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Theorem List for Metamath Proof Explorer - 31801-31900   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.3.26.3  The Ternary Goldbach Conjecture: Final Statement

Axiomax-hgt749 31801* Statement 7.49 of [Helfgott] p. 70. For a sufficiently big odd 𝑁, this postulates the existence of smoothing functions (eta star) and 𝑘 (eta plus) such that the lower bound for the circle integral is big enough. (Contributed by Thierry Arnoux, 15-Dec-2021.)
𝑛 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} ((10↑27) ≤ 𝑛 → ∃ ∈ ((0[,)+∞) ↑m ℕ)∃𝑘 ∈ ((0[,)+∞) ↑m ℕ)(∀𝑚 ∈ ℕ (𝑘𝑚) ≤ (1.079955) ∧ ∀𝑚 ∈ ℕ (𝑚) ≤ (1.414) ∧ ((0.00042248) · (𝑛↑2)) ≤ ∫(0(,)1)(((((Λ ∘f · )vts𝑛)‘𝑥) · ((((Λ ∘f · 𝑘)vts𝑛)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑛 · 𝑥)))) d𝑥))

Axiomax-ros335 31802 Theorem 12. of [RosserSchoenfeld] p. 71. Theorem chpo1ubb 25971 states that the ψ function is bounded by a linear term; this axiom postulates an upper bound for that linear term. This is stated as an axiom until a formal proof can be provided. (Contributed by Thierry Arnoux, 28-Dec-2021.)
𝑥 ∈ ℝ+ (ψ‘𝑥) < ((1.03883) · 𝑥)

Axiomax-ros336 31803 Theorem 13. of [RosserSchoenfeld] p. 71. Theorem chpchtlim 25969 states that the ψ and θ function are asymtotic to each other; this axiom postulates an upper bound for their difference. This is stated as an axiom until a formal proof can be provided. (Contributed by Thierry Arnoux, 28-Dec-2021.)
𝑥 ∈ ℝ+ ((ψ‘𝑥) − (θ‘𝑥)) < ((1.4262) · (√‘𝑥))

Theoremhgt750lemc 31804* An upper bound to the summatory function of the von Mangoldt function. (Contributed by Thierry Arnoux, 29-Dec-2021.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗) < ((1.03883) · 𝑁))

Theoremhgt750lemd 31805* An upper bound to the summatory function of the von Mangoldt function on non-primes. (Contributed by Thierry Arnoux, 29-Dec-2021.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑 → (10↑27) ≤ 𝑁)       (𝜑 → Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) < ((1.4263) · (√‘𝑁)))

Theoremhgt749d 31806* A deduction version of ax-hgt749 31801. (Contributed by Thierry Arnoux, 15-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁𝑂)    &   (𝜑 → (10↑27) ≤ 𝑁)       (𝜑 → ∃ ∈ ((0[,)+∞) ↑m ℕ)∃𝑘 ∈ ((0[,)+∞) ↑m ℕ)(∀𝑚 ∈ ℕ (𝑘𝑚) ≤ (1.079955) ∧ ∀𝑚 ∈ ℕ (𝑚) ≤ (1.414) ∧ ((0.00042248) · (𝑁↑2)) ≤ ∫(0(,)1)(((((Λ ∘f · )vts𝑁)‘𝑥) · ((((Λ ∘f · 𝑘)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥))

Theoremlogdivsqrle 31807 Conditions for ((log x ) / ( sqrt 𝑥)) to be decreasing. (Contributed by Thierry Arnoux, 20-Dec-2021.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑 → (exp‘2) ≤ 𝐴)    &   (𝜑𝐴𝐵)       (𝜑 → ((log‘𝐵) / (√‘𝐵)) ≤ ((log‘𝐴) / (√‘𝐴)))

Theoremhgt750lem 31808 Lemma for tgoldbachgtd 31819. (Contributed by Thierry Arnoux, 17-Dec-2021.)
((𝑁 ∈ ℕ0 ∧ (10↑27) ≤ 𝑁) → ((7.348) · ((log‘𝑁) / (√‘𝑁))) < (0.00042248))

Theoremhgt750lem2 31809 Decimal multiplication galore! (Contributed by Thierry Arnoux, 26-Dec-2021.)
(3 · ((((1.079955)↑2) · (1.414)) · ((1.4263) · (1.03883)))) < (7.348)

