completeness axiom. A Cauchy sequence with a modulus of convergence converges. 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).

Because we are stating this axiom before we have introduced notations for or division, we use 𝑁 for the natural numbers and express a reciprocal in terms of .

This construction-dependent theorem should not be referenced directly; instead, use ax-caucvg 7004. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.)

𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 < 𝑥 → ∃𝑗𝑁𝑘𝑁 (𝑗 < 𝑘 → ((𝐹𝑘) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹𝑘) + 𝑥)))))

7-Jul-2021ltadd1sr 6861 Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)
(𝐴R𝐴 <R (𝐴 +R 1R))

7-Jul-2021lteupri 6715 The difference from ltexpri 6711 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
(𝐴<P 𝐵 → ∃!𝑥P (𝐴 +P 𝑥) = 𝐵)

4-Jul-2021caucvgsrlemasr 6874 Lemma for caucvgsr 6886. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)
(𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))       (𝜑𝐴R)

3-Jul-2021caucvgsrlemoffres 6884 Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))    &   𝐺 = (𝑎N ↦ (((𝐹𝑎) +R 1R) +R (𝐴 ·R -1R)))       (𝜑 → ∃𝑦R𝑥R (0R <R 𝑥 → ∃𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹𝑘) +R 𝑥)))))

3-Jul-2021caucvgsrlemoffgt1 6883 Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))    &   𝐺 = (𝑎N ↦ (((𝐹𝑎) +R 1R) +R (𝐴 ·R -1R)))       (𝜑 → ∀𝑚N 1R <R (𝐺𝑚))

3-Jul-2021caucvgsrlemoffcau 6882 Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))    &   𝐺 = (𝑎N ↦ (((𝐹𝑎) +R 1R) +R (𝐴 ·R -1R)))       (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))

3-Jul-2021caucvgsrlemofff 6881 Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))    &   𝐺 = (𝑎N ↦ (((𝐹𝑎) +R 1R) +R (𝐴 ·R -1R)))       (𝜑𝐺:NR)

3-Jul-2021caucvgsrlemoffval 6880 Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))    &   𝐺 = (𝑎N ↦ (((𝐹𝑎) +R 1R) +R (𝐴 ·R -1R)))       ((𝜑𝐽N) → ((𝐺𝐽) +R 𝐴) = ((𝐹𝐽) +R 1R))

2-Jul-2021caucvgsrlemcl 6873 Lemma for caucvgsr 6886. Terms of the sequence from caucvgsrlemgt1 6879 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))       ((𝜑𝐴N) → (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)

29-Jun-2021ledivge1le 8652 If a number is less than or equal to another number, the number divided by a positive number greater than or equal to one is less than or equal to the other number. (Contributed by AV, 29-Jun-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ (𝐶 ∈ ℝ+ ∧ 1 ≤ 𝐶)) → (𝐴𝐵 → (𝐴 / 𝐶) ≤ 𝐵))

29-Jun-2021divle1le 8651 A real number divided by a positive real number is less than or equal to 1 iff the real number is less than or equal to the positive real number. (Contributed by AV, 29-Jun-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴𝐵))

29-Jun-2021caucvgsrlemfv 6875 Lemma for caucvgsr 6886. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))    &   𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))       ((𝜑𝐴N) → [⟨((𝐺𝐴) +P 1P), 1P⟩] ~R = (𝐹𝐴))

29-Jun-2021prsrriota 6872 Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
((𝐴R ∧ 0R <R 𝐴) → [⟨((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩] ~R = 𝐴)

29-Jun-2021prsrlt 6871 Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)
((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ [⟨(𝐴 +P 1P), 1P⟩] ~R <R [⟨(𝐵 +P 1P), 1P⟩] ~R ))

((𝐴P𝐵P) → [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R = ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ))

25-Jun-2021caucvgsrlembound 6878 Lemma for caucvgsr 6886. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))    &   𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))       (𝜑 → ∀𝑚N 1P<P (𝐺𝑚))

25-Jun-2021prsrpos 6869 Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.)
(𝐴P → 0R <R [⟨(𝐴 +P 1P), 1P⟩] ~R )

25-Jun-2021prsrcl 6868 Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.)
(𝐴P → [⟨(𝐴 +P 1P), 1P⟩] ~RR)

25-Jun-2021srpospr 6867 Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)
((𝐴R ∧ 0R <R 𝐴) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)

