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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-bj-ccbar 34501 | Definition of the set of extended complex numbers ℂ̅. (Contributed by BJ, 22-Jun-2019.) |
⊢ ℂ̅ = (ℂ ∪ ℂ∞) | ||
Theorem | bj-ccssccbar 34502 | Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.) |
⊢ ℂ ⊆ ℂ̅ | ||
Theorem | bj-ccinftyssccbar 34503 | Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.) |
⊢ ℂ∞ ⊆ ℂ̅ | ||
Syntax | cpinfty 34504 | Syntax for "plus infinity". |
class +∞ | ||
Definition | df-bj-pinfty 34505 | Definition of "plus infinity". (Contributed by BJ, 27-Jun-2019.) |
⊢ +∞ = (+∞ei‘0) | ||
Theorem | bj-pinftyccb 34506 | The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.) |
⊢ +∞ ∈ ℂ̅ | ||
Theorem | bj-pinftynrr 34507 | The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.) |
⊢ ¬ +∞ ∈ ℂ | ||
Syntax | cminfty 34508 | Syntax for "minus infinity". |
class -∞ | ||
Definition | df-bj-minfty 34509 | Definition of "minus infinity". (Contributed by BJ, 27-Jun-2019.) |
⊢ -∞ = (+∞ei‘π) | ||
Theorem | bj-minftyccb 34510 | The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.) |
⊢ -∞ ∈ ℂ̅ | ||
Theorem | bj-minftynrr 34511 | The extended complex number -∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.) |
⊢ ¬ -∞ ∈ ℂ | ||
Theorem | bj-pinftynminfty 34512 | The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.) |
⊢ +∞ ≠ -∞ | ||
Syntax | crrbar 34513 | Syntax for the set of extended real numbers. |
class ℝ̅ | ||
Definition | df-bj-rrbar 34514 | Definition of the set of extended real numbers. This aims to replace df-xr 10679. (Contributed by BJ, 29-Jun-2019.) |
⊢ ℝ̅ = (ℝ ∪ {-∞, +∞}) | ||
Syntax | cinfty 34515 | Syntax for ∞. |
class ∞ | ||
Definition | df-bj-infty 34516 | Definition of ∞, the point at infinity of the real or complex projective line. (Contributed by BJ, 27-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.) |
⊢ ∞ = 𝒫 ∪ ℂ | ||
Syntax | ccchat 34517 | Syntax for ℂ̂. |
class ℂ̂ | ||
Definition | df-bj-cchat 34518 | Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.) |
⊢ ℂ̂ = (ℂ ∪ {∞}) | ||
Syntax | crrhat 34519 | Syntax for ℝ̂. |
class ℝ̂ | ||
Definition | df-bj-rrhat 34520 | Define the real projective line. (Contributed by BJ, 27-Jun-2019.) |
⊢ ℝ̂ = (ℝ ∪ {∞}) | ||
Theorem | bj-rrhatsscchat 34521 | The real projective line is included in the complex projective line. (Contributed by BJ, 27-Jun-2019.) |
⊢ ℝ̂ ⊆ ℂ̂ | ||
We define the operations of addition and opposite on the extended complex numbers and on the complex projective line (Riemann sphere) simultaneously, thus "overloading" the operations. | ||
Syntax | caddcc 34522 | Syntax for the addition on extended complex numbers. |
class +ℂ̅ | ||
Definition | df-bj-addc 34523 | Define the additions on the extended complex numbers (on the subset of (ℂ̅ × ℂ̅) where it makes sense) and on the complex projective line (Riemann sphere). We use the plural in "additions" since these are two different operations, even though +ℂ̅ is overloaded. (Contributed by BJ, 22-Jun-2019.) |