Theoremhgt750lemf 31810* Lemma for the statement 7.50 of [Helfgott] p. 69. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝑃 ∈ ℝ)    &   (𝜑𝑄 ∈ ℝ)    &   (𝜑𝐻:ℕ⟶(0[,)+∞))    &   (𝜑𝐾:ℕ⟶(0[,)+∞))    &   ((𝜑𝑛𝐴) → (𝑛‘0) ∈ ℕ)    &   ((𝜑𝑛𝐴) → (𝑛‘1) ∈ ℕ)    &   ((𝜑𝑛𝐴) → (𝑛‘2) ∈ ℕ)    &   ((𝜑𝑚 ∈ ℕ) → (𝐾𝑚) ≤ 𝑃)    &   ((𝜑𝑚 ∈ ℕ) → (𝐻𝑚) ≤ 𝑄)       (𝜑 → Σ𝑛𝐴 (((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ≤ (((𝑃↑2) · 𝑄) · Σ𝑛𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))))

Theoremhgt750lemg 31811* Lemma for the statement 7.50 of [Helfgott] p. 69. Applying a permutation 𝑇 to the three factors of a product does not change the result. (Contributed by Thierry Arnoux, 1-Jan-2022.)
𝐹 = (𝑐𝑅 ↦ (𝑐𝑇))    &   (𝜑𝑇:(0..^3)–1-1-onto→(0..^3))    &   (𝜑𝑁:(0..^3)⟶ℕ)    &   (𝜑𝐿:ℕ⟶ℝ)    &   (𝜑𝑁𝑅)       (𝜑 → ((𝐿‘((𝐹𝑁)‘0)) · ((𝐿‘((𝐹𝑁)‘1)) · (𝐿‘((𝐹𝑁)‘2)))) = ((𝐿‘(𝑁‘0)) · ((𝐿‘(𝑁‘1)) · (𝐿‘(𝑁‘2)))))

Theoremoddprm2 31812* Two ways to write the set of odd primes. (Contributed by Thierry Arnoux, 27-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}       (ℙ ∖ {2}) = (𝑂 ∩ ℙ)

Theoremhgt750lemb 31813* An upper bound on the contribution of the non-prime terms in the Statement 7.50 of [Helfgott] p. 69. (Contributed by Thierry Arnoux, 28-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 2 ≤ 𝑁)    &   𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}       (𝜑 → Σ𝑛𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ ((log‘𝑁) · (Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗))))

Theoremhgt750lema 31814* An upper bound on the contribution of the non-prime terms in the Statement 7.50 of [Helfgott] p. 69. (Contributed by Thierry Arnoux, 1-Jan-2022.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 2 ≤ 𝑁)    &   𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}    &   𝐹 = (𝑑 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐𝑎) ∈ (𝑂 ∩ ℙ)} ↦ (𝑑 ∘ if(𝑎 = 0, ( I ↾ (0..^3)), ((pmTrsp‘(0..^3))‘{𝑎, 0}))))       (𝜑 → Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ (3 · Σ𝑛𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))))

Theoremhgt750leme 31815* An upper bound on the contribution of the non-prime terms in the Statement 7.50 of [Helfgott] p. 69. (Contributed by Thierry Arnoux, 29-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (10↑27) ≤ 𝑁)    &   (𝜑𝐻:ℕ⟶(0[,)+∞))    &   (𝜑𝐾:ℕ⟶(0[,)+∞))    &   ((𝜑𝑚 ∈ ℕ) → (𝐾𝑚) ≤ (1.079955))    &   ((𝜑𝑚 ∈ ℕ) → (𝐻𝑚) ≤ (1.414))       (𝜑 → Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ≤ (((7.348) · ((log‘𝑁) / (√‘𝑁))) · (𝑁↑2)))

Theoremtgoldbachgnn 31816* Lemma for tgoldbachgtd 31819. (Contributed by Thierry Arnoux, 15-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁𝑂)    &   (𝜑 → (10↑27) ≤ 𝑁)       (𝜑𝑁 ∈ ℕ)

Theoremtgoldbachgtde 31817* Lemma for tgoldbachgtd 31819. (Contributed by Thierry Arnoux, 15-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁𝑂)    &   (𝜑 → (10↑27) ≤ 𝑁)    &   (𝜑𝐻:ℕ⟶(0[,)+∞))    &   (𝜑𝐾:ℕ⟶(0[,)+∞))    &   ((𝜑𝑚 ∈ ℕ) → (𝐾𝑚) ≤ (1.079955))    &   ((𝜑𝑚 ∈ ℕ) → (𝐻𝑚) ≤ (1.414))    &   (𝜑 → ((0.00042248) · (𝑁↑2)) ≤ ∫(0(,)1)(((((Λ ∘f · 𝐻)vts𝑁)‘𝑥) · ((((Λ ∘f · 𝐾)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)       (𝜑 → 0 < Σ𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))))

Theoremtgoldbachgtda 31818* Lemma for tgoldbachgtd 31819. (Contributed by Thierry Arnoux, 15-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁𝑂)    &   (𝜑 → (10↑27) ≤ 𝑁)    &   (𝜑𝐻:ℕ⟶(0[,)+∞))    &   (𝜑𝐾:ℕ⟶(0[,)+∞))    &   ((𝜑𝑚 ∈ ℕ) → (𝐾𝑚) ≤ (1.079955))    &   ((𝜑𝑚 ∈ ℕ) → (𝐻𝑚) ≤ (1.414))    &   (𝜑 → ((0.00042248) · (𝑁↑2)) ≤ ∫(0(,)1)(((((Λ ∘f · 𝐻)vts𝑁)‘𝑥) · ((((Λ ∘f · 𝐾)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)       (𝜑 → 0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁)))

Theoremtgoldbachgtd 31819* Odd integers greater than (10↑27) have at least a representation as a sum of three odd primes. Final statement in section 7.4 of [Helfgott] p. 70 (Contributed by Thierry Arnoux, 15-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁𝑂)    &   (𝜑 → (10↑27) ≤ 𝑁)       (𝜑 → 0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁)))

Theoremtgoldbachgt 31820* Odd integers greater than (10↑27) have at least a representation as a sum of three odd primes. Final statement in section 7.4 of [Helfgott] p. 70 , expressed using the set 𝐺 of odd numbers which can be written as a sum of three odd primes. (Contributed by Thierry Arnoux, 22-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   𝐺 = {𝑧𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝𝑂𝑞𝑂𝑟𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}       𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑛𝑂 (𝑚 < 𝑛𝑛𝐺))

20.3.27  Elementary Geometry

20.3.27.1  Two-dimensional 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 31821 Extends class notation with the class of geometries fulfilling the planarity axioms.
class TarskiG2D

Definitiondf-trkg2d 31822* 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 31823* 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))))))

Theoremaxtglowdim2ALTV 31824* Alternate version of axtglowdim2 26170. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG2D)       (𝜑 → ∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))

Theoremaxtgupdim2ALTV 31825 Alternate version of axtgupdim2 26171. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage 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)))

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

Syntaxcafs 31826 Declare the syntax for the outer five segment configuration.
class AFS

Definitiondf-afs 31827* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (axtg5seg 26165). See df-ofs 33328. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.) (Revised 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 31828* 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 31829 Binary relation 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 31830 Rephrase axtg5seg 26165 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))

leftpad = (𝑐 ∈ V, 𝑤 ∈ V ↦ (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤)))

Theoremlpadval 31833 Value of the leftpad function. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → ((𝐶 leftpad 𝑊)‘𝐿) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊))

Theoremlpadlem1 31834 Lemma for the leftpad theorems. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐶𝑆)       (𝜑 → ((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ∈ Word 𝑆)

(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝐿 ≤ (♯‘𝑊))       (𝜑 → ((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) = ∅)

Theoremlpadlen1 31836 Length of a left-padded word, in the case the length of the given word 𝑊 is at least the desired length. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝐿 ≤ (♯‘𝑊))       (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = (♯‘𝑊))

Theoremlpadlem2 31837 Lemma for the leftpad theorems. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑 → (♯‘𝑊) ≤ 𝐿)       (𝜑 → (♯‘((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) = (𝐿 − (♯‘𝑊)))

Theoremlpadlen2 31838 Length of a left-padded word, in the case the given word 𝑊 is shorter than the desired length. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑 → (♯‘𝑊) ≤ 𝐿)       (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = 𝐿)

Theoremlpadmax 31839 Length of a left-padded word, in the general case, expressed with an if statement. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿))

Theoremlpadleft 31840 The contents of prefix of a left-padded word is always the letter 𝐶. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝑁 ∈ (0..^(𝐿 − (♯‘𝑊))))       (𝜑 → (((𝐶 leftpad 𝑊)‘𝐿)‘𝑁) = 𝐶)

Theoremlpadright 31841 The suffix of a left-padded word the original word 𝑊. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝑀 = if(𝐿 ≤ (♯‘𝑊), 0, (𝐿 − (♯‘𝑊))))    &   (𝜑𝑁 ∈ (0..^(♯‘𝑊)))       (𝜑 → (((𝐶 leftpad 𝑊)‘𝐿)‘(𝑁 + 𝑀)) = (𝑊𝑁))

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 31842 Extend wff notation with the 4-way conjunction. (New usage is discouraged.)
wff (𝜑𝜓𝜒𝜃)

Definitiondf-bnj17 31843 Define the 4-way conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓𝜒) ∧ 𝜃))

Syntaxc-bnj14 31844 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 31845* 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 31846 Extend wff notation with the following predicate: 𝑅 is set-like on 𝐴. (New usage is discouraged.)
wff 𝑅 Se 𝐴

Definitiondf-bnj13 31847* 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 31848 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 31849 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 31850 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 31851* 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 32084. (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 31852 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 31853* 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 31854 -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 31855 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓𝜓′)    &   (𝜒𝜒′)       ((𝜑𝜓𝜒) → (𝜓′𝜒′))

Theorembnj248 31856 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (((𝜑𝜓) ∧ 𝜒) ∧ 𝜃))

Theorembnj250 31857 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)))

Theorembnj251 31858 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ (𝜓 ∧ (𝜒𝜃))))

Theorembnj252 31859 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ (𝜓𝜒𝜃)))

Theorembnj253 31860 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓) ∧ 𝜒𝜃))

Theorembnj255 31861 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜓 ∧ (𝜒𝜃)))

Theorembnj256 31862 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))

Theorembnj257 31863 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜓𝜃𝜒))

Theorembnj258 31864 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓𝜃) ∧ 𝜒))

Theorembnj268 31865 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜒𝜓𝜃))

Theorembnj290 31866 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜒𝜃𝜓))

Theorembnj291 31867 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜒𝜃) ∧ 𝜓))

Theorembnj312 31868 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜓𝜑𝜒𝜃))

Theorembnj334 31869 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜒𝜑𝜓𝜃))

Theorembnj345 31870 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜃𝜑𝜓𝜒))

Theorembnj422 31871 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜒𝜃𝜑𝜓))

Theorembnj432 31872 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜒𝜃) ∧ (𝜑𝜓)))

Theorembnj446 31873 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜓𝜒𝜃) ∧ 𝜑))

Theorembnj23 31874* 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 31875 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 31876* 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 31877* 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 31878* 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 31879 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑𝜓)       𝑥𝜓

Theorembnj105 31880 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
1o ∈ V

Theorembnj115 31881 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 31882* 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 31883 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∃𝑥𝜓)    &   (𝜒𝜓)       (𝜑 ↔ ∃𝑥𝜒)

Theorembnj156 31884 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 31885* 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)

Theorembnj168 31886* First-order logic and set theory. Revised to remove dependence on ax-reg 9048. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by NM, 21-Dec-2016.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚𝐷 𝑛 = suc 𝑚)

Theorembnj206 31887 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑′[𝑀 / 𝑛]𝜑)    &   (𝜓′[𝑀 / 𝑛]𝜓)    &   (𝜒′[𝑀 / 𝑛]𝜒)    &   𝑀 ∈ V       ([𝑀 / 𝑛](𝜑𝜓𝜒) ↔ (𝜑′𝜓′𝜒′))

Theorembnj216 31888 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 ∈ V       (𝐴 = suc 𝐵𝐵𝐴)

Theorembnj219 31889 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑛 = suc 𝑚𝑚 E 𝑛)

Theorembnj226 31890* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵𝐶        𝑥𝐴 𝐵𝐶

Theorembnj228 31891 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)
(𝜑 ↔ ∀𝑥𝐴 𝜓)       ((𝑥𝐴𝜑) → 𝜓)

Theorembnj519 31892 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
𝐴 ∈ V       (𝐵 ∈ V → Fun {⟨𝐴, 𝐵⟩})

Theorembnj521 31893 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝐴 ∩ {𝐴}) = ∅

Theorembnj524 31894 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑𝜓)    &   𝐴 ∈ V       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)

Theorembnj525 31895* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐴 ∈ V       ([𝐴 / 𝑥]𝜑𝜑)

Theorembnj534 31896* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → (∃𝑥𝜑𝜓))       (𝜒 → ∃𝑥(𝜑𝜓))

Theorembnj538 31897* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof shortened by OpenAI, 30-Mar-2020.)
𝐴 ∈ V       ([𝐴 / 𝑦]𝑥𝐵 𝜑 ↔ ∀𝑥𝐵 [𝐴 / 𝑦]𝜑)

Theorembnj529 31898 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       (𝑀𝐷 → ∅ ∈ 𝑀)

Theorembnj551 31899 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)

Theorembnj563 31900 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))       ((𝜂𝜌) → suc 𝑖𝑚)

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