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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | setrec2 45901* |
This is the second of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(𝐹) is a subclass of all classes 𝐶 that
are closed
under 𝐹. Taken together, Theorems setrec1 45897 and setrec2v 45902
uniquely determine setrecs(𝐹) to be the minimal class closed
under 𝐹.
We express this by saying that if 𝐹 respects the ⊆ relation and 𝐶 is closed under 𝐹, then 𝐵 ⊆ 𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7599) to the other class. (Contributed by Emmett Weisz, 2-Sep-2021.) |
⊢ Ⅎ𝑎𝐹 & ⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | setrec2v 45902* | Version of setrec2 45901 with a disjoint variable condition instead of a nonfreeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.) |
⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | setis 45903* | Version of setrec2 45901 expressed as an induction schema. This theorem is a generalization of tfis3 7603. (Contributed by Emmett Weisz, 27-Feb-2022.) |
⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝑏 = 𝐴 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → ∀𝑎(∀𝑏 ∈ 𝑎 𝜓 → ∀𝑏 ∈ (𝐹‘𝑎)𝜓)) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝜒)) | ||
Theorem | elsetrecslem 45904* | Lemma for elsetrecs 45905. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 45902. To see why this lemma also requires setrec1 45897, consider what would happen if we replaced 𝐵 with {𝐴}. The antecedent would still hold, but the consequent would fail in general. Consider dispensing with the deduction form. (Contributed by Emmett Weisz, 11-Jul-2021.) (New usage is discouraged.) |
⊢ 𝐵 = setrecs(𝐹) ⇒ ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) | ||
Theorem | elsetrecs 45905* | A set 𝐴 is an element of setrecs(𝐹) iff 𝐴 is generated by some subset of setrecs(𝐹). The proof requires both setrec1 45897 and setrec2 45901, but this theorem is not strong enough to uniquely determine setrecs(𝐹). If 𝐹 respects the subset relation, the theorem still holds if both occurrences of ∈ are replaced by ⊆ for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.) |
⊢ 𝐵 = setrecs(𝐹) ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) | ||
Theorem | setrecsss 45906 | The setrecs operator respects the subset relation between two functions 𝐹 and 𝐺. (Contributed by Emmett Weisz, 13-Mar-2022.) |
⊢ (𝜑 → Fun 𝐺) & ⊢ (𝜑 → 𝐹 ⊆ 𝐺) ⇒ ⊢ (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺)) | ||
Theorem | setrecsres 45907 | A recursively generated class is unaffected when its input function is restricted to subsets of the class. (Contributed by Emmett Weisz, 14-Mar-2022.) |
⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → 𝐵 = setrecs((𝐹 ↾ 𝒫 𝐵))) | ||
Theorem | vsetrec 45908 | Construct V using set recursion. The proof indirectly uses trcl 9255, which relies on rec, but theoretically 𝐶 in trcl 9255 could be constructed using setrecs instead. The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable requirement between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 23-Jun-2021.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥) ⇒ ⊢ setrecs(𝐹) = V | ||
Theorem | 0setrec 45909 | If a function sends the empty set to itself, the function will not recursively generate any sets, regardless of its other values. (Contributed by Emmett Weisz, 23-Jun-2021.) |
⊢ (𝜑 → (𝐹‘∅) = ∅) ⇒ ⊢ (𝜑 → setrecs(𝐹) = ∅) | ||
Theorem | onsetreclem1 45910* | Lemma for onsetrec 45913. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) ⇒ ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} | ||
Theorem | onsetreclem2 45911* | Lemma for onsetrec 45913. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) ⇒ ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) | ||
Theorem | onsetreclem3 45912* | Lemma for onsetrec 45913. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) ⇒ ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) | ||
Theorem | onsetrec 45913 |
Construct On using set recursion. When 𝑥 ∈
On, the function
𝐹 constructs the least ordinal greater
than any of the elements of
𝑥, which is ∪ 𝑥 for a limit ordinal and suc ∪ 𝑥 for a
successor ordinal.
