P. 509 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50801-50900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	onsetrec 50801	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‘{1o, 2o}) = {∪ {1o, 2o}, suc ∪ {1o, 2o}} = {2o, 3o} which contains 3o, and (𝐹‘ω) = {∪ ω, suc ∪ ω} = {ω, ω +o 1o}, 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 50799 and onsetreclem3 50800 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 36554. 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
21.52.3  Construction of Games and Surreal Numbers Model organization after organization of reals - see TOC
Syntax	cpg 50802	Extend class notation to include the class of partisan game forms.
class Pg
Definition	df-pg 50803	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 50804*	Lemma for elpg 50807. (Contributed by Emmett Weisz, 28-Aug-2021.)
⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)) → ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg))
Theorem	elpglem2 50805*	Lemma for elpg 50807. (Contributed by Emmett Weisz, 28-Aug-2021.)
⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))
Theorem	elpglem3 50806*	Lemma for elpg 50807. (Contributed by Emmett Weisz, 28-Aug-2021.)
⊢ (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))))
Theorem	elpg 50807	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) ∧ (1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg))
Theorem	pgindlem 50808	Lemma for pgind 50810. (Contributed by Emmett Weisz, 27-May-2024.) (New usage is discouraged.)
⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st ‘𝑥) ∪ (2nd ‘𝑥)) ⊆ 𝑧)
Theorem	pgindnf 50809*	Version of pgind 50810 with extraneous not-free requirements. (Contributed by Emmett Weisz, 27-May-2024.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃))    &   ⊢ (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))    ⇒   ⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃))
Theorem	pgind 50810*	Induction on partizan games. (Contributed by Emmett Weisz, 27-May-2024.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐴 → (𝜒 ↔ 𝜃))    &   ⊢ (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st ‘𝑥) ∪ (2nd ‘𝑥))𝜒 → 𝜓))    ⇒   ⊢ (𝜑 → (𝐴 ∈ Pg → 𝜃))
21.53  Mathbox for David A. Wheeler 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/.
21.53.1  Natural deduction
Theorem	sbidd 50811	An identity theorem for substitution. See sbid 2290. 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 50812	An identity theorem for substitution. See sbid 2290. 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))
21.53.2  Greater than, greater than or equal to 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 50813	Extend wff notation to include the 'greater than or equal to' relation, see df-gte 50815.
class ≥
Syntax	cgt 50814	Extend wff notation to include the 'greater than' relation, see df-gt 50816.
class >
Definition	df-gte 50815	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 11349. 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 50816), 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 50817 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 50816	The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 11213. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 50815 for a discussion on why this approach is used for the definition. See gt-lt 50818 and gt-lth 50820 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 50817	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 50818	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 50819	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 50820	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 50821	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 50822	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
21.53.3  Hyperbolic trigonometric functions 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 50823	Extend class notation to include the hyperbolic sine function, see df-sinh 50826.
class sinh
Syntax	ccosh 50824	Extend class notation to include the hyperbolic cosine function. see df-cosh 50827.
class cosh
Syntax	ctanh 50825	Extend class notation to include the hyperbolic tangent function, see df-tanh 50828.
class tanh
Definition	df-sinh 50826	Define the hyperbolic sine function (sinh). We define it this way for cmpt 5185, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 50829 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 50832 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 50827	Define the hyperbolic cosine function (cosh). We define it this way for cmpt 5185, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (Contributed by David A. Wheeler, 10-May-2015.)
⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
Definition	df-tanh 50828	Define the hyperbolic tangent function (tanh). We define it this way for cmpt 5185, 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 50829	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 50826. See sinhval 16322 for a theorem to convert this further. See sinh-conventional 50832 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 50830	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 50827. See coshval 16323 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 50831	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 50828. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ (𝐴 ∈ (◡cosh “ (ℂ ∖ {0})) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i))
Theorem	sinh-conventional 50832	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 50833	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))
21.53.4  Reciprocal trigonometric functions (sec, csc, cot) Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them.
Syntax	csec 50834	Extend class notation to include the secant function, see df-sec 50837.
class sec
Syntax	ccsc 50835	Extend class notation to include the cosecant function, see df-csc 50838.
class csc
Syntax	ccot 50836	Extend class notation to include the cotangent function, see df-cot 50839.
class cot
Definition	df-sec 50837*	Define the secant function. We define it this way for cmpt 5185, 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 5185. (Contributed by David A. Wheeler, 14-Mar-2014.)
⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
Definition	df-csc 50838*	Define the cosecant function. We define it this way for cmpt 5185, 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 5185. (Contributed by David A. Wheeler, 14-Mar-2014.)
⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
Definition	df-cot 50839*	Define the cotangent function. We define it this way for cmpt 5185, 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 5185. (Contributed by David A. Wheeler, 14-Mar-2014.)
⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
Theorem	secval 50840	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 50841	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 50842	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 50843	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 50844	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 50845	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 50846	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 50847	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 50848	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 50849	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 50850	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 50851	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 50852	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 50853	The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
⊢ (sec‘0) = 1
Theorem	onetansqsecsq 50854	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 50855	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))
Theorem	dvsec 50856	Derivative of the secant function. (Contributed by Jon Pennant, 28-Aug-2026.)
⊢ (ℂ D sec) = (𝑥 ∈ dom sec ↦ ((sec‘𝑥) · (tan‘𝑥)))
Theorem	dvcsc 50857	Derivative of the cosecant function. (Contributed by Jon Pennant, 28-Aug-2026.)
⊢ (ℂ D csc) = (𝑥 ∈ dom csc ↦ ( -(csc‘𝑥) · (cot‘𝑥)))
Theorem	dvcot 50858	Derivative of the cotangent function. (Contributed by Jon Pennant, 28-Aug-2026.)
⊢ (ℂ D cot) = (𝑥 ∈ dom cot ↦ -((csc‘𝑥)↑2))
21.53.5  Identities for "if" Utility theorems for "if".
Theorem	ifnmfalse 50859	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 4490 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 ∉ 𝐵 → if(𝐴 ∈ 𝐵, 𝐶, 𝐷) = 𝐷)
21.53.6  Logarithms generalized to arbitrary base using ` logb ` Most of this subsection was moved to main set.mm, section "Logarithms to an arbitrary base".
Theorem	logb2aval 50860	Define the value of the logb function, the logarithm generalized to an arbitrary base, when used in the 2-argument form logb ⟨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})) → ( logb ‘⟨𝐵, 𝑋⟩) = ((log‘𝑋) / (log‘𝐵)))
21.53.7  Logarithm laws generalized to an arbitrary base - log_ 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 logb 𝑋) form defined separately.
Syntax	clog- 50861	Extend class notation to include the logarithm generalized to an arbitrary base.
class log_
Definition	df-logbALT 50862*	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))))
21.53.8  Formally define notions such as reflexivity 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 50863	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 50864. (Contributed by David A. Wheeler, 1-Dec-2019.)
wff 𝑅 Reflexive 𝐴
Definition	df-reflexive 50864*	Define reflexive relation; relation 𝑅 is reflexive over the set 𝐴 iff ∀𝑥 ∈ 𝐴𝑥𝑅𝑥. (Contributed by David A. Wheeler, 1-Dec-2019.)
⊢ (𝑅 Reflexive 𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥))
Syntax	wirreflexive 50865	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 50866. (Contributed by David A. Wheeler, 1-Dec-2019.)
wff 𝑅 Irreflexive 𝐴
Definition	df-irreflexive 50866*	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.)
⊢ (𝑅 Irreflexive 𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
21.53.9  Algebra helpers 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 11502. We might want to switch to a naming convention like addcomli 11502.
