![]() |
Metamath
Proof Explorer Theorem List (p. 362 of 491) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30946) |
![]() (30947-32469) |
![]() (32470-49035) |
Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-outsideof 36101 | The outside of relationship. This relationship expresses that 𝑃, 𝐴, and 𝐵 fall on a line, but 𝑃 is not on the segment 𝐴𝐵. This definition is taken from theorem 6.4 of [Schwabhauser] p. 43, since it requires no dummy variables. (Contributed by Scott Fenton, 17-Oct-2013.) |
⊢ OutsideOf = ( Colinear ∖ Btwn ) | ||
Theorem | broutsideof 36102 | Binary relation form of OutsideOf. Theorem 6.4 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ (𝑃 Colinear 〈𝐴, 𝐵〉 ∧ ¬ 𝑃 Btwn 〈𝐴, 𝐵〉)) | ||
Theorem | broutsideof2 36103 | Alternate form of OutsideOf. Definition 6.1 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ (𝐴 Btwn 〈𝑃, 𝐵〉 ∨ 𝐵 Btwn 〈𝑃, 𝐴〉)))) | ||
Theorem | outsidene1 36104 | Outsideness implies inequality. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf〈𝐴, 𝐵〉 → 𝐴 ≠ 𝑃)) | ||
Theorem | outsidene2 36105 | Outsideness implies inequality. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf〈𝐴, 𝐵〉 → 𝐵 ≠ 𝑃)) | ||
Theorem | btwnoutside 36106 | A principle linking outsideness to betweenness. Theorem 6.2 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → (((𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ 𝐶 ≠ 𝑃) ∧ 𝑃 Btwn 〈𝐴, 𝐶〉) → (𝑃 Btwn 〈𝐵, 𝐶〉 ↔ 𝑃OutsideOf〈𝐴, 𝐵〉))) | ||
Theorem | broutsideof3 36107* | Characterization of outsideness in terms of relationship to a fourth point. Theorem 6.3 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ ∃𝑐 ∈ (𝔼‘𝑁)(𝑐 ≠ 𝑃 ∧ 𝑃 Btwn 〈𝐴, 𝑐〉 ∧ 𝑃 Btwn 〈𝐵, 𝑐〉)))) | ||
Theorem | outsideofrflx 36108 | Reflexivity of outsideness. Theorem 6.5 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → (𝐴 ≠ 𝑃 → 𝑃OutsideOf〈𝐴, 𝐴〉)) | ||
Theorem | outsideofcom 36109 | Commutativity law for outsideness. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ 𝑃OutsideOf〈𝐵, 𝐴〉)) | ||
Theorem | outsideoftr 36110 | Transitivity law for outsideness. Theorem 6.7 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → ((𝑃OutsideOf〈𝐴, 𝐵〉 ∧ 𝑃OutsideOf〈𝐵, 𝐶〉) → 𝑃OutsideOf〈𝐴, 𝐶〉)) | ||
Theorem | outsideofeq 36111 | Uniqueness law for OutsideOf. Analogue of segconeq 35991. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → (((𝐴OutsideOf〈𝑋, 𝑅〉 ∧ 〈𝐴, 𝑋〉Cgr〈𝐵, 𝐶〉) ∧ (𝐴OutsideOf〈𝑌, 𝑅〉 ∧ 〈𝐴, 𝑌〉Cgr〈𝐵, 𝐶〉)) → 𝑋 = 𝑌)) | ||
Theorem | outsideofeu 36112* | Given a nondegenerate ray, there is a unique point congruent to the segment 𝐵𝐶 lying on the ray 𝐴𝑅. Theorem 6.11 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 23-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ((𝑅 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶) → ∃!𝑥 ∈ (𝔼‘𝑁)(𝐴OutsideOf〈𝑥, 𝑅〉 ∧ 〈𝐴, 𝑥〉Cgr〈𝐵, 𝐶〉))) | ||
Theorem | outsidele 36113 | Relate OutsideOf to Seg≤. Theorem 6.13 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf〈𝐴, 𝐵〉 → (〈𝑃, 𝐴〉 Seg≤ 〈𝑃, 𝐵〉 ↔ 𝐴 Btwn 〈𝑃, 𝐵〉))) | ||
Theorem | outsideofcol 36114 | Outside of implies colinearity. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (𝑃OutsideOf〈𝑄, 𝑅〉 → 𝑃 Colinear 〈𝑄, 𝑅〉) | ||
Syntax | cline2 36115 | Declare the constant for the line function. |
class Line | ||
Syntax | cray 36116 | Declare the constant for the ray function. |
class Ray | ||
Syntax | clines2 36117 | Declare the constant for the set of all lines. |
class LinesEE | ||
Definition | df-line2 36118* | Define the Line function. This function generates the line passing through the distinct points 𝑎 and 𝑏. Adapted from definition 6.14 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 25-Oct-2013.) |
⊢ Line = {〈〈𝑎, 𝑏〉, 𝑙〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear )} | ||
Definition | df-ray 36119* | Define the Ray function. This function generates the set of all points that lie on the ray starting at 𝑝 and passing through 𝑎. Definition 6.8 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 21-Oct-2013.) |
⊢ Ray = {〈〈𝑝, 𝑎〉, 𝑟〉 ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf〈𝑎, 𝑥〉})} | ||
Definition | df-lines2 36120 | Define the set of all lines. Definition 6.14, part 2 of [Schwabhauser] p. 45. See ellines 36133 for membership. (Contributed by Scott Fenton, 28-Oct-2013.) |
⊢ LinesEE = ran Line | ||