For example, (𝐹‘{1_{o}, 2_{o}}) = {∪ {1_{o}, 2_{o}}, suc ∪ {1_{o}, 2_{o}}} = {2_{o}, 3_{o}} which contains 3_{o}, and (𝐹‘ω) = {∪ ω, suc ∪ ω} = {ω, ω +_{o} 1_{o}}, which contains ω. If we start with the empty set and keep applying 𝐹 transfinitely many times, all ordinal numbers will be generated. Any function 𝐹 fulfilling lemmas onsetreclem2 45911 and onsetreclem3 45912 will recursively generate On; for example, 𝐹 = (𝑥 ∈ V ↦ suc suc ∪ 𝑥}) also works. Whether this function or the function in the theorem is used, taking this theorem as a definition of On is unsatisfying because it relies on the different properties of limit and successor ordinals. A different approach could be to let 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝑥 ∣ Tr 𝑦}), based on dfon2 33354. The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable condition between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 22-Jun-2021.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) ⇒ ⊢ setrecs(𝐹) = On | ||
Model organization after organization of reals - see TOC | ||
Syntax | cpg 45914 | Extend class notation to include the class of partisan game forms. |
class Pg | ||
Definition | df-pg 45915 | Define the class of partisan games. More precisely, this is the class of partisan game forms, many of which represent equal partisan games. In Metamath, equality between partisan games is represented by a different equivalence relation than class equality. (Contributed by Emmett Weisz, 22-Aug-2021.) |
⊢ Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥))) | ||
Theorem | elpglem1 45916* | Lemma for elpg 45919. (Contributed by Emmett Weisz, 28-Aug-2021.) |
⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1^{st} ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^{nd} ‘𝐴) ∈ 𝒫 𝑥)) → ((1^{st} ‘𝐴) ⊆ Pg ∧ (2^{nd} ‘𝐴) ⊆ Pg)) | ||
Theorem | elpglem2 45917* | Lemma for elpg 45919. (Contributed by Emmett Weisz, 28-Aug-2021.) |
⊢ (((1^{st} ‘𝐴) ⊆ Pg ∧ (2^{nd} ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1^{st} ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^{nd} ‘𝐴) ∈ 𝒫 𝑥))) | ||
Theorem | elpglem3 45918* | Lemma for elpg 45919. (Contributed by Emmett Weisz, 28-Aug-2021.) |
⊢ (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1^{st} ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^{nd} ‘𝐴) ∈ 𝒫 𝑥)))) | ||
Theorem | elpg 45919 | Membership in the class of partisan games. In John Horton Conway's On Numbers and Games, this is stated as "If 𝐿 and 𝑅 are any two sets of games, then there is a game {𝐿 ∣ 𝑅}. All games are constructed in this way." The first sentence corresponds to the backward direction of our theorem, and the second to the forward direction. (Contributed by Emmett Weisz, 27-Aug-2021.) |
⊢ (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1^{st} ‘𝐴) ⊆ Pg ∧ (2^{nd} ‘𝐴) ⊆ Pg)) | ||
This is the mathbox of David A. Wheeler, dwheeler at dwheeler dot com. Among other things, I have added a number of formal definitions for widely-used functions, e.g., those defined in ISO 80000-2:2009(E) Quantities and units - Part 2: Mathematical signs and symbols used in the natural sciences and technology and the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/. | ||
Theorem | sbidd 45920 | An identity theorem for substitution. See sbid 2257. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.) |
⊢ (𝜑 → [𝑥 / 𝑥]𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | sbidd-misc 45921 | An identity theorem for substitution. See sbid 2257. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.) |
⊢ ((𝜑 → [𝑥 / 𝑥]𝜓) ↔ (𝜑 → 𝜓)) | ||
As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Here we formally define the widely-used relations 'greater than' and 'greater than or equal to', so that we have formal definitions of them, as well as a few related theorems. | ||
Syntax | cge-real 45922 | Extend wff notation to include the 'greater than or equal to' relation, see df-gte 45924. |
class ≥ | ||
Syntax | cgt 45923 | Extend wff notation to include the 'greater than' relation, see df-gt 45925. |
class > | ||
Definition | df-gte 45924 |
Define the 'greater than or equal' predicate over the reals. Defined in
ISO 80000-2:2009(E) operation 2-7.10. It is used as a primitive in the
"NIST Digital Library of Mathematical Functions" , front
introduction,
"Common Notations and Definitions" section at
http://dlmf.nist.gov/front/introduction#Sx4.
This relation is merely
the converse of the 'less than or equal to' relation defined by df-le 10771.
