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Theorem List for Metamath Proof Explorer - 29801-29900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremplyrecld 29801 Closure of a polynomial with real coefficients. (Contributed by Thierry Arnoux, 18-Sep-2018.)
(𝜑𝐹 ∈ (Poly‘ℝ))    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → (𝐹𝑋) ∈ ℝ)

Theoremsignsplypnf 29802* The quotient of a polynomial 𝐹 by a monic monomial of same degree 𝐺 converges to the highest coefficient of 𝐹. (Contributed by Thierry Arnoux, 18-Sep-2018.)
𝐷 = (deg‘𝐹)    &   𝐶 = (coeff‘𝐹)    &   𝐵 = (𝐶𝐷)    &   𝐺 = (𝑥 ∈ ℝ+ ↦ (𝑥𝐷))       (𝐹 ∈ (Poly‘ℝ) → (𝐹𝑓 / 𝐺) ⇝𝑟 𝐵)

Theoremsignsply0 29803* Lemma for the rule of signs, based on Bolzano's intermediate value theorem for polynomials : If the lowest and highest coefficient 𝐴 and 𝐵 are of opposite signs, the polynomial admits a positive root. (Contributed by Thierry Arnoux, 19-Sep-2018.)
𝐷 = (deg‘𝐹)    &   𝐶 = (coeff‘𝐹)    &   𝐵 = (𝐶𝐷)    &   𝐴 = (𝐶‘0)    &   (𝜑𝐹 ∈ (Poly‘ℝ))    &   (𝜑𝐹 ≠ 0𝑝)    &   (𝜑 → (𝐴 · 𝐵) < 0)       (𝜑 → ∃𝑧 ∈ ℝ+ (𝐹𝑧) = 0)

20.3.24  Descartes's rule of signs

20.3.24.1  Sign changes in a word over real numbers

Theoremsignspval 29804* The value of the skipping 0 sign operation. (Contributed by Thierry Arnoux, 9-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))       ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → (𝑋 𝑌) = if(𝑌 = 0, 𝑋, 𝑌))

Theoremsignsw0glem 29805* Neutral element property of . (Contributed by Thierry Arnoux, 9-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))       𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)

Theoremsignswbase 29806 The base of 𝑊 is the triplet reprensenting the possible signs. (Contributed by Thierry Arnoux, 9-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       {-1, 0, 1} = (Base‘𝑊)

Theoremsignswplusg 29807* The operation of 𝑊. (Contributed by Thierry Arnoux, 9-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}        = (+g𝑊)

Theoremsignsw0g 29808* The neutral element of 𝑊. (Contributed by Thierry Arnoux, 9-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       0 = (0g𝑊)

Theoremsignswmnd 29809* 𝑊 is a monoid structure on {-1, 0, 1} which operation retains the right side, but skips zeroes. This will be used for skipping zeroes when counting sign changes. (Contributed by Thierry Arnoux, 9-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       𝑊 ∈ Mnd

Theoremsignswrid 29810* The zero-skipping operation propagages nonzeros. (Contributed by Thierry Arnoux, 11-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       (𝑋 ∈ {-1, 0, 1} → (𝑋 0) = 𝑋)

Theoremsignswlid 29811* The zero-skipping operation keeps nonzeros. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       (((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) ∧ 𝑌 ≠ 0) → (𝑋 𝑌) = 𝑌)

Theoremsignswn0 29812* The zero-skipping operation propagages nonzeros. (Contributed by Thierry Arnoux, 11-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       (((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) ∧ 𝑋 ≠ 0) → (𝑋 𝑌) ≠ 0)

Theoremsignswch 29813* The zero-skipping operation changes value when the operands change signs. (Contributed by Thierry Arnoux, 9-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       ((𝑋 ∈ {-1, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → ((𝑋 𝑌) ≠ 𝑋 ↔ (𝑋 · 𝑌) < 0))