23-Jun-2021caucvgsrlemcau 6877 Lemma for caucvgsr 6886. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))    &   𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))       (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛)<P ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐺𝑘)<P ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))

23-Jun-2021caucvgsrlemf 6876 Lemma for caucvgsr 6886. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))    &   𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))       (𝜑𝐺:NP)

22-Jun-2021caucvgsrlemgt1 6879 Lemma for caucvgsr 6886. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))       (𝜑 → ∃𝑦R𝑥R (0R <R 𝑥 → ∃𝑗N𝑖N (𝑗 <N 𝑖 → ((𝐹𝑖) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹𝑖) +R 𝑥)))))

20-Jun-2021caucvgsr 6886 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 6810 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 6885).

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 6881).

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 6810 to get a limit (see caucvgsrlemgt1 6879).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6879).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6884). (Contributed by Jim Kingdon, 20-Jun-2021.)

(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))       (𝜑 → ∃𝑦R𝑥R (0R <R 𝑥 → ∃𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹𝑘) +R 𝑥)))))

19-Jun-20212tnp1ge0ge0 9143 Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.)
(𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁))

19-Jun-2021caucvgsrlembnd 6885 Lemma for caucvgsr 6886. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))       (𝜑 → ∃𝑦R𝑥R (0R <R 𝑥 → ∃𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹𝑘) +R 𝑥)))))

19-Jun-2021caucvgprprlemclphr 6803 Lemma for caucvgprpr 6810. The putative limit is a positive real. Like caucvgprprlemcl 6802 but without a distinct variable constraint between 𝜑 and 𝑟. (Contributed by Jim Kingdon, 19-Jun-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑𝐿P)

19-Jun-2021ltnqpr 6691 We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 19-Jun-2021.)
((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))

17-Jun-2021caucvgprprlemnbj 6791 Lemma for caucvgprpr 6810. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑𝐵N)    &   (𝜑𝐽N)       (𝜑 → ¬ (((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹𝐽))

16-Jun-2021caucvgprprlemexbt 6804 Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩    &   (𝜑𝑄Q)    &   (𝜑𝑇P)    &   (𝜑 → (𝐿 +P ⟨{𝑝𝑝 <Q 𝑄}, {𝑞𝑄 <Q 𝑞}⟩)<P 𝑇)       (𝜑 → ∃𝑏N (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q 𝑄}, {𝑞𝑄 <Q 𝑞}⟩)<P 𝑇)

16-Jun-2021prplnqu 6718 Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)
(𝜑𝑋P)    &   (𝜑𝑄Q)    &   (𝜑𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))       (𝜑 → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)

15-Jun-2021caucvgprprlemexb 6805 Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩    &   (𝜑𝑄P)    &   (𝜑𝑅N)       (𝜑 → (((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑅) +P 𝑄) → ∃𝑏N (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹𝑅) +P 𝑄)))

5-Jun-2021caucvgprprlemaddq 6806 Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩    &   (𝜑𝑋P)    &   (𝜑𝑄P)    &   (𝜑 → ∃𝑟N (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))       (𝜑𝑋<P (𝐿 +P 𝑄))

27-Mar-2021notnotd 560 Deduction associated with notnot 559 and notnoti 574. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
(𝜑𝜓)       (𝜑 → ¬ ¬ 𝜓)

20-Mar-2021subfzo0 9097 The difference between two elements in a half-open range of nonnegative integers is greater than the negation of the upper bound and less than the upper bound of the range. (Contributed by AV, 20-Mar-2021.)
((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁)) → (-𝑁 < (𝐼𝐽) ∧ (𝐼𝐽) < 𝑁))

3-Mar-2021caucvgprprlemval 6786 Lemma for caucvgprpr 6810. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))       ((𝜑𝐴 <N 𝐵) → ((𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))

27-Feb-2021recnnpr 6646 The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.)
(𝐴N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)

12-Feb-2021caucvgprprlemnjltk 6789 Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       ((𝜑𝐽 <N 𝐾) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))

12-Feb-2021caucvgprprlemnkeqj 6788 Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       ((𝜑𝐾 = 𝐽) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))

12-Feb-2021caucvgprprlemnkltj 6787 Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       ((𝜑𝐾 <N 𝐽) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))

12-Feb-2021caucvgprprlemcbv 6785 Lemma for caucvgprpr 6810. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))       (𝜑 → ∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))