We do not write this as (𝑥 ≥ 𝑦 ↔ 𝑦 ≤ 𝑥), and similarly we do not write ` > ` as (𝑥 > 𝑦 ↔ 𝑦 < 𝑥), because these are not definitional axioms as understood by mmj2 (those definitions will be flagged as being "potentially non-conservative"). We could write them this way: ⊢ > = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ^{*} ∧ 𝑦 ∈ ℝ^{*}) ∧ 𝑦 < 𝑥)} and ⊢ ≥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ^{*} ∧ 𝑦 ∈ ℝ^{*}) ∧ 𝑦 ≤ 𝑥)} but these are very complicated. This definition of ≥, and the similar one for > (df-gt 45925), are a bit strange when you see them for the first time, but these definitions are much simpler for us to process and are clearly conservative definitions. (My thanks to Mario Carneiro for pointing out this simpler approach.) See gte-lte 45926 for a more conventional expression of the relationship between < and >. As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
⊢ ≥ = ^{◡} ≤ | ||
Definition | df-gt 45925 |
The 'greater than' relation is merely the converse of the 'less than or
equal to' relation defined by df-lt 10640. Defined in ISO 80000-2:2009(E)
operation 2-7.12. See df-gte 45924 for a discussion on why this approach is
used for the definition. See gt-lt 45927 and gt-lth 45929 for more conventional
expression of the relationship between < and
>.
As a stylistic issue, set.mm prefers 'less than or equal' instead of 'greater than or equal' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
⊢ > = ^{◡} < | ||
Theorem | gte-lte 45926 | Simple relationship between ≤ and ≥. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | gt-lt 45927 | Simple relationship between < and >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵 ↔ 𝐵 < 𝐴)) | ||
Theorem | gte-lteh 45928 | Relationship between ≤ and ≥ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴) | ||
Theorem | gt-lth 45929 | Relationship between < and > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 > 𝐵 ↔ 𝐵 < 𝐴) | ||
Theorem | ex-gt 45930 | Simple example of >, in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.) |
⊢ ¬ 0 > 0 | ||
Theorem | ex-gte 45931 | Simple example of ≥, in this case, 0 is greater than or equal to 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.) |
⊢ 0 ≥ 0 | ||
It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as (cos‘(i · 𝑥)). However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved. | ||
Syntax | csinh 45932 | Extend class notation to include the hyperbolic sine function, see df-sinh 45935. |
class sinh | ||
Syntax | ccosh 45933 | Extend class notation to include the hyperbolic cosine function. see df-cosh 45936. |
class cosh | ||
Syntax | ctanh 45934 | Extend class notation to include the hyperbolic tangent function, see df-tanh 45937. |
class tanh | ||
Definition | df-sinh 45935 | Define the hyperbolic sine function (sinh). We define it this way for cmpt 5120, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 45938 for a simple way to evaluate it. We define this function by dividing by i, which uses fewer operations than many conventional definitions (and thus is more convenient to use in set.mm). See sinh-conventional 45941 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.) |
⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i)) | ||
Definition | df-cosh 45936 | Define the hyperbolic cosine function (cosh). We define it this way for cmpt 5120, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (Contributed by David A. Wheeler, 10-May-2015.) |
⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥))) | ||
Definition | df-tanh 45937 | Define the hyperbolic tangent function (tanh). We define it this way for cmpt 5120, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (Contributed by David A. Wheeler, 10-May-2015.) |
⊢ tanh = (𝑥 ∈ (^{◡}cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i)) | ||
Theorem | sinhval-named 45938 | Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 45935. See sinhval 15611 for a theorem to convert this further. See sinh-conventional 45941 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.) |
⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i)) | ||
Theorem | coshval-named 45939 | Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 45936. See coshval 15612 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.) |
⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴))) | ||
Theorem | tanhval-named 45940 | Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 45937. (Contributed by David A. Wheeler, 10-May-2015.) |
⊢ (𝐴 ∈ (^{◡}cosh “ (ℂ ∖ {0})) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i)) | ||
Theorem | sinh-conventional 45941 | Conventional definition of sinh. Here we show that the sinh definition we're using has the same meaning as the conventional definition used in some other sources. We choose a slightly different definition of sinh because it has fewer operations, and thus is more convenient to manipulate using set.mm. (Contributed by David A. Wheeler, 10-May-2015.) |
⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) = (-i · (sin‘(i · 𝐴)))) | ||