⊢ +ℂ̅ = (𝑥 ∈ (((ℂ × ℂ̅) ∪ (ℂ̅ × ℂ)) ∪ ((ℂ̂ × ℂ̂) ∪ ( I ↾ ℂ∞))) ↦ if(((1st ‘𝑥) = ∞ ∨ (2nd ‘𝑥) = ∞), ∞, if((1st ‘𝑥) ∈ ℂ, if((2nd ‘𝑥) ∈ ℂ, 〈((1st ‘(1st ‘𝑥)) +R (1st ‘(2nd ‘𝑥))), ((2nd ‘(1st ‘𝑥)) +R (2nd ‘(2nd ‘𝑥)))〉, (2nd ‘𝑥)), (1st ‘𝑥)))) | ||
Syntax | coppcc 34524 | Syntax for negation on the set of extended complex numbers and the complex projective line (Riemann sphere). |
class -ℂ̅ | ||
Definition | df-bj-oppc 34525* | Define the negation (operation giving the opposite) on the set of extended complex numbers and the complex projective line (Riemann sphere). (Contributed by BJ, 22-Jun-2019.) |
⊢ -ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = ∞, ∞, if(𝑥 ∈ ℂ, (℩𝑦 ∈ ℂ (𝑥 +ℂ̅ 𝑦) = 0), (+∞eiτ‘(𝑥 +ℂ̅ 〈1/2, 0R〉))))) | ||
In this section, we redefine df-ltxr 10680 without the intermediate step of df-lt 10550. | ||
Syntax | cltxr 34526 | Syntax for the standard (strict) order on the extended reals. |
class <ℝ̅ | ||
Definition | df-bj-lt 34527* | Define the standard (strict) order on the extended reals. (Contributed by BJ, 4-Feb-2023.) |
⊢ <ℝ̅ = ({𝑥 ∈ (ℝ̅ × ℝ̅) ∣ ∃𝑦∃𝑧(((1st ‘𝑥) = 〈𝑦, 0R〉 ∧ (2nd ‘𝑥) = 〈𝑧, 0R〉) ∧ 𝑦 <R 𝑧)} ∪ ((({-∞} × ℝ) ∪ (ℝ × {+∞})) ∪ ({-∞} × {+∞}))) | ||
Since one needs arguments in order to define multiplication in ℂ̅, and one needs complex multiplication in order to define arguments, it would be contrieved to construct a whole theory for a temporary multiplication (and temporary powers, then temporary logarithm, and finally temporary argument) before redefining the extended complex multiplication. Therefore, we adopt a two-step process, see df-bj-mulc 34531. | ||
Syntax | carg 34528 | Syntax for the argument of a nonzero extended complex number. |
class Arg | ||
Definition | df-bj-arg 34529 | Define the argument of a nonzero extended complex number. By convention, it has values in (-π, π]. Another convention chooses values in [0, 2π) but the present convention simplifies formulas giving the argument as an arctangent. (Contributed by BJ, 22-Jun-2019.) The "else" case of the second conditional operator, corresponding to infinite extended complex numbers other than -∞, gives a definition depending on the specific definition chosen for these numbers (df-bj-inftyexpitau 34484), and therefore should not be relied upon. (New usage is discouraged.) |
⊢ Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), if(𝑥<ℝ̅0, π, (((1st ‘𝑥) / (2 · π)) − π)))) | ||
Syntax | cmulc 34530 | Syntax for the multiplication of extended complex numbers. |
class ·ℂ̅ | ||
Definition | df-bj-mulc 34531 |
Define the multiplication of extended complex numbers and of the complex
projective line (Riemann sphere). In our convention, a product with 0 is
0, even when the other factor is infinite. An alternate convention leaves
products of 0 with an infinite number undefined since the multiplication
is not continuous at these points. Note that our convention entails
(0 / 0) = 0 (given df-bj-invc 34533).