8-Feb-2021caucvgprprlemloccalc 6782 Lemma for caucvgprpr 6810. Rearranging some expressions for caucvgprprlemloc 6801. (Contributed by Jim Kingdon, 8-Feb-2021.)
(𝜑𝑆 <Q 𝑇)    &   (𝜑𝑌Q)    &   (𝜑 → (𝑆 +Q 𝑌) = 𝑇)    &   (𝜑𝑋Q)    &   (𝜑 → (𝑋 +Q 𝑋) <Q 𝑌)    &   (𝜑𝑀N)    &   (𝜑 → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋)       (𝜑 → (⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝑇}, {𝑢𝑇 <Q 𝑢}⟩)

5-Feb-2021fvinim0ffz 9096 The function values for the borders of a finite interval of integers, which is the domain of the function, are not in the image of the interior of the interval iff the intersection of the images of the interior and the borders is empty. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.)
((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))

28-Jan-2021caucvgprprlemelu 6784 Lemma for caucvgprpr 6810. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.)
𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝑋 ∈ (2nd𝐿) ↔ (𝑋Q ∧ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩))

21-Jan-2021caucvgprprlemell 6783 Lemma for caucvgprpr 6810. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)
𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝑋 ∈ (1st𝐿) ↔ (𝑋Q ∧ ∃𝑏N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))

20-Jan-2021caucvgprprlemnkj 6790 Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       (𝜑 → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))

8-Jan-2021ltnqpri 6692 We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 8-Jan-2021.)
(𝐴 <Q 𝐵 → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)

29-Dec-2020caucvgprprlemmu 6793 Lemma for caucvgprpr 6810. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑 → ∃𝑡Q 𝑡 ∈ (2nd𝐿))

29-Dec-2020caucvgprprlemml 6792 Lemma for caucvgprpr 6810. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑 → ∃𝑠Q 𝑠 ∈ (1st𝐿))

21-Dec-2020caucvgprprlemloc 6801 Lemma for caucvgprpr 6810. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑 → ∀𝑠Q𝑡Q (𝑠 <Q 𝑡 → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿))))

21-Dec-2020caucvgprprlemdisj 6800 Lemma for caucvgprpr 6810. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))

21-Dec-2020caucvgprprlemrnd 6799 Lemma for caucvgprpr 6810. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿))) ∧ ∀𝑡Q (𝑡 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))))

21-Dec-2020caucvgprprlemupu 6798 Lemma for caucvgprpr 6810. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       ((𝜑𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)) → 𝑡 ∈ (2nd𝐿))

21-Dec-2020caucvgprprlemopu 6797 Lemma for caucvgprpr 6810. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       ((𝜑𝑡 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))

21-Dec-2020caucvgprprlemlol 6796 Lemma for caucvgprpr 6810. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       ((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))

21-Dec-2020caucvgprprlemopl 6795 Lemma for caucvgprpr 6810. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)))

21-Dec-2020caucvgprprlemm 6794 Lemma for caucvgprpr 6810. The putative limit is inhabited. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑡Q 𝑡 ∈ (2nd𝐿)))

29-Nov-2020nqpru 6650 Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by <P. (Contributed by Jim Kingdon, 29-Nov-2020.)
((𝐴Q𝐵P) → (𝐴 ∈ (2nd𝐵) ↔ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))

28-Nov-2020caucvgprprlemk 6781 Lemma for caucvgprpr 6810. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.)
(𝜑𝐽 <N 𝐾)    &   (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)       (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)

25-Nov-2020caucvgprprlem2 6808 Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩    &   (𝜑𝑄P)    &   (𝜑𝐽 <N 𝐾)    &   (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)       (𝜑𝐿<P ((𝐹𝐾) +P 𝑄))

25-Nov-2020caucvgprprlem1 6807 Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩    &   (𝜑𝑄P)    &   (𝜑𝐽 <N 𝐾)    &   (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)       (𝜑 → (𝐹𝐾)<P (𝐿 +P 𝑄))

25-Nov-2020archrecpr 6762 Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)
(𝐴P → ∃𝑗N ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝐴)

21-Nov-2020caucvgprprlemlim 6809 Lemma for caucvgprpr 6810. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑 → ∀𝑥P𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹𝑘) +P 𝑥))))

21-Nov-2020caucvgprprlemcl 6802 Lemma for caucvgprpr 6810. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑𝐿P)

17-Nov-2020pm3.2 126 Join antecedents with conjunction. Theorem *3.2 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) (Proof shortened by Jia Ming, 17-Nov-2020.)
(𝜑 → (𝜓 → (𝜑𝜓)))

14-Nov-2020caucvgprpr 6810 A Cauchy sequence of positive reals with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, 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). We also specify that every term needs to be larger than a given value 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This is similar to caucvgpr 6780 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 6760) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.)