Theorem | sinhpcosh 45942 | Prove that (sinh‘𝐴) + (cosh‘𝐴) = (exp‘𝐴) using the conventional hyperbolic trigonometric functions. (Contributed by David A. Wheeler, 27-May-2015.) |
⊢ (𝐴 ∈ ℂ → ((sinh‘𝐴) + (cosh‘𝐴)) = (exp‘𝐴)) | ||
Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them. | ||
Syntax | csec 45943 | Extend class notation to include the secant function, see df-sec 45946. |
class sec | ||
Syntax | ccsc 45944 | Extend class notation to include the cosecant function, see df-csc 45947. |
class csc | ||
Syntax | ccot 45945 | Extend class notation to include the cotangent function, see df-cot 45948. |
class cot | ||
Definition | df-sec 45946* | Define the secant function. We define it this way for cmpt 5120, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). The sec function is defined in ISO 80000-2:2009(E) operation 2-13.6 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 5120. (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥))) | ||
Definition | df-csc 45947* | Define the cosecant function. We define it this way for cmpt 5120, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 5120. (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥))) | ||
Definition | df-cot 45948* | Define the cotangent function. We define it this way for cmpt 5120, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 5120. (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥))) | ||
Theorem | secval 45949 | Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) = (1 / (cos‘𝐴))) | ||
Theorem | cscval 45950 | Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴))) | ||
Theorem | cotval 45951 | Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴))) | ||
Theorem | seccl 45952 | The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) ∈ ℂ) | ||
Theorem | csccl 45953 | The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) ∈ ℂ) | ||
Theorem | cotcl 45954 | The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) ∈ ℂ) | ||
Theorem | reseccl 45955 | The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) ∈ ℝ) | ||
Theorem | recsccl 45956 | The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) ∈ ℝ) | ||
Theorem | recotcl 45957 | The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) ∈ ℝ) | ||
Theorem | recsec 45958 | The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (cos‘𝐴) = (1 / (sec‘𝐴))) | ||
Theorem | reccsc 45959 | The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (sin‘𝐴) = (1 / (csc‘𝐴))) | ||
Theorem | reccot 45960 | The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0 ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = (1 / (cot‘𝐴))) | ||
Theorem | rectan 45961 | The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0 ∧ (cos‘𝐴) ≠ 0) → (cot‘𝐴) = (1 / (tan‘𝐴))) | ||
Theorem | sec0 45962 | The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.) |
⊢ (sec‘0) = 1 | ||
Theorem | onetansqsecsq 45963 | Prove the tangent squared secant squared identity (1 + ((tan A ) ^ 2 ) ) = ( ( sec 𝐴)↑2)). (Contributed by David A. Wheeler, 25-May-2015.) |
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (1 + ((tan‘𝐴)↑2)) = ((sec‘𝐴)↑2)) | ||
Theorem | cotsqcscsq 45964 | Prove the tangent squared cosecant squared identity (1 + ((cot A ) ^ 2 ) ) = ( ( csc 𝐴)↑2)). (Contributed by David A. Wheeler, 27-May-2015.) |
⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (1 + ((cot‘𝐴)↑2)) = ((csc‘𝐴)↑2)) | ||
Utility theorems for "if". | ||
Theorem | ifnmfalse 45965 | If A is not a member of B, but an "if" condition requires it, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs versus applying iffalse 4433 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (𝐴 ∉ 𝐵 → if(𝐴 ∈ 𝐵, 𝐶, 𝐷) = 𝐷) | ||
Most of this subsection was moved to main set.mm, section "Logarithms to an arbitrary base". | ||
Theorem | logb2aval 45966 | Define the value of the log_{b} function, the logarithm generalized to an arbitrary base, when used in the 2-argument form log_{b} ⟨𝐵, 𝑋⟩ (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.) |
⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → ( log_{b} ‘⟨𝐵, 𝑋⟩) = ((log‘𝑋) / (log‘𝐵))) | ||
Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear. This supports the notational form ((log_‘𝐵)‘𝑋); that looks a little more like traditional notation, but is different from other 2-parameter functions. E.g., ((log_‘;10)‘;;100) = 2. This form is less convenient to work with inside set.mm as compared to the (𝐵 log_{b} 𝑋) form defined separately. | ||
Syntax | clog- 45967 | Extend class notation to include the logarithm generalized to an arbitrary base. |
class log_ | ||
Definition | df-logbALT 45968* | Define the log_ operator. This is the logarithm generalized to an arbitrary base. It can be used as ((log_‘𝐵)‘𝑋) for "log base B of X". This formulation suggested by Mario Carneiro. (Contributed by David A. Wheeler, 14-Jul-2017.) (New usage is discouraged.) |
⊢ log_ = (𝑏 ∈ (ℂ ∖ {0, 1}) ↦ (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝑏)))) | ||
