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Theorem List for Intuitionistic Logic Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcj0 9501 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
(∗‘0) = 0

Theoremcji 9502 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
(∗‘i) = -i

Theoremcjreim 9503 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 + (i · 𝐵))) = (𝐴 − (i · 𝐵)))

Theoremcjreim2 9504 The conjugate of the representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 − (i · 𝐵))) = (𝐴 + (i · 𝐵)))

Theoremcj11 9505 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) = (∗‘𝐵) ↔ 𝐴 = 𝐵))

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

Theoremcjap0 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))

Theoremcjne0 9508 A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 9507 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)
(𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ (∗‘𝐴) ≠ 0))

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

Theoremcnrecnv 9510* The inverse to the canonical bijection from (ℝ × ℝ) to from cnref1o 8582. (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))       𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)

Theoremrecli 9511 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (ℜ‘𝐴) ∈ ℝ

Theoremimcli 9512 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (ℑ‘𝐴) ∈ ℝ

Theoremcjcli 9513 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (∗‘𝐴) ∈ ℂ

Theoremreplimi 9514 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
𝐴 ∈ ℂ       𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))

Theoremcjcji 9515 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (∗‘(∗‘𝐴)) = 𝐴

Theoremreim0bi 9516 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)

Theoremrerebi 9517 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)
𝐴 ∈ ℂ       (𝐴 ∈ ℝ ↔ (ℜ‘𝐴) = 𝐴)

Theoremcjrebi 9518 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.)
𝐴 ∈ ℂ       (𝐴 ∈ ℝ ↔ (∗‘𝐴) = 𝐴)

Theoremrecji 9519 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)

Theoremimcji 9520 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)

Theoremcjmulrcli 9521 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (𝐴 · (∗‘𝐴)) ∈ ℝ

Theoremcjmulvali 9522 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (𝐴 · (∗‘𝐴)) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))

Theoremcjmulge0i 9523 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
𝐴 ∈ ℂ       0 ≤ (𝐴 · (∗‘𝐴))

Theoremrenegi 9524 Real part of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (ℜ‘-𝐴) = -(ℜ‘𝐴)

Theoremimnegi 9525 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (ℑ‘-𝐴) = -(ℑ‘𝐴)

Theoremcjnegi 9526 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (∗‘-𝐴) = -(∗‘𝐴)

Theoremaddcji 9527 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (𝐴 + (∗‘𝐴)) = (2 · (ℜ‘𝐴))

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (ℜ‘(𝐴 + 𝐵)) = ((ℜ‘𝐴) + (ℜ‘𝐵))

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵))

Theoremremuli 9530 Real part of a product. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))

Theoremimmuli 9531 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵)))

Theoremcjaddi 9532 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (∗‘(𝐴 + 𝐵)) = ((∗‘𝐴) + (∗‘𝐵))

Theoremcjmuli 9533 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (∗‘(𝐴 · 𝐵)) = ((∗‘𝐴) · (∗‘𝐵))

Theoremipcni 9534 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (ℜ‘(𝐴 · (∗‘𝐵))) = (((ℜ‘𝐴) · (ℜ‘𝐵)) + ((ℑ‘𝐴) · (ℑ‘𝐵)))

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

Theoremcrrei 9536 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (ℜ‘(𝐴 + (i · 𝐵))) = 𝐴

Theoremcrimi 9537 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (ℑ‘(𝐴 + (i · 𝐵))) = 𝐵

Theoremrecld 9538 The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (ℜ‘𝐴) ∈ ℝ)

Theoremimcld 9539 The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (ℑ‘𝐴) ∈ ℝ)

Theoremcjcld 9540 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (∗‘𝐴) ∈ ℂ)

Theoremreplimd 9541 Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴))))

Theoremremimd 9542 Value of the conjugate of a complex number. The value is the real part minus i times the imaginary part. Definition 10-3.2 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (∗‘𝐴) = ((ℜ‘𝐴) − (i · (ℑ‘𝐴))))

Theoremcjcjd 9543 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (∗‘(∗‘𝐴)) = 𝐴)

Theoremreim0bd 9544 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (ℑ‘𝐴) = 0)       (𝜑𝐴 ∈ ℝ)

Theoremrerebd 9545 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (ℜ‘𝐴) = 𝐴)       (𝜑𝐴 ∈ ℝ)