(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))       (𝜑 → ∃𝑦P𝑥P𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘)<P (𝑦 +P 𝑥) ∧ 𝑦<P ((𝐹𝑘) +P 𝑥))))

27-Oct-2020bj-omssonALT 10088 Alternate proof of bj-omsson 10087. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
ω ⊆ On

27-Oct-2020bj-omsson 10087 Constructive proof of omsson 4335. See also bj-omssonALT 10088. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
ω ⊆ On

27-Oct-2020bj-nnelon 10084 A natural number is an ordinal. Constructive proof of nnon 4332. Can also be proved from bj-omssonALT 10088. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → 𝐴 ∈ On)

27-Oct-2020bj-nnord 10083 A natural number is an ordinal. Constructive proof of nnord 4334. Can also be proved from bj-nnelon 10084 if the latter is proved from bj-omssonALT 10088. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Ord 𝐴)

27-Oct-2020bj-axempty2 10014 Axiom of the empty set from bounded separation, alternate version to bj-axempty 10013. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.)
𝑥𝑦 ¬ 𝑦𝑥

25-Oct-2020bj-indind 10056 If 𝐴 is inductive and 𝐵 is "inductive in 𝐴", then (𝐴𝐵) is inductive. (Contributed by BJ, 25-Oct-2020.)
((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → Ind (𝐴𝐵))

25-Oct-2020bj-axempty 10013 Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a non-empty universe. See axnul 3882. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.)
𝑥𝑦𝑥

25-Oct-2020bj-axemptylem 10012 Lemma for bj-axempty 10013 and bj-axempty2 10014. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.)
𝑥𝑦(𝑦𝑥 → ⊥)

23-Oct-2020caucvgprlemnkj 6764 Lemma for caucvgpr 6780. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       (𝜑 → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))

20-Oct-2020caucvgprlemupu 6770 Lemma for caucvgpr 6780. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿))

20-Oct-2020caucvgprlemopu 6769 Lemma for caucvgpr 6780. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       ((𝜑𝑟 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))

20-Oct-2020caucvgprlemlol 6768 Lemma for caucvgpr 6780. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))

20-Oct-2020caucvgprlemopl 6767 Lemma for caucvgpr 6780. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))

18-Oct-2020caucvgprlemnbj 6765 Lemma for caucvgpr 6780. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑𝐵N)    &   (𝜑𝐽N)       (𝜑 → ¬ (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽))

9-Oct-2020caucvgprlemladdfu 6775 Lemma for caucvgpr 6780. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ⊆ {𝑢Q ∣ ∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑢})

9-Oct-2020caucvgprlemk 6763 Lemma for caucvgpr 6780. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.)
(𝜑𝐽 <N 𝐾)    &   (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑄)       (𝜑 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑄)

8-Oct-2020caucvgprlemladdrl 6776 Lemma for caucvgpr 6780. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))

3-Oct-2020caucvgprlem2 6778 Lemma for caucvgpr 6780. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    &   (𝜑𝑄Q)    &   (𝜑𝐽 <N 𝐾)    &   (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑄)       (𝜑𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩)

3-Oct-2020caucvgprlem1 6777 Lemma for caucvgpr 6780. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    &   (𝜑𝑄Q)    &   (𝜑𝐽 <N 𝐾)    &   (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑄)       (𝜑 → ⟨{𝑙𝑙 <Q (𝐹𝐾)}, {𝑢 ∣ (𝐹𝐾) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))

3-Oct-2020ltnnnq 6521 Ordering of positive integers via <N or <Q is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.)
((𝐴N𝐵N) → (𝐴 <N 𝐵 ↔ [⟨𝐴, 1𝑜⟩] ~Q <Q [⟨𝐵, 1𝑜⟩] ~Q ))

1-Oct-2020caucvgprlemlim 6779 Lemma for caucvgpr 6780. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       (𝜑 → ∀𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))