Note that this definition uses · and Arg and /. Indeed, it would be contrieved to bypass ordinary complex multiplication, and the present two-step definition looks like a good compromise. (Contributed by BJ, 22-Jun-2019.) |
⊢ ·ℂ̅ = (𝑥 ∈ ((ℂ̅ × ℂ̅) ∪ (ℂ̂ × ℂ̂)) ↦ if(((1st ‘𝑥) = 0 ∨ (2nd ‘𝑥) = 0), 0, if(((1st ‘𝑥) = ∞ ∨ (2nd ‘𝑥) = ∞), ∞, if(𝑥 ∈ (ℂ × ℂ), ((1st ‘𝑥) · (2nd ‘𝑥)), (+∞eiτ‘(((Arg‘(1st ‘𝑥)) +ℂ̅ (Arg‘(2nd ‘𝑥))) / τ)))))) | ||
Syntax | cinvc 34532 | Syntax for the inverse of nonzero extended complex numbers. |
class -1ℂ̅ | ||
Definition | df-bj-invc 34533* | Define inversion, which maps a nonzero extended complex number or element of the complex projective line (Riemann sphere) to its inverse. Beware of the overloading: the equality (-1ℂ̅‘0) = ∞ is to be understood in the complex projective line, but 0 as an extended complex number does not have an inverse, which we can state as (-1ℂ̅‘0) ∉ ℂ̅. Note that this definition relies on df-bj-mulc 34531, which does not bypass ordinary complex multiplication, but defines extended complex multiplication on top of it. Therefore, we could have used directly / instead of (℩... ·ℂ̅ ...). (Contributed by BJ, 22-Jun-2019.) |
⊢ -1ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = 0, ∞, if(𝑥 ∈ ℂ, (℩𝑦 ∈ ℂ (𝑥 ·ℂ̅ 𝑦) = 1), 0))) | ||
Syntax | ciomnn 34534 | Syntax for the canonical bijection from (ω ∪ {ω}) onto (ℕ0 ∪ {+∞}). |
class iω↪ℕ | ||
Definition | df-bj-iomnn 34535* |
Definition of the canonical bijection from (ω ∪
{ω}) onto
(ℕ0 ∪ {+∞}).
To understand this definition, recall that set.mm constructs reals as couples whose first component is a prereal and second component is the zero prereal (in order that one have ℝ ⊆ ℂ), that prereals are equivalence classes of couples of positive reals, the latter are Dedekind cuts of positive rationals, which are equivalence classes of positive ordinals. In partiular, we take the successor ordinal at the beginning and subtract 1 at the end since the intermediate systems contain only (strictly) positive numbers. Note the similarity with df-bj-fractemp 34482 but we did not use the present definition there since we wanted to have defined +∞ first. See bj-iomnnom 34544 for its value at +∞. TODO: Prove ⊢ (iω↪ℕ‘∅) = 0. Define ⊢ ℕ0 = (iω↪ℕ “ ω) and ⊢ ℕ = (ℕ0 ∖ {0}). Prove ⊢ iω↪ℕ:(ω ∪ {ω})–1-1-onto→(ℕ0 ∪ {+∞}) and ⊢ (iω↪ℕ ↾ ω):ω–1-1-onto→ℕ0. Prove that these bijections are respectively an isomorphism of ordered "extended rigs" and of ordered rigs. Prove ⊢ (iω↪ℕ ↾ ω) = rec((𝑥 ∈ ℝ ↦ (𝑥 + 1)), 0). (Contributed by BJ, 18-Feb-2023.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.) |
⊢ iω↪ℕ = ((𝑛 ∈ ω ↦ 〈[〈{𝑟 ∈ Q ∣ 𝑟 <Q 〈suc 𝑛, 1o〉}, 1P〉] ~R , 0R〉) ∪ {〈ω, +∞〉}) | ||
Theorem | bj-imafv 34536 | If the direct image of a singleton under any of two functions is the same, then the values of these functions at the corresponding point agree. (Contributed by BJ, 18-Mar-2023.) |
⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹‘𝐴) = (𝐺‘𝐴)) | ||
Theorem | bj-funun 34537 | Value of a function expressed as a union of two functions at a point not in the domain of one of them. (Contributed by BJ, 18-Mar-2023.) |