20.3.24.2  Counting sign changes in a word over real numbers

Theoremsignslema 29814 Computational part of signwlemn . (Contributed by Thierry Arnoux, 29-Sep-2018.)
(𝜑𝐸 ∈ ℕ0)    &   (𝜑𝐹 ∈ ℕ0)    &   (𝜑𝐺 ∈ ℕ0)    &   (𝜑𝐻 ∈ ℕ0)    &   (𝜑 → (𝐸 < 𝐺 ∧ ¬ 2 ∥ (𝐺𝐸)))    &   (𝜑 → ((𝐻𝐺) − (𝐹𝐸)) ∈ {0, 2})       (𝜑 → (𝐹 < 𝐻 ∧ ¬ 2 ∥ (𝐻𝐹)))

Theoremsignstfv 29815* Value of the zero-skipping sign word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝐹 ∈ Word ℝ → (𝑇𝐹) = (𝑛 ∈ (0..^(#‘𝐹)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝐹𝑖))))))

Theoremsignstfval 29816* Value of the zero-skipping sign word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(#‘𝐹))) → ((𝑇𝐹)‘𝑁) = (𝑊 Σg (𝑖 ∈ (0...𝑁) ↦ (sgn‘(𝐹𝑖)))))

Theoremsignstcl 29817* Closure of the zero skipping sign word. (Contributed by Thierry Arnoux, 9-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(#‘𝐹))) → ((𝑇𝐹)‘𝑁) ∈ {-1, 0, 1})

Theoremsignstf 29818* The zero skipping sign word is a word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝐹 ∈ Word ℝ → (𝑇𝐹) ∈ Word ℝ)

Theoremsignstlen 29819* Length of the zero skipping sign word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝐹 ∈ Word ℝ → (#‘(𝑇𝐹)) = (#‘𝐹))

Theoremsignstf0 29820* Sign of a single letter word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝐾 ∈ ℝ → (𝑇‘⟨“𝐾”⟩) = ⟨“(sgn‘𝐾)”⟩)

Theoremsignstfvn 29821* Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(#‘𝐹)) = (((𝑇𝐹)‘((#‘𝐹) − 1)) (sgn‘𝐾)))

Theoremsignsvtn0 29822* If the last letter is non zero, then this is the zero-skipping sign. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝑁 = (#‘𝐹)       ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘(𝑁 − 1)) ≠ 0) → ((𝑇𝐹)‘(𝑁 − 1)) = (sgn‘(𝐹‘(𝑁 − 1))))

Theoremsignstfvp 29823* Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(#‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑁) = ((𝑇𝐹)‘𝑁))

Theoremsignstfvneq0 29824* In case the first letter is not zero, the zero skipping sign is never zero. (Contributed by Thierry Arnoux, 10-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝑁 ∈ (0..^(#‘𝐹))) → ((𝑇𝐹)‘𝑁) ≠ 0)

Theoremsignstfvcl 29825* Closure of the zero skipping sign in case the first letter is not zero. (Contributed by Thierry Arnoux, 10-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝑁 ∈ (0..^(#‘𝐹))) → ((𝑇𝐹)‘𝑁) ∈ {-1, 1})

Theoremsignstfvc 29826* Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 11-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ Word ℝ ∧ 𝐺 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(#‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇𝐹)‘𝑁))

Theoremsignstres 29827* Restriction of a zero skipping sign to a subword. (Contributed by Thierry Arnoux, 11-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → ((𝑇𝐹) ↾ (0..^𝑁)) = (𝑇‘(𝐹 ↾ (0..^𝑁))))

Theoremsignstfveq0a 29828* Lemma for signstfveq0 29829. (Contributed by Thierry Arnoux, 11-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝑁 = (#‘𝐹)       (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → 𝑁 ∈ (ℤ‘2))

Theoremsignstfveq0 29829* In case the last letter is zero, the zero skipping sign is the same as the previous letter. (Contributed by Thierry Arnoux, 11-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝑁 = (#‘𝐹)       (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → ((𝑇𝐹)‘(𝑁 − 1)) = ((𝑇𝐹)‘(𝑁 − 2)))

Theoremsignsvvfval 29830* The value of 𝑉, which represents the number of times the sign changes in a word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝐹 ∈ Word ℝ → (𝑉𝐹) = Σ𝑗 ∈ (1..^(#‘𝐹))if(((𝑇𝐹)‘𝑗) ≠ ((𝑇𝐹)‘(𝑗 − 1)), 1, 0))