27-Sep-2020caucvgprlemloc 6773 Lemma for caucvgpr 6780. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       (𝜑 → ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))

27-Sep-2020caucvgprlemdisj 6772 Lemma for caucvgpr 6780. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))

27-Sep-2020caucvgprlemrnd 6771 Lemma for caucvgpr 6780. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))))

27-Sep-2020caucvgprlemm 6766 Lemma for caucvgpr 6780. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))

27-Sep-2020archrecnq 6761 Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝐴Q → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝐴)

26-Sep-2020caucvgprlemcl 6774 Lemma for caucvgpr 6780. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       (𝜑𝐿P)

26-Aug-2020sqrt0rlem 9601 Lemma for sqrt0 9602. (Contributed by Jim Kingdon, 26-Aug-2020.)
((𝐴 ∈ ℝ ∧ ((𝐴↑2) = 0 ∧ 0 ≤ 𝐴)) ↔ 𝐴 = 0)

23-Aug-2020sqrtrval 9598 Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.)
(𝐴 ∈ ℝ → (√‘𝐴) = (𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))

23-Aug-2020df-rsqrt 9596 Define a function whose value is the square root of a nonnegative real number.

Defining the square root for complex numbers has one difficult part: choosing between the two roots. The usual way to define a principal square root for all complex numbers relies on excluded middle or something similar. But in the case of a nonnegative real number, we don't have the complications presented for general complex numbers, and we can choose the nonnegative root.

(Contributed by Jim Kingdon, 23-Aug-2020.)

√ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℝ ((𝑦↑2) = 𝑥 ∧ 0 ≤ 𝑦)))

19-Aug-2020addnqpr 6659 Addition of fractions embedded into positive reals. One can either add the fractions as fractions, or embed them into positive reals and add them as positive reals, and get the same result. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → ⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))

19-Aug-2020addnqprlemfu 6658 Lemma for addnqpr 6659. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))

19-Aug-2020addnqprlemfl 6657 Lemma for addnqpr 6659. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))

19-Aug-2020addnqprlemru 6656 Lemma for addnqpr 6659. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))

19-Aug-2020addnqprlemrl 6655 Lemma for addnqpr 6659. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))

15-Aug-2020caucvgprlemcanl 6742 Lemma for cauappcvgprlemladdrl 6755. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)
(𝜑𝐿P)    &   (𝜑𝑆Q)    &   (𝜑𝑅Q)    &   (𝜑𝑄Q)       (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ 𝑅 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))

4-Aug-2020cauappcvgprlemupu 6747 Lemma for cauappcvgpr 6760. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿))

4-Aug-2020cauappcvgprlemopu 6746 Lemma for cauappcvgpr 6760. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       ((𝜑𝑟 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))

4-Aug-2020cauappcvgprlemlol 6745 Lemma for cauappcvgpr 6760. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))

4-Aug-2020cauappcvgprlemopl 6744 Lemma for cauappcvgpr 6760. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))

18-Jul-2020cauappcvgprlemloc 6750 Lemma for cauappcvgpr 6760. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       (𝜑 → ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))

18-Jul-2020cauappcvgprlemdisj 6749 Lemma for cauappcvgpr 6760. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))

18-Jul-2020cauappcvgprlemrnd 6748 Lemma for cauappcvgpr 6760. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))))

18-Jul-2020cauappcvgprlemm 6743 Lemma for cauappcvgpr 6760. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))

11-Jul-2020cauappcvgprlemladdrl 6755 Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))

11-Jul-2020cauappcvgprlemladdru 6754 Lemma for cauappcvgprlemladd 6756. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) ⊆ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))

11-Jul-2020cauappcvgprlemladdfl 6753 Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))

11-Jul-2020cauappcvgprlemladdfu 6752 Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))

8-Jul-2020nqprl 6649 Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by <P. (Contributed by Jim Kingdon, 8-Jul-2020.)
((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) ↔ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))

24-Jun-2020nqprlu 6645 The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.)
(𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)

23-Jun-2020cauappcvgprlem2 6758 Lemma for cauappcvgpr 6760. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑄Q)    &   (𝜑𝑅Q)       (𝜑𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)

23-Jun-2020cauappcvgprlem1 6757 Lemma for cauappcvgpr 6760. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑄Q)    &   (𝜑𝑅Q)       (𝜑 → ⟨{𝑙𝑙 <Q (𝐹𝑄)}, {𝑢 ∣ (𝐹𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩))