⊢ (𝜑 → 𝐹 = (𝐺 ∪ 𝐻)) & ⊢ (𝜑 → ¬ 𝐴 ∈ dom 𝐻) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) | ||
Theorem | bj-fununsn1 34538 | Value of a function expressed as a union of a function and a singleton on a couple (with disjoint domain) at a point not equal to the first component of that couple. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝐹 = (𝐺 ∪ {〈𝐵, 𝐶〉})) & ⊢ (𝜑 → ¬ 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) | ||
Theorem | bj-fununsn2 34539 | Value of a function expressed as a union of a function and a singleton on a couple (with disjoint domain) at the first component of that couple. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝐹 = (𝐺 ∪ {〈𝐵, 𝐶〉})) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐹‘𝐵) = 𝐶) | ||
Theorem | bj-fvsnun1 34540 | The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. (Contributed by NM, 23-Sep-2007.) Put in deduction form and remove two sethood hypotheses. (Revised by BJ, 18-Mar-2023.) |
⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) & ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∖ {𝐴})) ⇒ ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷)) | ||
Theorem | bj-fvsnun2 34541 | The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6945. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 18-Mar-2023.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵) | ||
Theorem | bj-fvmptunsn1 34542* | Value of a function expressed as a union of a mapsto expression and a singleton on a couple (with disjoint domain) at the first component of that couple. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝐹 = ((𝑥 ∈ 𝐴 ↦ 𝐵) ∪ {〈𝐶, 𝐷〉})) & ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐹‘𝐶) = 𝐷) | ||
Theorem | bj-fvmptunsn2 34543* | Value of a function expressed as a union of a mapsto expression and a singleton on a couple (with disjoint domain) at a point in the domain of the mapsto construction. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝐹 = ((𝑥 ∈ 𝐴 ↦ 𝐵) ∪ {〈𝐶, 𝐷〉})) & ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐸 ∈ 𝐴) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐸) → 𝐵 = 𝐺) ⇒ ⊢ (𝜑 → (𝐹‘𝐸) = 𝐺) | ||
Theorem | bj-iomnnom 34544 | The canonical bijection from (ω ∪ {ω}) onto (ℕ0 ∪ {+∞}) maps ω to +∞. (Contributed by BJ, 18-Feb-2023.) |
⊢ (iω↪ℕ‘ω) = +∞ | ||
Syntax | cnnbar 34545 | Syntax for the extended natural numbers. |
class ℕ̅ | ||
Definition | df-bj-nnbar 34546 | Definition of the extended natural numbers. (Contributed by BJ, 28-Jul-2023.) |
⊢ ℕ̅ = (ℕ0 ∪ {+∞}) | ||
Syntax | czzbar 34547 | Syntax for the extended integers. |
class ℤ̅ | ||
Definition | df-bj-zzbar 34548 | Definition of the extended integers. (Contributed by BJ, 28-Jul-2023.) |
⊢ ℤ̅ = (ℤ ∪ {-∞, +∞}) | ||
Syntax | czzhat 34549 | Syntax for the one-point-compactified integers. |
class ℤ̂ | ||
Definition | df-bj-zzhat 34550 | Definition of the one-point-compactified. (Contributed by BJ, 28-Jul-2023.) |
⊢ ℤ̂ = (ℤ ∪ {∞}) | ||
Syntax | cdivc 34551 | Syntax for the divisibility relation. |
class ∥ℂ | ||
Definition | df-bj-divc 34552* |
Definition of the divisibility relation (compare df-dvds 15608).