Theoremsignsvvf 29831* 𝑉 is a function. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       𝑉:Word ℝ⟶ℕ0

Theoremsignsvf0 29832* There is no change of sign in the empty word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝑉‘∅) = 0

Theoremsignsvf1 29833* In a single-letter word, which represents a constant polynomial, there is no change of sign. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝐾 ∈ ℝ → (𝑉‘⟨“𝐾”⟩) = 0)

Theoremsignsvfn 29834* Number of changes in a word compared to a shorter word. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (𝑉‘(𝐹 ++ ⟨“𝐾”⟩)) = ((𝑉𝐹) + if((((𝑇𝐹)‘((#‘𝐹) − 1)) · 𝐾) < 0, 1, 0)))

Theoremsignsvtp 29835* Adding a letter of the same sign as the highest coefficient does not change the sign. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   (𝜑𝐸 ∈ (Word ℝ ∖ {∅}))    &   (𝜑 → (𝐸‘0) ≠ 0)    &   (𝜑𝐹 = (𝐸 ++ ⟨“𝐴”⟩))    &   (𝜑𝐴 ∈ ℝ)    &   𝑁 = (#‘𝐸)    &   𝐵 = ((𝑇𝐸)‘(𝑁 − 1))       ((𝜑 ∧ 0 < (𝐴 · 𝐵)) → (𝑉𝐹) = (𝑉𝐸))

Theoremsignsvtn 29836* Adding a letter of a different sign as the highest coefficient changes the sign. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   (𝜑𝐸 ∈ (Word ℝ ∖ {∅}))    &   (𝜑 → (𝐸‘0) ≠ 0)    &   (𝜑𝐹 = (𝐸 ++ ⟨“𝐴”⟩))    &   (𝜑𝐴 ∈ ℝ)    &   𝑁 = (#‘𝐸)    &   𝐵 = ((𝑇𝐸)‘(𝑁 − 1))       ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑉𝐹) − (𝑉𝐸)) = 1)

Theoremsignsvfpn 29837* Adding a letter of the same sign as the highest coefficient does not change the sign. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   (𝜑𝐸 ∈ (Word ℝ ∖ {∅}))    &   (𝜑 → (𝐸‘0) ≠ 0)    &   (𝜑𝐹 = (𝐸 ++ ⟨“𝐴”⟩))    &   (𝜑𝐴 ∈ ℝ)    &   𝑁 = (#‘𝐸)    &   𝐵 = (𝐸‘(𝑁 − 1))       ((𝜑 ∧ 0 < (𝐵 · 𝐴)) → (𝑉𝐹) = (𝑉𝐸))

Theoremsignsvfnn 29838* Adding a letter of a different sign as the highest coefficient changes the sign. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   (𝜑𝐸 ∈ (Word ℝ ∖ {∅}))    &   (𝜑 → (𝐸‘0) ≠ 0)    &   (𝜑𝐹 = (𝐸 ++ ⟨“𝐴”⟩))    &   (𝜑𝐴 ∈ ℝ)    &   𝑁 = (#‘𝐸)    &   𝐵 = (𝐸‘(𝑁 − 1))       ((𝜑 ∧ (𝐵 · 𝐴) < 0) → ((𝑉𝐹) − (𝑉𝐸)) = 1)

Theoremsignlem0 29839* Adding a zero as the highest coefficient does not change the parity of the sign changes. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → (𝑉‘(𝐹 ++ ⟨“0”⟩)) = (𝑉𝐹))

Theoremsignshf 29840* 𝐻, corresponding to the word 𝐹 multiplied by (𝑥𝐶), as a function. (Contributed by Thierry Arnoux, 29-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝐻 = ((⟨“0”⟩ ++ 𝐹) ∘𝑓 − ((𝐹 ++ ⟨“0”⟩)∘𝑓/𝑐 · 𝐶))       ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → 𝐻:(0..^((#‘𝐹) + 1))⟶ℝ)