23-Jun-2020cauappcvgprlemladd 6756 Lemma for cauappcvgpr 6760. This takes 𝐿 and offsets it by the positive fraction 𝑆. (Contributed by Jim Kingdon, 23-Jun-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)

20-Jun-2020cauappcvgprlemlim 6759 Lemma for cauappcvgpr 6760. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       (𝜑 → ∀𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))

20-Jun-2020cauappcvgprlemcl 6751 Lemma for cauappcvgpr 6760. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       (𝜑𝐿P)

19-Jun-2020cauappcvgpr 6760 A Cauchy approximation has a limit. A Cauchy approximation, here 𝐹, is similar to a Cauchy sequence but is indexed by the desired tolerance (that is, how close together terms needs to be) rather than by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p. (varies) with a few differences such as that we are proving the existence of a limit without anything about how fast it converges (that is, mere existence instead of existence, in HoTT terms), and that the codomain of 𝐹 is Q rather than P. We also specify that every term needs to be larger than a fraction 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of caucvgpr 6780 and caucvgprpr 6810 but is somewhat simpler, so reading this one first may help understanding the other two.

(Contributed by Jim Kingdon, 19-Jun-2020.)

(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))       (𝜑 → ∃𝑦P𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))

18-Jun-2020caucvgpr 6780 A Cauchy sequence of positive fractions with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, 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). We also specify that every term needs to be larger than a fraction 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of cauappcvgpr 6760 and caucvgprpr 6810. Reading cauappcvgpr 6760 first (the simplest of the three) might help understanding the other two.

(Contributed by Jim Kingdon, 18-Jun-2020.)

(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))       (𝜑 → ∃𝑦P𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))

15-Jun-2020imdivapd 9575 Imaginary part of a division. Related to remul2 9473. (Contributed by Jim Kingdon, 15-Jun-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (ℑ‘(𝐵 / 𝐴)) = ((ℑ‘𝐵) / 𝐴))

15-Jun-2020redivapd 9574 Real part of a division. Related to remul2 9473. (Contributed by Jim Kingdon, 15-Jun-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (ℜ‘(𝐵 / 𝐴)) = ((ℜ‘𝐵) / 𝐴))

15-Jun-2020cjdivapd 9568 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))

15-Jun-2020riotaexg 5472 Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
(𝐴𝑉 → (𝑥𝐴 𝜓) ∈ V)

14-Jun-2020cjdivapi 9535 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))

14-Jun-2020cjdivap 9509 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))

14-Jun-2020cjap0 9507 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
(𝐴 ∈ ℂ → (𝐴 # 0 ↔ (∗‘𝐴) # 0))

14-Jun-2020cjap 9506 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) # (∗‘𝐵) ↔ 𝐴 # 𝐵))

14-Jun-2020imdivap 9481 Imaginary part of a division. Related to immul2 9480. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (ℑ‘(𝐴 / 𝐵)) = ((ℑ‘𝐴) / 𝐵))

14-Jun-2020redivap 9474 Real part of a division. Related to remul2 9473. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (ℜ‘(𝐴 / 𝐵)) = ((ℜ‘𝐴) / 𝐵))

14-Jun-2020mulreap 9464 A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 ∈ ℝ ↔ (𝐵 · 𝐴) ∈ ℝ))

13-Jun-2020sqgt0apd 9408 The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 # 0)       (𝜑 → 0 < (𝐴↑2))

13-Jun-2020reexpclzapd 9405 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ∈ ℝ)

13-Jun-2020expdivapd 9395 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝐴 / 𝐵)↑𝑁) = ((𝐴𝑁) / (𝐵𝑁)))

13-Jun-2020sqdivapd 9394 Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))

12-Jun-2020expsubapd 9392 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐴↑(𝑀𝑁)) = ((𝐴𝑀) / (𝐴𝑁)))

12-Jun-2020expm1apd 9391 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴↑(𝑁 − 1)) = ((𝐴𝑁) / 𝐴))

12-Jun-2020expp1zapd 9390 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴𝑁) · 𝐴))

12-Jun-2020exprecapd 9389 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → ((1 / 𝐴)↑𝑁) = (1 / (𝐴𝑁)))

12-Jun-2020expnegapd 9388 Value of a complex number raised to a negative power. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴↑-𝑁) = (1 / (𝐴𝑁)))

12-Jun-2020expap0d