Since 0 is absorbing, ⊢ (𝐴 ∈ (ℂ̅ ∪ ℂ̂) → (𝐴 ∥ℂ 0)) and ⊢ ((0 ∥ℂ 𝐴) ↔ 𝐴 = 0). (Contributed by BJ, 28-Jul-2023.) |
⊢ ∥ℂ = {〈𝑥, 𝑦〉 ∣ (〈𝑥, 𝑦〉 ∈ ((ℂ̅ × ℂ̅) ∪ (ℂ̂ × ℂ̂)) ∧ ∃𝑛 ∈ (ℤ̅ ∪ ℤ̂)(𝑛 ·ℂ̅ 𝑥) = 𝑦)} | ||
See ccmn 18906 and subsequents. The first few statements of this subsection can be put very early after ccmn 18906. Proposal: in the main part, make separate subsections of commutative monoids and abelian groups. Relabel cabl 18907 to "cabl" or, preferably, other labels containing "abl" to "abel", for consistency. | ||
Theorem | bj-smgrpssmgm 34553 | Semigroups are magmas. (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.) |
⊢ Smgrp ⊆ Mgm | ||
Theorem | bj-smgrpssmgmel 34554 | Semigroups are magmas (elemental version). (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.) |
⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm) | ||
Theorem | bj-mndsssmgrp 34555 | Monoids are semigroups. (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.) |
⊢ Mnd ⊆ Smgrp | ||
Theorem | bj-mndsssmgrpel 34556 | Monoids are semigroups (elemental version). (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.) |
⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp) | ||
Theorem | bj-cmnssmnd 34557 | Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
⊢ CMnd ⊆ Mnd | ||
Theorem | bj-cmnssmndel 34558 | Commutative monoids are monoids (elemental version). This is a more direct proof of cmnmnd 18922, which relies on iscmn 18914. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ CMnd → 𝐴 ∈ Mnd) | ||
Theorem | bj-grpssmnd 34559 | Groups are monoids. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.) |
⊢ Grp ⊆ Mnd | ||
Theorem | bj-grpssmndel 34560 | Groups are monoids (elemental version). Shorter proof of grpmnd 18110. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ Grp → 𝐴 ∈ Mnd) | ||
Theorem | bj-ablssgrp 34561 | Abelian groups are groups. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
⊢ Abel ⊆ Grp | ||
Theorem | bj-ablssgrpel 34562 | Abelian groups are groups (elemental version). This is a shorter proof of ablgrp 18911. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ Abel → 𝐴 ∈ Grp) | ||
Theorem | bj-ablsscmn 34563 | Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
⊢ Abel ⊆ CMnd | ||
Theorem | bj-ablsscmnel 34564 | Abelian groups are commutative monoids (elemental version). This is a shorter proof of ablcmn 18913. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ Abel → 𝐴 ∈ CMnd) | ||
Theorem | bj-modssabl 34565 | (The additive groups of) modules are abelian groups. (The elemental version is lmodabl 19681; see also lmodgrp 19641 and lmodcmn 19682.) (Contributed by BJ, 9-Jun-2019.) |
⊢ LMod ⊆ Abel | ||
Theorem | bj-vecssmod 34566 | Vector spaces are modules. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
⊢ LVec ⊆ LMod | ||
Theorem | bj-vecssmodel 34567 | Vector spaces are modules (elemental version). This is a shorter proof of lveclmod 19878. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ LVec → 𝐴 ∈ LMod) | ||
UPDATE: a similar summation is already defined as df-gsum 16716 (although it mixes finite and infinite sums, which makes it harder to understand). | ||
Syntax | cfinsum 34568 | Syntax for the class "finite summation in monoids". |
class FinSum | ||
Definition | df-bj-finsum 34569* | Finite summation in commutative monoids. This finite summation function can be extended to pairs 〈𝑦, 𝑧〉 where 𝑦 is a left-unital magma and 𝑧 is defined on a totally ordered set (choosing left-associative composition), or dropping unitality and requiring nonempty families, or on any monoids for families of permutable elements, etc. We use the term "summation", even though the definition stands for any unital, commutative and associative composition law. (Contributed by BJ, 9-Jun-2019.) |