EXPERIMENTAL. Several terms are used in comments but not directly defined in set.mm. For example, there are proofs that a number of specific relations are reflexive, but there is no formal definition of what being reflexive actually *means*. Stating the relationships directly, instead of defining a broader property such as being reflexive, can reduce proof size (because the definition of that property does not need to be expanded later). A disadvantage, however, is that there are several terms that are widely used in comments but do not have a clear formal definition. Here we define wffs that formally define some of these key terms. The intent isn't to use these directly, but to instead provide a clear formal definition of widely-used mathematical terminology (we even use this terminology within the comments of set.mm itself). We could define these using extensible structures, but doing so appears overly restrictive. These definitions don't require the use of extensible structures; requiring something to be in an extensible structure to use them is too restrictive. Even if an extensible structure is already in use, it may in use for other things. For example, in geometry, there is a "less-than" relation, but while the geometry itself is an extensible structure, we would have to build a new structure to state "the geometric less-than relation is transitive" (which is more work than it's probably worth). By creating definitions that aren't tied to extensible structures we create definitions that can be applied to anything, including extensible structures, in whatever way we'd like. BJ suggests that it might be better to define these as functions. There are many advantages to doing that, but they won't work for proper classes. I'm currently trying to also support proper classes, so I have not taken that approach, but if that turns out to be unreasonable then BJ's approach is very much worth considering. Examples would be: BinRel = (𝑥 ∈ V ↦ {𝑟 ∣ 𝑟 ⊆ (𝑥 × 𝑥)}), ReflBinRel = (𝑥 ∈ V ↦ {𝑟 ∈ ( BinRel ‘𝑥) ∣ ( I ↾ 𝑥) ⊆ 𝑟}), and IrreflBinRel = (𝑥 ∈ V ↦ {𝑟 ∈ ( BinRel ‘𝑥) ∣ (𝑟 ∩ ( I ↾ 𝑥)) = ∅}). For more discussion see: https://github.com/metamath/set.mm/pull/1286 | ||
Syntax | wreflexive 45969 | Extend wff definition to include "Reflexive" applied to a class, which is true iff class R is a reflexive relation over the set A. See df-reflexive 45970. (Contributed by David A. Wheeler, 1-Dec-2019.) |
wff 𝑅Reflexive𝐴 | ||
Definition | df-reflexive 45970* | Define reflexive relation; relation 𝑅 is reflexive over the set 𝐴 iff ∀𝑥 ∈ 𝐴𝑥𝑅𝑥. (Contributed by David A. Wheeler, 1-Dec-2019.) |
⊢ (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)) | ||
Syntax | wirreflexive 45971 | Extend wff definition to include "Irreflexive" applied to a class, which is true iff class R is an irreflexive relation over the set A. See df-irreflexive 45972. (Contributed by David A. Wheeler, 1-Dec-2019.) |
wff 𝑅Irreflexive𝐴 | ||
Definition | df-irreflexive 45972* | Define irreflexive relation; relation 𝑅 is irreflexive over the set 𝐴 iff ∀𝑥 ∈ 𝐴¬ 𝑥𝑅𝑥. Note that a relation can be neither reflexive nor irreflexive. (Contributed by David A. Wheeler, 1-Dec-2019.) |
⊢ (𝑅Irreflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)) | ||
This is an experimental approach to make it clearer (and easier) to do basic algebra in set.mm. These little theorems support basic algebra on equations at a slightly higher conceptual level. Instead of always having to "build up" equivalent expressions for one side of an equation, these theorems allow you to directly manipulate an equality. These higher-level steps lead to easier to understand proofs when they can be used, as well as proofs that are slightly shorter (when measured in steps). There are disadvantages. In particular, this approach requires many theorems (for many permutations to provide all of the operations). It can also only handle certain cases; more complex approaches must still be approached by "building up" equalities as is done today. However, I expect that we can create enough theorems to make it worth doing. I'm trying this out to see if this is helpful and if the number of permutations is manageable. To commute LHS for addition, use addcomli 10922. We might want to switch to a naming convention like addcomli 10922. | ||
Theorem | comraddi 45973 | Commute RHS addition. See addcomli 10922 to commute addition on LHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐴 = (𝐵 + 𝐶) ⇒ ⊢ 𝐴 = (𝐶 + 𝐵) | ||
Theorem | mvlraddi 45974 | Move the right term in a sum on the LHS to the RHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 + 𝐵) = 𝐶 ⇒ ⊢ 𝐴 = (𝐶 − 𝐵) | ||
Theorem | mvrladdi 45975 | Move the left term in a sum on the RHS to the LHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐴 = (𝐵 + 𝐶) ⇒ ⊢ (𝐴 − 𝐵) = 𝐶 | ||
Theorem | assraddsubi 45976 | Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ 𝐴 = ((𝐵 + 𝐶) − 𝐷) ⇒ ⊢ 𝐴 = (𝐵 + (𝐶 − 𝐷)) | ||
Theorem | joinlmuladdmuli 45977 | Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷 ⇒ ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷 | ||
Theorem | joinlmulsubmuld 45978 | Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 15-Oct-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → ((𝐴 · 𝐵) − (𝐶 · 𝐵)) = 𝐷) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐶) · 𝐵) = 𝐷) | ||
Theorem | joinlmulsubmuli 45979 | Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ ((𝐴 · 𝐵) − (𝐶 · 𝐵)) = 𝐷 ⇒ ⊢ ((𝐴 − 𝐶) · 𝐵) = 𝐷 | ||
Theorem | mvlrmuld 45980 | Move the right term in a product on the LHS to the RHS, deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) & ⊢ (𝜑 → (𝐴 · 𝐵) = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = (𝐶 / 𝐵)) | ||
Theorem | mvlrmuli 45981 | Move the right term in a product on the LHS to the RHS, inference form. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 ≠ 0 & ⊢ (𝐴 · 𝐵) = 𝐶 ⇒ ⊢ 𝐴 = (𝐶 / 𝐵) | ||
Examples using the algebra helpers. | ||
Theorem | i2linesi 45982 | Solve for the intersection of two lines expressed in Y = MX+B form (note that the lines cannot be vertical). Here we use inference form. We just solve for X, since Y can be trivially found by using X. This is an example of how to use the algebra helpers. Notice that because this proof uses algebra helpers, the main steps of the proof are higher level and easier to follow by a human reader. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ 𝑋 ∈ ℂ & ⊢ 𝑌 = ((𝐴 · 𝑋) + 𝐵) & ⊢ 𝑌 = ((𝐶 · 𝑋) + 𝐷) & ⊢ (𝐴 − 𝐶) ≠ 0 ⇒ ⊢ 𝑋 = ((𝐷 − 𝐵) / (𝐴 − 𝐶)) | ||
Theorem | i2linesd 45983 | Solve for the intersection of two lines expressed in Y = MX+B form (note that the lines cannot be vertical). Here we use deduction form. We just solve for X, since Y can be trivially found by using X. This is an example of how to use the algebra helpers. Notice that because this proof uses algebra helpers, the main steps of the proof are higher level and easier to follow by a human reader. (Contributed by David A. Wheeler, 15-Oct-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑌 = ((𝐴 · 𝑋) + 𝐵)) & ⊢ (𝜑 → 𝑌 = ((𝐶 · 𝑋) + 𝐷)) & ⊢ (𝜑 → (𝐴 − 𝐶) ≠ 0) ⇒ ⊢ (𝜑 → 𝑋 = ((𝐷 − 𝐵) / (𝐴 − 𝐶))) | ||
Prove that some formal expressions using classical logic have meanings that might not be obvious to some lay readers. I find these are common mistakes and are worth pointing out to new people. In particular we prove alimp-surprise 45984, empty-surprise 45986, and eximp-surprise 45988. | ||
Theorem | alimp-surprise 45984 |
Demonstrate that when using "for all" and material implication the
consequent can be both always true and always false if there is no case
where the antecedent is true.
Those inexperienced with formal notations of classical logic can be surprised with what "for all" and material implication do together when the implication's antecedent is never true. This can happen, for example, when the antecedent is set membership but the set is the empty set (e.g., 𝑥 ∈ 𝑀 and 𝑀 = ∅). This is perhaps best explained using an example. The sentence "All Martians are green" would typically be represented formally using the expression ∀𝑥(𝜑 → 𝜓). In this expression 𝜑 is true iff 𝑥 is a Martian and 𝜓 is true iff 𝑥 is green. Similarly, "All Martians are not green" would typically be represented as ∀𝑥(𝜑 → ¬ 𝜓). However, if there are no Martians (¬ ∃𝑥𝜑), then both of those expressions are true. That is surprising to the inexperienced, because the two expressions seem to be the opposite of each other. The reason this occurs is because in classical logic the implication (𝜑 → 𝜓) is equivalent to ¬ 𝜑 ∨ 𝜓 (as proven in imor 852). When 𝜑 is always false, ¬ 𝜑 is always true, and an or with true is always true. Here are a few technical notes. In this notation, 𝜑 and 𝜓 are predicates that return a true or false value and may depend on 𝑥. We only say may because it actually doesn't matter for our proof. In Metamath this simply means that we do not require that 𝜑, 𝜓, and 𝑥 be distinct (so 𝑥 can be part of 𝜑 or 𝜓). In natural language the term "implies" often presumes that the antecedent can occur in at one least circumstance and that there is some sort of causality. However, exactly what causality means is complex and situation-dependent. Modern logic typically uses material implication instead; this has a rigorous definition, but it is important for new users of formal notation to precisely understand it. There are ways to solve this, e.g., expressly stating that the antecedent exists (see alimp-no-surprise 45985) or using the allsome quantifier (df-alsi 45992) . For other "surprises" for new users of classical logic, see empty-surprise 45986 and eximp-surprise 45988. (Contributed by David A. Wheeler, 17-Oct-2018.) |
⊢ ¬ ∃𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓)) | ||
Theorem | alimp-no-surprise 45985 | There is no "surprise" in a for-all with implication if there exists a value where the antecedent is true. This is one way to prevent for-all with implication from allowing anything. For a contrast, see alimp-surprise 45984. The allsome quantifier also counters this problem, see df-alsi 45992. (Contributed by David A. Wheeler, 27-Oct-2018.) |
⊢ ¬ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑) | ||
Theorem | empty-surprise 45986 |
Demonstrate that when using restricted "for all" over a class the
expression can be both always true and always false if the class is
empty.
Those inexperienced with formal notations of classical logic can be surprised with what restricted "for all" does over an empty set. It is important to note that ∀𝑥 ∈ 𝐴𝜑 is simply an abbreviation for ∀𝑥(𝑥 ∈ 𝐴 → 𝜑) (per df-ral 3059). Thus, if 𝐴 is the empty set, this expression is always true regardless of the value of 𝜑 (see alimp-surprise 45984). If you want the expression ∀𝑥 ∈ 𝐴𝜑 to not be vacuously true, you need to ensure that set 𝐴 is inhabited (e.g., ∃𝑥 ∈ 𝐴). (Technical note: You can also assert that 𝐴 ≠ ∅; this is an equivalent claim in classical logic as proven in n0 4245, but in intuitionistic logic the statement 𝐴 ≠ ∅ is a weaker claim than ∃𝑥 ∈ 𝐴.) Some materials on logic (particularly those that discuss "syllogisms") are based on the much older work by Aristotle, but Aristotle expressly excluded empty sets from his system. Aristotle had a specific goal; he was trying to develop a "companion-logic" for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature... This is why he leaves no room for such nonexistent entities in his logic." (Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy, http://www.iep.utm.edu/aris-log/ 4245). While this made sense for his purposes, it is less flexible than modern (classical) logic which does permit empty sets. If you wish to make claims that require a nonempty set, you must expressly include that requirement, e.g., by stating ∃𝑥𝜑. Examples of proofs that do this include barbari 2672, celaront 2674, and cesaro 2681. For another "surprise" for new users of classical logic, see alimp-surprise 45984 and eximp-surprise 45988. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ ¬ ∃𝑥 𝑥 ∈ 𝐴 ⇒ ⊢ ∀𝑥 ∈ 𝐴 𝜑 | ||
Theorem | empty-surprise2 45987 |
"Prove" that false is true when using a restricted "for
all" over the
empty set, to demonstrate that the expression is always true if the
value ranges over the empty set.
Those inexperienced with formal notations of classical logic can be surprised with what restricted "for all" does over an empty set. We proved the general case in empty-surprise 45986. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1570); the problem is that restricted "for all" works in ways that might seem counterintuitive to the inexperienced when given an empty set. Solutions to this can include requiring that the set not be empty or by using the allsome quantifier df-alsc 45993. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ ¬ ∃𝑥 𝑥 ∈ 𝐴 ⇒ ⊢ ∀𝑥 ∈ 𝐴 ⊥ | ||
Theorem | eximp-surprise 45988 |
Show what implication inside "there exists" really expands to (using
implication directly inside "there exists" is usually a
mistake).
Those inexperienced with formal notations of classical logic may use expressions combining "there exists" with implication. That is usually a mistake, because as proven using imor 852, such an expression can be rewritten using not with or - and that is often not what the author intended. New users of formal notation who use "there exists" with an implication should consider if they meant "and" instead of "implies". A stark example is shown in eximp-surprise2 45989. See also alimp-surprise 45984 and empty-surprise 45986. (Contributed by David A. Wheeler, 17-Oct-2018.) |
⊢ (∃𝑥(𝜑 → 𝜓) ↔ ∃𝑥(¬ 𝜑 ∨ 𝜓)) | ||
Theorem | eximp-surprise2 45989 |
Show that "there exists" with an implication is always true if there
exists a situation where the antecedent is false.
Those inexperienced with formal notations of classical logic may use expressions combining "there exists" with implication. This is usually a mistake, because that combination does not mean what an inexperienced person might think it means. For example, if there is some object that does not meet the precondition 𝜑, then the expression ∃𝑥(𝜑 → 𝜓) as a whole is always true, no matter what 𝜓 is (𝜓 could even be false, ⊥). New users of formal notation who use "there exists" with an implication should consider if they meant "and" instead of "implies". See eximp-surprise 45988, which shows what implication really expands to. See also empty-surprise 45986. (Contributed by David A. Wheeler, 18-Oct-2018.) |
⊢ ∃𝑥 ¬ 𝜑 ⇒ ⊢ ∃𝑥(𝜑 → 𝜓) | ||
These are definitions and proofs involving an experimental "allsome" quantifier (aka "all some"). In informal language, statements like "All Martians are green" imply that there is at least one Martian. But it's easy to mistranslate informal language into formal notations because similar statements like ∀𝑥𝜑 → 𝜓 do not imply that 𝜑 is ever true, leading to vacuous truths. See alimp-surprise 45984 and empty-surprise 45986 as examples of the problem. Some systems include a mechanism to counter this, e.g., PVS allows types to be appended with "+" to declare that they are nonempty. This section presents a different solution to the same problem. The "allsome" quantifier expressly includes the notion of both "all" and "there exists at least one" (aka some), and is defined to make it easier to more directly express both notions. The hope is that if a quantifier more directly expresses this concept, it will be used instead and reduce the risk of creating formal expressions that look okay but in fact are mistranslations. The term "allsome" was chosen because it's short, easy to say, and clearly hints at the two concepts it combines. I do not expect this to be used much in Metamath, because in Metamath there's a general policy of avoiding the use of new definitions unless there are very strong reasons to do so. Instead, my goal is to rigorously define this quantifier and demonstrate a few basic properties of it. The syntax allows two forms that look like they would be problematic, but they are fine. When applied to a top-level implication we allow ∀!𝑥(𝜑 → 𝜓), and when restricted (applied to a class) we allow ∀!𝑥 ∈ 𝐴𝜑. The first symbol after the setvar variable must always be ∈ if it is the form applied to a class, and since ∈ cannot begin a wff, it is unambiguous. The → looks like it would be a problem because 𝜑 or 𝜓 might include implications, but any implication arrow → within any wff must be surrounded by parentheses, so only the implication arrow of ∀! can follow the wff. The implication syntax would work fine without the parentheses, but I added the parentheses because it makes things clearer inside larger complex expressions, and it's also more consistent with the rest of the syntax. For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html 45986 I hope that others will eventually agree that allsome is awesome. | ||
Syntax | walsi 45990 | Extend wff definition to include "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true, and there is at least least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) |
wff ∀!𝑥(𝜑 → 𝜓) | ||
Syntax | walsc 45991 | Extend wff definition to include "all some" applied to a class, which means 𝜑 is true for all 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.) |
wff ∀!𝑥 ∈ 𝐴𝜑 | ||
Definition | df-alsi 45992 | Define "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true and there is at least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (∀!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑)) | ||
Definition | df-alsc 45993 | Define "all some" applied to a class, which means 𝜑 is true for all 𝑥 in 𝐴 and there is at least one 𝑥 in 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (∀!𝑥 ∈ 𝐴𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴)) | ||
Theorem | alsconv 45994 | There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.) |
⊢ (∀!𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀!𝑥 ∈ 𝐴𝜑) | ||
Theorem | alsi1d 45995 | Deduction rule: Given "all some" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) | ||
Theorem | alsi2d 45996 | Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | alsc1d 45997 | Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) | ||
Theorem | alsc2d 45998 | Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓) ⇒ ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) | ||
Theorem | alscn0d 45999* | Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018.) |
⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓) ⇒ ⊢ (𝜑 → 𝐴 ≠ ∅) | ||
Theorem | alsi-no-surprise 46000 | Demonstrate that there is never a "surprise" when using the allsome quantifier, that is, it is never possible for the consequent to be both always true and always false. This uses the definition of df-alsi 45992; the proof itself builds on alimp-no-surprise 45985. For a contrast, see alimp-surprise 45984. (Contributed by David A. Wheeler, 27-Oct-2018.) |
⊢ ¬ (∀!𝑥(𝜑 → 𝜓) ∧ ∀!𝑥(𝜑 → ¬ 𝜓)) |
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