Theorem | funray 36121 | Show that the Ray relationship is a function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ Fun Ray | ||
Theorem | fvray 36122* | Calculate the value of the Ray function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → (𝑃Ray𝐴) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf〈𝐴, 𝑥〉}) | ||
Theorem | funline 36123 | Show that the Line relationship is a function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ Fun Line | ||
Theorem | linedegen 36124 | When Line is applied with the same argument, the result is the empty set. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (𝐴Line𝐴) = ∅ | ||
Theorem | fvline 36125* | Calculate the value of the Line function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = {𝑥 ∣ 𝑥 Colinear 〈𝐴, 𝐵〉}) | ||
Theorem | liness 36126 | A line is a subset of the space its two points lie in. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) ⊆ (𝔼‘𝑁)) | ||
Theorem | fvline2 36127* | Alternate definition of a line. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear 〈𝐴, 𝐵〉}) | ||
Theorem | lineunray 36128 | A line is composed of a point and the two rays emerging from it. Theorem 6.15 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → (𝑃 Btwn 〈𝑄, 𝑅〉 → (𝑃Line𝑄) = (((𝑃Ray𝑄) ∪ {𝑃}) ∪ (𝑃Ray𝑅)))) | ||
Theorem | lineelsb2 36129 | If 𝑆 lies on 𝑃𝑄, then 𝑃𝑄 = 𝑃𝑆. Theorem 6.16 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → (𝑆 ∈ (𝑃Line𝑄) → (𝑃Line𝑄) = (𝑃Line𝑆))) | ||
Theorem | linerflx1 36130 | Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → 𝑃 ∈ (𝑃Line𝑄)) | ||
Theorem | linecom 36131 | Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → (𝑃Line𝑄) = (𝑄Line𝑃)) | ||
Theorem | linerflx2 36132 | Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → 𝑄 ∈ (𝑃Line𝑄)) | ||
Theorem | ellines 36133* | Membership in the set of all lines. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))) | ||
Theorem | linethru 36134 | If 𝐴 is a line containing two distinct points 𝑃 and 𝑄, then 𝐴 is the line through 𝑃 and 𝑄. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ LinesEE ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄)) | ||
Theorem | hilbert1.1 36135* | There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → ∃𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) | ||
Theorem | hilbert1.2 36136* | There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by NM, 17-Jun-2017.) |
⊢ (𝑃 ≠ 𝑄 → ∃*𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) | ||
Theorem | linethrueu 36137* | There is a unique line going through any two distinct points. Theorem 6.19 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → ∃!𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) | ||
Theorem | lineintmo 36138* | Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE ∧ 𝐴 ≠ 𝐵) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | ||
Syntax | cfwddif 36139 | Declare the syntax for the forward difference operator. |
class △ | ||
Definition | df-fwddif 36140* | Define the forward difference operator. This is a discrete analogue of the derivative operator. Definition 2.42 of [GramKnuthPat], p. 47. (Contributed by Scott Fenton, 18-May-2020.) |
⊢ △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥)))) | ||
Syntax | cfwddifn 36141 | Declare the syntax for the nth forward difference operator. |
class △n | ||
Definition | df-fwddifn 36142* | Define the nth forward difference operator. This works out to be the forward difference operator iterated 𝑛 times. (Contributed by Scott Fenton, 28-May-2020.) |
⊢ △n = (𝑛 ∈ ℕ0, 𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘)))))) | ||
Theorem | fwddifval 36143 | Calculate the value of the forward difference operator at a point. (Contributed by Scott Fenton, 18-May-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → (𝑋 + 1) ∈ 𝐴) ⇒ ⊢ (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋))) | ||
Theorem | fwddifnval 36144* | The value of the forward difference operator at a point. (Contributed by Scott Fenton, 28-May-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ 𝐴) ⇒ ⊢ (𝜑 → ((𝑁 △n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘))))) | ||
Theorem | fwddifn0 36145 | The value of the n-iterated forward difference operator at zero is just the function value. (Contributed by Scott Fenton, 28-May-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → ((0 △n 𝐹)‘𝑋) = (𝐹‘𝑋)) | ||