Theoremsignshwrd 29841* 𝐻, corresponding to the word 𝐹 multiplied by (𝑥𝐶), is a word. (Contributed by Thierry Arnoux, 29-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝐻 = ((⟨“0”⟩ ++ 𝐹) ∘𝑓 − ((𝐹 ++ ⟨“0”⟩)∘𝑓/𝑐 · 𝐶))       ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → 𝐻 ∈ Word ℝ)

Theoremsignshlen 29842* Length of 𝐻, corresponding to the word 𝐹 multiplied by (𝑥𝐶). (Contributed by Thierry Arnoux, 14-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝐻 = ((⟨“0”⟩ ++ 𝐹) ∘𝑓 − ((𝐹 ++ ⟨“0”⟩)∘𝑓/𝑐 · 𝐶))       ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → (#‘𝐻) = ((#‘𝐹) + 1))

Theoremsignshnz 29843* 𝐻 is not the empty word. (Contributed by Thierry Arnoux, 14-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝐻 = ((⟨“0”⟩ ++ 𝐹) ∘𝑓 − ((𝐹 ++ ⟨“0”⟩)∘𝑓/𝑐 · 𝐶))       ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → 𝐻 ≠ ∅)

20.3.25  Elementary Geometry

20.3.25.1  Two-dimension geometry

This definition has been superseded by DimTarskiG and is no longer needed in the main part of set.mm. It is only kept here for reference.

Syntaxcstrkg2d 29844 Extends class notation with the class of geometries fulfilling the planarity axioms.
class TarskiG2D

Definitiondf-trkg2d 29845* Define the class of geometries fulfilling the lower dimension axiom, Axiom A8 of [Schwabhauser] p. 12, and the upper dimension axiom, Axiom A9 of [Schwabhauser] p. 13, for dimension 2. (Contributed by Thierry Arnoux, 14-Mar-2019.) (New usage is discouraged.)
TarskiG2D = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∃𝑥𝑝𝑦𝑝𝑧𝑝 ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))}

Theoremistrkg2d 29846* Property of fulfilling dimension 2 axiom. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)       (𝐺 ∈ TarskiG2D ↔ (𝐺 ∈ V ∧ (∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))

Theoremaxtglowdim2OLD 29847* Lower dimension axiom for dimension 2, Axiom A8 of [Schwabhauser] p. 13. There exist 3 non-colinear points. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG2D)       (𝜑 → ∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))

Theoremaxtgupdim2OLD 29848 Upper dimension axiom for dimension 2, Axiom A9 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑈𝑃)    &   (𝜑𝑉𝑃)    &   (𝜑𝑈𝑉)    &   (𝜑 → (𝑋 𝑈) = (𝑋 𝑉))    &   (𝜑 → (𝑌 𝑈) = (𝑌 𝑉))    &   (𝜑 → (𝑍 𝑈) = (𝑍 𝑉))    &   (𝜑𝐺 ∈ TarskiG2D)       (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))

20.3.25.2  Outer Five Segment (not used, no need to move to main)

Syntaxcafs 29849 Declare the syntax for the outer five segment configuration.
class AFS

Definitiondf-afs 29850* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (axtg5seg 25057). See df-ofs 31099. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.) (Modified by Thierry Arnoux, 15-Mar-2019.)
AFS = (𝑔 ∈ TarskiG ↦ {⟨𝑒, 𝑓⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / ][(Itv‘𝑔) / 𝑖]𝑎𝑝𝑏𝑝𝑐𝑝𝑑𝑝𝑥𝑝𝑦𝑝𝑧𝑝𝑤𝑝 (𝑒 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑓 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ ((𝑏 ∈ (𝑎𝑖𝑐) ∧ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ((𝑎𝑏) = (𝑥𝑦) ∧ (𝑏𝑐) = (𝑦𝑧)) ∧ ((𝑎𝑑) = (𝑥𝑤) ∧ (𝑏𝑑) = (𝑦𝑤))))})

Theoremafsval 29851* Value of the AFS relation for a given geometry structure. (Contributed by Thierry Arnoux, 20-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)       (𝜑 → (AFS‘𝐺) = {⟨𝑒, 𝑓⟩ ∣ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃𝑥𝑃𝑦𝑃𝑧𝑃𝑤𝑃 (𝑒 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑓 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ ((𝑏 ∈ (𝑎𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝑎 𝑏) = (𝑥 𝑦) ∧ (𝑏 𝑐) = (𝑦 𝑧)) ∧ ((𝑎 𝑑) = (𝑥 𝑤) ∧ (𝑏 𝑑) = (𝑦 𝑤))))})

Theorembrafs 29852 Binary relationship form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-2013.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑂 = (AFS‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑊𝑃)       (𝜑 → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑂⟨⟨𝑋, 𝑌⟩, ⟨𝑍, 𝑊⟩⟩ ↔ ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ ((𝐴 𝐵) = (𝑋 𝑌) ∧ (𝐵 𝐶) = (𝑌 𝑍)) ∧ ((𝐴 𝐷) = (𝑋 𝑊) ∧ (𝐵 𝐷) = (𝑌 𝑊)))))

Theoremtg5segofs 29853 Rephrase axtg5seg 25057 using the outer five segment predicate. Theorem 2.10 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝑂 = (AFS‘𝐺)    &   (𝜑𝐻𝑃)    &   (𝜑𝐼𝑃)    &   (𝜑 → ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑂⟨⟨𝐸, 𝐹⟩, ⟨𝐻, 𝐼⟩⟩)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐶 𝐷) = (𝐻 𝐼))

20.4  Mathbox for Jonathan Ben-Naim

Note: On 4-Sep-2016 and after, 745 unused theorems were deleted from this mathbox, and 359 theorems used only once or twice were merged into their referencing theorems. The originals can be recovered from set.mm versions prior to this date.

Syntaxw-bnj17 29854 Extend wff notation with the 4-way conjunction. (New usage is discouraged.)
wff (𝜑𝜓𝜒𝜃)

Definitiondf-bnj17 29855 Define the 4-way conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓𝜒) ∧ 𝜃))

Syntaxc-bnj14 29856 Extend class notation with the function giving: the class of all elements of 𝐴 that are "smaller" than 𝑋 according to 𝑅. (New usage is discouraged.)
class pred(𝑋, 𝐴, 𝑅)

Definitiondf-bnj14 29857* Define the function giving: the class of all elements of 𝐴 that are "smaller" than 𝑋 according to 𝑅. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
pred(𝑋, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑋}

Syntaxw-bnj13 29858 Extend wff notation with the following predicate: 𝑅 is set-like on 𝐴. (New usage is discouraged.)
wff 𝑅 Se 𝐴

Definitiondf-bnj13 29859* Define the following predicate: 𝑅 is set-like on 𝐴. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑅 Se 𝐴 ↔ ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)

Syntaxw-bnj15 29860 Extend wff notation with the following predicate: 𝑅 is both well-founded and set-like on 𝐴. (New usage is discouraged.)
wff 𝑅 FrSe 𝐴

Definitiondf-bnj15 29861 Define the following predicate: 𝑅 is both well-founded and set-like on 𝐴. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Se 𝐴))

Syntaxc-bnj18 29862 Extend class notation with the function giving: the transitive closure of 𝑋 in 𝐴 by 𝑅. (New usage is discouraged.)
class trCl(𝑋, 𝐴, 𝑅)

Definitiondf-bnj18 29863* Define the function giving: the transitive closure of 𝑋 in 𝐴 by 𝑅. This definition has been designed for facilitating verification that it is eliminable and that the \$d restrictions are sound and complete. For a more readable definition see bnj882 30099. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
trCl(𝑋, 𝐴, 𝑅) = 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} 𝑖 ∈ dom 𝑓(𝑓𝑖)

Syntaxw-bnj19 29864 Extend wff notation with the following predicate: 𝐵 is transitive for 𝐴 and 𝑅. (New usage is discouraged.)
wff TrFo(𝐵, 𝐴, 𝑅)

Definitiondf-bnj19 29865* Define the following predicate: 𝐵 is transitive for 𝐴 and 𝑅. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵)