⊢ FinSum = (𝑥 ∈ {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↦ (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)))) | ||
Theorem | bj-finsumval0 34570* | Value of a finite sum. (Contributed by BJ, 9-Jun-2019.) (Proof shortened by AV, 5-May-2021.) |
⊢ (𝜑 → 𝐴 ∈ CMnd) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵:𝐼⟶(Base‘𝐴)) ⇒ ⊢ (𝜑 → (𝐴 FinSum 𝐵) = (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼))))) | ||
A few basic theorems to start affine, Euclidean, and Cartesian geometry. The first step is to define real vector spaces, then barycentric coordinates and convex hulls. | ||
In this section, we introduce real vector spaces. | ||
Theorem | bj-fvimacnv0 34571 | Variant of fvimacnv 6823 where membership of 𝐴 in the domain is not needed provided the containing class 𝐵 does not contain the empty set. Note that this antecedent would not be needed with definition df-afv 43339. (Contributed by BJ, 7-Jan-2024.) |
⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵))) | ||
Theorem | bj-isvec 34572 | The predicate "is a vector space". (Contributed by BJ, 6-Jan-2024.) |
⊢ (𝜑 → 𝐾 = (Scalar‘𝑉)) ⇒ ⊢ (𝜑 → (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing))) | ||
Theorem | bj-flddrng 34573 | Fields are division rings. (Contributed by BJ, 6-Jan-2024.) |
⊢ Field ⊆ DivRing | ||
Theorem | bj-rrdrg 34574 | The field of real numbers is a division ring. (Contributed by BJ, 6-Jan-2024.) |
⊢ ℝfld ∈ DivRing | ||
Theorem | bj-isclm 34575 | The predicate "is a subcomplex module". (Contributed by BJ, 6-Jan-2024.) |
⊢ (𝜑 → 𝐹 = (Scalar‘𝑊)) & ⊢ (𝜑 → 𝐾 = (Base‘𝐹)) ⇒ ⊢ (𝜑 → (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))) | ||
Syntax | crrvec 34576 | Syntax for the class of real vector spaces. |
class ℝ-Vec | ||
Definition | df-bj-rvec 34577 | Definition of the class of real vector spaces. The previous definition, ⊢ ℝ-Vec = {𝑥 ∈ LMod ∣ (Scalar‘𝑥) = ℝfld}, can be recovered using bj-isrvec 34578. The present one is preferred since it does not use any dummy variable. That ℝ-Vec could be defined with LVec in place of LMod is a consequence of bj-isrvec2 34584. (Contributed by BJ, 9-Jun-2019.) |
⊢ ℝ-Vec = (LMod ∩ (◡Scalar “ {ℝfld})) | ||
Theorem | bj-isrvec 34578 | The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.) |
⊢ (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LMod ∧ (Scalar‘𝑉) = ℝfld)) | ||
Theorem | bj-rvecmod 34579 | Real vector spaces are modules (elemental version). (Contributed by BJ, 6-Jan-2024.) |
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ LMod) | ||
Theorem | bj-rvecssmod 34580 | Real vector spaces are modules. (Contributed by BJ, 6-Jan-2024.) |
⊢ ℝ-Vec ⊆ LMod | ||
Theorem | bj-rvecrr 34581 | The field of scalars of a real vector space is the field of real numbers. (Contributed by BJ, 6-Jan-2024.) |
⊢ (𝑉 ∈ ℝ-Vec → (Scalar‘𝑉) = ℝfld) | ||
Theorem | bj-isrvecd 34582 | The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.) |
⊢ (𝜑 → (Scalar‘𝑉) = 𝐾) ⇒ ⊢ (𝜑 → (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LMod ∧ 𝐾 = ℝfld))) | ||
Theorem | bj-rvecvec 34583 | Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.) |
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ LVec) | ||
Theorem | bj-isrvec2 34584 | The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.) |
⊢ (𝜑 → (Scalar‘𝑉) = 𝐾) ⇒ ⊢ (𝜑 → (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LVec ∧ 𝐾 = ℝfld))) | ||
Theorem | bj-rvecssvec 34585 | Real vector spaces are vector spaces. (Contributed by BJ, 6-Jan-2024.) |
⊢ ℝ-Vec ⊆ LVec | ||
Theorem | bj-rveccmod 34586 | Real vector spaces are subcomplex modules (elemental version). (Contributed by BJ, 6-Jan-2024.) |