Theoremcjrebd 9546 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (∗‘𝐴) = 𝐴)       (𝜑𝐴 ∈ ℝ)

Theoremcjne0d 9547 A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 9548 which is similar but for apartness. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (∗‘𝐴) ≠ 0)

Theoremcjap0d 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)

Theoremrecjd 9549 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴))

Theoremimcjd 9550 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴))

Theoremcjmulrcld 9551 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · (∗‘𝐴)) ∈ ℝ)

Theoremcjmulvald 9552 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · (∗‘𝐴)) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2)))

Theoremcjmulge0d 9553 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → 0 ≤ (𝐴 · (∗‘𝐴)))

Theoremrenegd 9554 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (ℜ‘-𝐴) = -(ℜ‘𝐴))

Theoremimnegd 9555 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (ℑ‘-𝐴) = -(ℑ‘𝐴))

Theoremcjnegd 9556 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (∗‘-𝐴) = -(∗‘𝐴))

Theoremaddcjd 9557 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 + (∗‘𝐴)) = (2 · (ℜ‘𝐴)))

Theoremcjexpd 9558 Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (∗‘(𝐴𝑁)) = ((∗‘𝐴)↑𝑁))

Theoremreaddd 9559 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (ℜ‘(𝐴 + 𝐵)) = ((ℜ‘𝐴) + (ℜ‘𝐵)))

Theoremimaddd 9560 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵)))

Theoremresubd 9561 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (ℜ‘(𝐴𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵)))

Theoremimsubd 9562 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (ℑ‘(𝐴𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))

Theoremremuld 9563 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵))))

Theoremimmuld 9564 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵))))

Theoremcjaddd 9565 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (∗‘(𝐴 + 𝐵)) = ((∗‘𝐴) + (∗‘𝐵)))

Theoremcjmuld 9566 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (∗‘(𝐴 · 𝐵)) = ((∗‘𝐴) · (∗‘𝐵)))

Theoremipcnd 9567 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (ℜ‘(𝐴 · (∗‘𝐵))) = (((ℜ‘𝐴) · (ℜ‘𝐵)) + ((ℑ‘𝐴) · (ℑ‘𝐵))))

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

Theoremrered 9569 A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (ℜ‘𝐴) = 𝐴)

Theoremreim0d 9570 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (ℑ‘𝐴) = 0)

Theoremcjred 9571 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (∗‘𝐴) = 𝐴)

Theoremremul2d 9572 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵)))

Theoremimmul2d 9573 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵)))

Theoremredivapd 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))

Theoremimdivapd 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))

Theoremcrred 9576 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (ℜ‘(𝐴 + (i · 𝐵))) = 𝐴)

Theoremcrimd 9577 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (ℑ‘(𝐴 + (i · 𝐵))) = 𝐵)

3.7.3  Sequence convergence

Theoremcaucvgrelemrec 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))

Theoremcaucvgrelemcau 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)))))

Theoremcaucvgre 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)))

Theoremcvg1nlemcxze 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))

Theoremcvg1nlemf 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)

Theoremcvg1nlemcau 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))))

Theoremcvg1nlemres 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)))

Theoremcvg1n 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)))

Theoremuzin2 9586 The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)
((𝐴 ∈ ran ℤ𝐵 ∈ ran ℤ) → (𝐴𝐵) ∈ ran ℤ)

Theoremrexanuz 9587* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)
(∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))

Theoremrexuz3 9588* Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))

Theoremrexanuz2 9589* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓))

Theoremr19.29uz 9590* A version of 19.29 1511 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)
𝑍 = (ℤ𝑀)       ((∀𝑘𝑍 𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓))

Theoremr19.2uz 9591* A version of r19.2m 3309 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)
𝑍 = (ℤ𝑀)       (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 → ∃𝑘𝑍 𝜑)

Theoremrecvguniqlem 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 → ⊥)

Theoremrecvguniq 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)

3.7.4  Square root; absolute value

Syntaxcsqrt 9594 Extend class notation to include square root of a complex number.
class

Syntaxcabs 9595 Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs

Definitiondf-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)))

Definitiondf-abs 9597 Define the function for the absolute value (modulus) of a complex number. (Contributed by NM, 27-Jul-1999.)
abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))

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

Theoremabsval 9599 The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
(𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴))))

Theoremrennim 9600 A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.)
(𝐴 ∈ ℝ → (i · 𝐴) ∉ ℝ+)

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