20.4.1  First-order logic and set theory

Theorembnj170 29866 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒) ↔ ((𝜓𝜒) ∧ 𝜑))

Theorembnj240 29867 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓𝜓′)    &   (𝜒𝜒′)       ((𝜑𝜓𝜒) → (𝜓′𝜒′))

Theorembnj248 29868 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (((𝜑𝜓) ∧ 𝜒) ∧ 𝜃))

Theorembnj250 29869 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)))

Theorembnj251 29870 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ (𝜓 ∧ (𝜒𝜃))))

Theorembnj252 29871 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ (𝜓𝜒𝜃)))

Theorembnj253 29872 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓) ∧ 𝜒𝜃))

Theorembnj255 29873 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜓 ∧ (𝜒𝜃)))

Theorembnj256 29874 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))

Theorembnj257 29875 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜓𝜃𝜒))

Theorembnj258 29876 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓𝜃) ∧ 𝜒))

Theorembnj268 29877 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜒𝜓𝜃))

Theorembnj290 29878 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜒𝜃𝜓))

Theorembnj291 29879 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜒𝜃) ∧ 𝜓))

Theorembnj312 29880 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜓𝜑𝜒𝜃))

Theorembnj334 29881 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜒𝜑𝜓𝜃))

Theorembnj345 29882 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜃𝜑𝜓𝜒))

Theorembnj422 29883 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜒𝜃𝜑𝜓))

Theorembnj432 29884 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜒𝜃) ∧ (𝜑𝜓)))

Theorembnj446 29885 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜓𝜒𝜃) ∧ 𝜑))

Theorembnj21 29886* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑥𝐴𝜑}       𝐵𝐴

Theorembnj23 29887* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
𝐵 = {𝑥𝐴 ∣ ¬ 𝜑}       (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤𝐴 (𝑤𝑅𝑦[𝑤 / 𝑥]𝜑))

Theorembnj31 29888 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝐴 𝜓)    &   (𝜓𝜒)       (𝜑 → ∃𝑥𝐴 𝜒)

Theorembnj62 29889* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
([𝑧 / 𝑥]𝑥 Fn 𝐴𝑧 Fn 𝐴)

Theorembnj89 29890* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑍 ∈ V       ([𝑍 / 𝑦]∃!𝑥𝜑 ↔ ∃!𝑥[𝑍 / 𝑦]𝜑)

Theorembnj90 29891* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
𝑌 ∈ V       ([𝑌 / 𝑥]𝑧 Fn 𝑥𝑧 Fn 𝑌)

Theorembnj101 29892 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑𝜓)       𝑥𝜓

Theorembnj105 29893 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
1𝑜 ∈ V

Theorembnj115 29894 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ ∀𝑛𝐷 (𝜏𝜃))       (𝜂 ↔ ∀𝑛((𝑛𝐷𝜏) → 𝜃))

Theorembnj132 29895* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∃𝑥(𝜓𝜒))       (𝜑 ↔ (𝜓 → ∃𝑥𝜒))

Theorembnj133 29896 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∃𝑥𝜓)    &   (𝜒𝜓)       (𝜑 ↔ ∃𝑥𝜒)

Theorembnj142OLD 29897 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.) Obsolete as of 29-Dec-2018. This is now incorporated into the proof of fnsnb 6219.
(𝐹 Fn {𝐴} → (𝑢𝐹𝑢 = ⟨𝐴, (𝐹𝐴)⟩))

Theorembnj145OLD 29898 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.) Obsolete as of 29-Dec-2018. This is now incorporated into the proof of fnsnb 6219.
𝐴 ∈ V    &   (𝐹𝐴) ∈ V       (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})

Theorembnj156 29899 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜁0 ↔ (𝑓 Fn 1𝑜𝜑′𝜓′))    &   (𝜁1[𝑔 / 𝑓]𝜁0)    &   (𝜑1[𝑔 / 𝑓]𝜑′)    &   (𝜓1[𝑔 / 𝑓]𝜓′)       (𝜁1 ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))

Theorembnj158 29900* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       (𝑚𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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