Theorem	mvlraddi 50867	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	assraddsubi 50868	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 50869	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 50870	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 50871	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 50872	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 50873	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)
21.53.10  Algebra helper examples Examples using the algebra helpers.
Theorem	i2linesi 50874	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 50875	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)))
21.53.11  Formal methods "surprises" 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 50876, empty-surprise 50878, and eximp-surprise 50880.
Theorem	alimp-surprise 50876	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 867). 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 50877) or using the allsome quantifier (df-als 50884) . For other "surprises" for new users of classical logic, see empty-surprise 50878 and eximp-surprise 50880. (Contributed by David A. Wheeler, 17-Oct-2018.)
⊢ ¬ ∃𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓))
Theorem	alimp-no-surprise 50877	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 50876. The allsome quantifier also counters this problem, see df-als 50884. (Contributed by David A. Wheeler, 27-Oct-2018.)
⊢ ¬ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑)
Theorem	empty-surprise 50878	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 3077). Thus, if 𝐴 is the empty set, this expression is always true regardless of the value of 𝜑 (see alimp-surprise 50876). 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 4299, 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/ 4299). 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 2693, celaront 2695, and cesaro 2702. For another "surprise" for new users of classical logic, see alimp-surprise 50876 and eximp-surprise 50880. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ ¬ ∃𝑥 𝑥 ∈ 𝐴    ⇒   ⊢ ∀𝑥 ∈ 𝐴 𝜑
Theorem	empty-surprise2 50879	"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 50878. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1597); 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-rals 50885. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ ¬ ∃𝑥 𝑥 ∈ 𝐴    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ⊥
Theorem	eximp-surprise 50880	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 867, 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 50881. See also alimp-surprise 50876 and empty-surprise 50878. (Contributed by David A. Wheeler, 17-Oct-2018.)
⊢ (∃𝑥(𝜑 → 𝜓) ↔ ∃𝑥(¬ 𝜑 ∨ 𝜓))
Theorem	eximp-surprise2 50881	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 50880, which shows what implication really expands to. See also empty-surprise 50878. (Contributed by David A. Wheeler, 18-Oct-2018.)
⊢ ∃𝑥 ¬ 𝜑    ⇒   ⊢ ∃𝑥(𝜑 → 𝜓)
21.53.12  Allsome quantifier These are definitions and proofs involving the "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 50876 and empty-surprise 50878 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 ∈ 𝐴(𝜑 → 𝜓). 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. Naming: "als" is allsome. The form restricted to a class is prefixed with "r", following the way set.mm names the restricted quantifiers it is built from: ∀ gives df-ral 3077 and ∃ gives df-rex 3087, so df-als 50884 (the general form) gives df-rals 50885 (the restricted form). Earlier versions of this material differed, so old references may not match. They wrote the quantifier as an "inverted A" followed by an exclamation point, and they named the general form df-alsi and the restricted form df-alsc. The symbol is now an "inverted A" followed by a "backwards E", which more readers can correctly guess without being taught it. The restricted definition also changed, and the older one was a mistake; see df-rals 50885 for what was wrong with it. Soundness of the two definitions below. Definitions are required to be eliminable and conservative (see the section comment for df-bi 210). Both df-als 50884 and df-rals 50885 meet these requirements directly, and so neither needs a justification theorem. Each is stated as a biconditional whose left side is a new syntax construct (wals 50882 or wrals 50883) applied to distinct metavariables, and whose right side uses only constructs introduced earlier (∀, ∃, ∧, →, and the restricted quantifiers df-ral 3077 and df-rex 3087). Any occurrence of the new construct can therefore be replaced by the right side, which is eliminability; conservativity follows, since a proof of a statement not mentioning ∀∃ can have every use of the definition replaced in this way. Note in particular that every variable occurring on the right side already occurs on the left side, so no dummy variable is introduced. A justification theorem is needed only when that fails, that is, when a definition introduces a dummy variable and the choice of that variable must be shown not to matter (as in eujust 2596 for df-eu 2594), or when the definition cannot use ↔ because it is defining ↔ itself (as in bijust 208 for df-bi 210). Neither case arises here. The restricted quantifier definitions df-ral 3077 and df-rex 3087 have the same shape as these and likewise need no justification. For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html 3087 I hope that others will eventually agree that allsome is awesome.
Syntax	wals 50882	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 one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
wff ∀∃𝑥(𝜑 → 𝜓)
Syntax	wrals 50883	Extend wff definition to include "all some" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
wff ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓)
Definition	df-als 50884	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-rals 50885	Define "all some" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴 where 𝜑 is true. An older definition of the "all some" quantifier when scoped to a class, named df-alsc and now removed, instead applied a bare formula 𝜑 to the members of a class, asserting only (∀𝑥 ∈ 𝐴𝜑 ∧ ∃𝑥𝑥 ∈ 𝐴), that is, that the formula held throughout 𝐴 and that 𝐴 had at least one member. I've now decided that that was a mistake. Its older existence conjunct ∃𝑥𝑥 ∈ 𝐴 did not require any member of 𝐴 to satisfy the antecedent, so if the formula was itself an implication, that inner implication could still be vacuously true, which is precisely what the allsome quantifier exists to prevent. For example, the older definition meant that "among Martians, all tall ones are green" could be considered true if there are Martians, but no tall Martians. This version of the definition instead ensures that claims of the form "among Martians, all tall ones are green" can only be true if all tall Martians are green and that there is at least one tall Martian. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))
Theorem	dfrals2 50886	The bounded "all some" form is the general form with the class membership folded into the antecedent. (Contributed by David A. Wheeler, 22-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
Theorem	alsd 50887	Introduction rule: "all some" holds if the "for all" part holds and the antecedent has a witness. This is the converse of als1d 50889 and als2d 50890 taken together, and is what lets an "all some" statement be proved rather than merely taken apart. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))    &   ⊢ (𝜑 → ∃𝑥𝜓)    ⇒   ⊢ (𝜑 → ∀∃𝑥(𝜓 → 𝜒))
Theorem	ralsd 50888	Introduction rule for "all some" restricted to a class. This is the converse of rals1d 50891 and rals2d 50892 taken together. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))
Theorem	als1d 50889	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	als2d 50890	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	rals1d 50891	Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
Theorem	rals2d 50892	Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. Note that the witness must satisfy the antecedent 𝜓, not merely be a member of 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	ralsn0d 50893*	Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → 𝐴 ≠ ∅)
Theorem	alsex 50894	The consequent of an "all some" is witnessed: if 𝜓 holds of every 𝑥 satisfying 𝜑, and some 𝑥 satisfies 𝜑, then some 𝑥 satisfies 𝜓. This is the positive counterpart of als-no-surprise 50902, and it is the property that ordinary "for all" with implication lacks: from ∀𝑥(𝜑 → 𝜓) alone nothing whatever follows about 𝜓, as alimp-surprise 50876 shows. It is the reason the allsome quantifier says what a speaker of "all Martians are green" usually means. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓)
Theorem	ralsex 50895	The consequent of an "all some" restricted to a class is witnessed: some member of 𝐴 satisfying 𝜑 also satisfies 𝜓. Restricted counterpart of alsex 50894. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	alsbii 50896	Congruence: equivalents may be substituted inside an "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ ∀∃𝑥(𝜒 → 𝜃))
Theorem	ralsbii 50897	Congruence for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥 ∈ 𝐴(𝜒 → 𝜃))
Theorem	alsbid 50898	Deduction form of alsbii 50896. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (∀∃𝑥(𝜓 → 𝜒) ↔ ∀∃𝑥(𝜃 → 𝜏)))
Theorem	nfals 50899	Bound-variable hypothesis builder for "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)
Theorem	nfrals 50900*	Bound-variable hypothesis builder for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥∀∃𝑦 ∈ 𝐴(𝜑 → 𝜓)