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂMod) | ||
Theorem | bj-rvecsscmod 34587 | Real vector spaces are subcomplex modules. (Contributed by BJ, 6-Jan-2024.) |
⊢ ℝ-Vec ⊆ ℂMod | ||
Theorem | bj-rvecsscvec 34588 | Real vector spaces are subcomplex vector spaces. (Contributed by BJ, 6-Jan-2024.) |
⊢ ℝ-Vec ⊆ ℂVec | ||
Theorem | bj-rveccvec 34589 | Real vector spaces are subcomplex vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.) |
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂVec) | ||
Theorem | bj-rvecssabl 34590 | (The additive groups of) real vector spaces are commutative groups. (Contributed by BJ, 9-Jun-2019.) |
⊢ ℝ-Vec ⊆ Abel | ||
Theorem | bj-rvecabl 34591 | (The additive groups of) real vector spaces are commutative groups (elemental version). (Contributed by BJ, 9-Jun-2019.) |
⊢ (𝐴 ∈ ℝ-Vec → 𝐴 ∈ Abel) | ||
Some lemmas to ease algebraic manipulations. | ||
Theorem | bj-subcom 34592 | A consequence of commutativity of multiplication. (Contributed by BJ, 6-Jun-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) − (𝐵 · 𝐴)) = 0) | ||
Theorem | bj-lineqi 34593 | Solution of a (scalar) linear equation. (Contributed by BJ, 6-Jun-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑌 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝐵) = 𝑌) ⇒ ⊢ (𝜑 → 𝑋 = ((𝑌 − 𝐵) / 𝐴)) | ||
Lemmas about barycentric coordinates. For the moment, this is limited to the one-dimensional case (complex line), where existence and uniqueness of barycentric coordinates are proved by bj-bary1 34596 (which computes them). It would be nice to prove the two-dimensional case (is it easier to use ad hoc computations, or Cramer formulas?), in order to do some planar geometry. | ||
Theorem | bj-bary1lem 34594 | Lemma for bj-bary1 34596: expression for a barycenter of two points in one dimension (complex line). (Contributed by BJ, 6-Jun-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 = ((((𝐵 − 𝑋) / (𝐵 − 𝐴)) · 𝐴) + (((𝑋 − 𝐴) / (𝐵 − 𝐴)) · 𝐵))) | ||
Theorem | bj-bary1lem1 34595 | Lemma for bj-bary1: computation of one of the two barycentric coordinates of a barycenter of two points in one dimension (complex line). (Contributed by BJ, 6-Jun-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ ℂ) & ⊢ (𝜑 → 𝑇 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) → 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴)))) | ||
Theorem | bj-bary1 34596 | Barycentric coordinates in one dimension (complex line). In the statement, 𝑋 is the barycenter of the two points 𝐴, 𝐵 with respective normalized coefficients 𝑆, 𝑇. (Contributed by BJ, 6-Jun-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ ℂ) & ⊢ (𝜑 → 𝑇 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) ↔ (𝑆 = ((𝐵 − 𝑋) / (𝐵 − 𝐴)) ∧ 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴))))) | ||
Syntax | cend 34597 | Token for the monoid of endomorphisms. |
class End | ||
Definition | df-bj-end 34598* | The monoid of endomorphisms on an object of a category. (Contributed by BJ, 4-Apr-2024.) |
⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {〈(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)〉, 〈(+g‘ndx), (〈𝑥, 𝑥〉(comp‘𝑐)𝑥)〉})) | ||
Theorem | bj-endval 34599 | Value of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) |
⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) = {〈(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)〉, 〈(+g‘ndx), (〈𝑋, 𝑋〉(comp‘𝐶)𝑋)〉}) | ||
Theorem | bj-endbase 34600 | Base set of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) |
⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → (Base‘((End ‘𝐶)‘𝑋)) = (𝑋(Hom ‘𝐶)𝑋)) |
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