Theorem | fwddifnp1 36146* | The value of the n-iterated forward difference at a successor. (Contributed by Scott Fenton, 28-May-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑋 + 𝑘) ∈ 𝐴) ⇒ ⊢ (𝜑 → (((𝑁 + 1) △n 𝐹)‘𝑋) = (((𝑁 △n 𝐹)‘(𝑋 + 1)) − ((𝑁 △n 𝐹)‘𝑋))) | ||
Theorem | rankung 36147 | The rank of the union of two sets. Closed form of rankun 9893. (Contributed by Scott Fenton, 15-Jul-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))) | ||
Theorem | ranksng 36148 | The rank of a singleton. Closed form of ranksn 9891. (Contributed by Scott Fenton, 15-Jul-2015.) |
⊢ (𝐴 ∈ 𝑉 → (rank‘{𝐴}) = suc (rank‘𝐴)) | ||
Theorem | rankelg 36149 | The membership relation is inherited by the rank function. Closed form of rankel 9876. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝐵) → (rank‘𝐴) ∈ (rank‘𝐵)) | ||
Theorem | rankpwg 36150 | The rank of a power set. Closed form of rankpw 9880. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ (𝐴 ∈ 𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴)) | ||
Theorem | rank0 36151 | The rank of the empty set is ∅. (Contributed by Scott Fenton, 17-Jul-2015.) |
⊢ (rank‘∅) = ∅ | ||
Theorem | rankeq1o 36152 | The only set with rank 1o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.) |
⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = {∅}) | ||
Syntax | chf 36153 | The constant Hf is a class. |
class Hf | ||
Definition | df-hf 36154 | Define the hereditarily finite sets. These are the finite sets whose elements are finite, and so forth. (Contributed by Scott Fenton, 9-Jul-2015.) |
⊢ Hf = ∪ (𝑅1 “ ω) | ||
Theorem | elhf 36155* | Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.) |
⊢ (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥)) | ||
Theorem | elhf2 36156 | Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω) | ||
Theorem | elhf2g 36157 | Hereditarily finiteness via rank. Closed form of elhf2 36156. (Contributed by Scott Fenton, 15-Jul-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)) | ||
Theorem | 0hf 36158 | The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.) |
⊢ ∅ ∈ Hf | ||
Theorem | hfun 36159 | The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.) |
⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ 𝐵) ∈ Hf ) | ||
Theorem | hfsn 36160 | The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.) |
⊢ (𝐴 ∈ Hf → {𝐴} ∈ Hf ) | ||
Theorem | hfadj 36161 | Adjoining one HF element to an HF set preserves HF status. (Contributed by Scott Fenton, 15-Jul-2015.) |
⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ {𝐵}) ∈ Hf ) | ||
Theorem | hfelhf 36162 | Any member of an HF set is itself an HF set. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ Hf ) → 𝐴 ∈ Hf ) | ||
Theorem | hftr 36163 | The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ Tr Hf | ||
Theorem | hfext 36164* | Extensionality for HF sets depends only on comparison of HF elements. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ Hf (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) | ||
Theorem | hfuni 36165 | The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ Hf ) | ||
Theorem | hfpw 36166 | The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ (𝐴 ∈ Hf → 𝒫 𝐴 ∈ Hf ) | ||
Theorem | hfninf 36167 | ω is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) |
⊢ ¬ ω ∈ Hf | ||
Theorem | rmoeqi 36168 | Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓) | ||
Theorem | rmoeqbii 36169 | Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜓 ↔ 𝜒) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒) | ||
Theorem | reueqi 36170 | Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓) | ||
Theorem | reueqbii 36171 | Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜓 ↔ 𝜒) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒) | ||
Theorem | sbceqbii 36172 | Formula-building inference for class substitution. General version of sbcbii 3851. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜓) | ||
Theorem | disjeq1i 36173 | Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶) | ||
Theorem | disjeq12i 36174 | Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷) | ||
Theorem | rabeqbii 36175 | Equality theorem for restricted class abstractions. Inference version. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓} | ||
Theorem | iuneq12i 36176 | Equality theorem for indexed union. Inference version. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 | ||
Theorem | iineq1i 36177 | Equality theorem for indexed intersection. Inference version. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶 | ||
Theorem | iineq12i 36178 | Equality theorem for indexed intersection. Inference version. General version of iineq1i 36177. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷 | ||
Theorem | riotaeqbii 36179 | Equivalent wff's and equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓) | ||
Theorem | riotaeqi 36180 | Equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑) | ||
Theorem | ixpeq1i 36181 | Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶 | ||
Theorem | ixpeq12i 36182 | Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷 | ||
Theorem | sumeq2si 36183 | Equality inference for sum. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐵 = 𝐶 ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 | ||
Theorem | sumeq12si 36184 | Equality inference for sum. General version of sumeq2si 36183. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ Σ𝑥 ∈ 𝐴 𝐶 = Σ𝑥 ∈ 𝐵 𝐷 | ||
Theorem | prodeq2si 36185 | Equality inference for product. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐵 = 𝐶 ⇒ ⊢ ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 | ||
Theorem | prodeq12si 36186 | Equality inference for product. General version of prodeq2si 36185. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐷 | ||
Theorem | itgeq12i 36187 | Equality inference for an integral. General version of itgeq1i 36188 and itgeq2i 36189. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥 | ||
Theorem | itgeq1i 36188 | Equality inference for an integral. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥 | ||
Theorem | itgeq2i 36189 | Equality inference for an integral. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐵 = 𝐶 ⇒ ⊢ ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥 | ||
Theorem | ditgeq123i 36190 | Equality inference for the directed integral. General version of ditgeq12i 36191 and ditgeq3i 36192. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 & ⊢ 𝐸 = 𝐹 ⇒ ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑥 | ||
Theorem | ditgeq12i 36191 | Equality inference for the directed integral. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥 | ||
Theorem | ditgeq3i 36192 | Equality inference for the directed integral. (Contributed by GG, 1-Sep-2025.) |
⊢ 𝐶 = 𝐷 ⇒ ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑥 | ||
Theorem | rmoeqdv 36193* | Formula-building rule for restricted at-most-one quantifier. Deduction form. (Contributed by GG, 1-Sep-2025.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | rmoeqbidv 36194* | Formula-building rule for restricted at-most-one quantifier. Deduction form. General version of rmobidv 3394. (Contributed by GG, 1-Sep-2025.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | reueqbidv 36195* | Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reubidv 3395. (Contributed by GG, 1-Sep-2025.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | sbequbidv 36196* | Deduction substituting both sides of a biconditional. (Contributed by GG, 1-Sep-2025.) |
⊢ (𝜑 → 𝑢 = 𝑣) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝑢 / 𝑥]𝜓 ↔ [𝑣 / 𝑥]𝜒)) | ||
Theorem | disjeq12dv 36197* | Equality theorem for disjoint collection. Deduction version. (Contributed by GG, 1-Sep-2025.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) | ||
Theorem | ixpeq12dv 36198* | Equality theorem for infinite Cartesian product. Deduction version. (Contributed by GG, 1-Sep-2025.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷) | ||
Theorem | sumeq12sdv 36199* | Equality deduction for sum. General version of sumeq2sdv 15735. (Contributed by GG, 1-Sep-2025.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷) | ||
Theorem | prodeq12sdv 36200* | Equality deduction for product. General version of prodeq2sdv 15955. (Contributed by GG, 1-Sep-2025.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐷) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |