P. 513 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 51201-51300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	setrecsres 51201	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 51202	Construct V using set recursion. The proof indirectly uses trcl 9766, which relies on rec, but theoretically 𝐶 in trcl 9766 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 51203	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 51204*	Lemma for onsetrec 51207. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})    ⇒   ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}
Theorem	onsetreclem2 51205*	Lemma for onsetrec 51207. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})    ⇒   ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)
Theorem	onsetreclem3 51206*	Lemma for onsetrec 51207. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})    ⇒   ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎))
Theorem	onsetrec 51207	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 51205 and onsetreclem3 51206 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 36769. 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
21.54.3  Construction of Games and Surreal Numbers Model organization after organization of reals - see TOC
Syntax	cpg 51208	Extend class notation to include the class of partisan game forms.
class Pg
Definition	df-pg 51209	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 51210*	Lemma for elpg 51213. (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 51211*	Lemma for elpg 51213. (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 51212*	Lemma for elpg 51213. (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 51213	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 51214	Lemma for pgind 51216. (Contributed by Emmett Weisz, 27-May-2024.) (New usage is discouraged.)
⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st ‘𝑥) ∪ (2nd ‘𝑥)) ⊆ 𝑧)
Theorem	pgindnf 51215*	Version of pgind 51216 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 51216*	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))
21.55  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/.
21.55.1  Natural deduction
Theorem	sbidd 51217	An identity theorem for substitution. See sbid 2294. 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 51218	An identity theorem for substitution. See sbid 2294. 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))
21.55.2  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 51219	Extend wff notation to include the 'greater than or equal to' relation, see df-gte 51221.
class ≥
Syntax	cgt 51220	Extend wff notation to include the 'greater than' relation, see df-gt 51222.
class >
Definition	df-gte 51221	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 11388. 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 51222), 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 51223 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 51222	The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 11252. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 51221 for a discussion on why this approach is used for the definition. See gt-lt 51224 and gt-lth 51226 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 51223	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 51224	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 51225	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 51226	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 51227	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 51228	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
21.55.3  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 51229	Extend class notation to include the hyperbolic sine function, see df-sinh 51232.
class sinh
Syntax	ccosh 51230	Extend class notation to include the hyperbolic cosine function. see df-cosh 51233.
class cosh
Syntax	ctanh 51231	Extend class notation to include the hyperbolic tangent function, see df-tanh 51234.
class tanh
Definition	df-sinh 51232	Define the hyperbolic sine function (sinh). We define it this way for cmpt 5192, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 51235 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 51237 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 51233	Define the hyperbolic cosine function (cosh). We define it this way for cmpt 5192, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (Contributed by David A. Wheeler, 10-May-2015.)
⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
Definition	df-tanh 51234	Define the hyperbolic tangent function (tanh). We define it this way for cmpt 5192, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (Contributed by David A. Wheeler, 10-May-2015.) (Revised by Jon Pennant, 22-Oct-2026.)
⊢ tanh = (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((sinh‘𝑥) / (cosh‘𝑥)))
Theorem	sinhval-named 51235	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 51232. See sinhval 16366 for a theorem to convert this further. See sinh-conventional 51237 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 51236	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 51233. See coshval 16367 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴)))
Theorem	sinh-conventional 51237	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 51238	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))
Theorem	sinhf 51239	Domain and codomain of the sinh function. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ sinh:ℂ⟶ℂ
Theorem	sinhcl 51240	The closure of the sinh function with complex argument. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) ∈ ℂ)
Theorem	sinh0 51241	Value of the sinh function at 0. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ (sinh‘0) = 0
Theorem	sinhneg 51242	The sinh function is odd. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ (𝐴 ∈ ℂ → (sinh‘ -𝐴) = -(sinh‘𝐴))
Theorem	coshf 51243	Domain and codomain of the cosh function. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ cosh:ℂ⟶ℂ
Theorem	coshcl 51244	The closure of the cosh function with complex argument. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) ∈ ℂ)
Theorem	cosh0 51245	Value of the cosh function at 0. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ (cosh‘0) = 1
Theorem	coshneg 51246	The cosh function is even. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ (𝐴 ∈ ℂ → (cosh‘ -𝐴) = (cosh‘𝐴))
Theorem	tanhval 51247	Value of the tanh function in terms of sinh and cosh. (Contributed by Jon Pennant, 22-Oct-2026.)
⊢ ((𝐴 ∈ ℂ ∧ (cosh‘𝐴) ≠ 0) → (tanh‘𝐴) = ((sinh‘𝐴) / (cosh‘𝐴)))
Theorem	tanhvaltan 51248	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 51234. (Contributed by David A. Wheeler, 10-May-2015.) (Revised by Jon Pennant, 22-Oct-2026.)
⊢ ((𝐴 ∈ ℂ ∧ (cosh‘𝐴) ≠ 0) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i))
Theorem	tanhf 51249	Domain and codomain of the tanh function. (Contributed by Jon Pennant, 22-Oct-2026.)
⊢ tanh:(◡cosh “ (ℂ ∖ {0}))⟶ℂ
Theorem	tanhcl 51250	The closure of the tanh function with complex argument. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ ((𝐴 ∈ ℂ ∧ (cosh‘𝐴) ≠ 0) → (tanh‘𝐴) ∈ ℂ)
Theorem	tanh0 51251	Value of the tanh function at 0. (Contributed by Jon Pennant, 22-Oct-2026.)
⊢ (tanh‘0) = 0
Theorem	tanhneg 51252	The tanh function is odd. (Contributed by Jon Pennant, 22-Oct-2026.)
⊢ ((𝐴 ∈ ℂ ∧ (cosh‘𝐴) ≠ 0) → (tanh‘ -𝐴) = -(tanh‘𝐴))
Theorem	tanhvalnegitan 51253	Value of the tanh function in terms of tan and multiplication by -i. (Contributed by Jon Pennant, 22-Oct-2026.)
⊢ ((𝐴 ∈ ℂ ∧ (cosh‘𝐴) ≠ 0) → (tanh‘𝐴) = ( -i · (tan‘(i · 𝐴))))
Theorem	sinhadd 51254	Addition formula for sinh. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (sinh‘(𝐴 + 𝐵)) = (((sinh‘𝐴) · (cosh‘𝐵)) + ((cosh‘𝐴) · (sinh‘𝐵))))
Theorem	coshadd 51255	Addition formula for cosh. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cosh‘(𝐴 + 𝐵)) = (((cosh‘𝐴) · (cosh‘𝐵)) + ((sinh‘𝐴) · (sinh‘𝐵))))
Theorem	sinhsub 51256	Subtraction formula for sinh. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (sinh‘(𝐴 − 𝐵)) = (((sinh‘𝐴) · (cosh‘𝐵)) − ((cosh‘𝐴) · (sinh‘𝐵))))
Theorem	coshsub 51257	Subtraction formula for cosh. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cosh‘(𝐴 − 𝐵)) = (((cosh‘𝐴) · (cosh‘𝐵)) − ((sinh‘𝐴) · (sinh‘𝐵))))
Theorem	tanhadd 51258	Addition formula for tanh. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (cosh‘𝐴) ≠ 0) ∧ ((cosh‘𝐵) ≠ 0 ∧ (cosh‘(𝐴 + 𝐵)) ≠ 0)) → (tanh‘(𝐴 + 𝐵)) = (((tanh‘𝐴) + (tanh‘𝐵)) / (1 + ((tanh‘𝐴) · (tanh‘𝐵)))))
Theorem	tanhaddd 51259	Addition formula for tanh. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (cosh‘𝐴) ≠ 0)    &   ⊢ (𝜑 → (cosh‘𝐵) ≠ 0)    &   ⊢ (𝜑 → (cosh‘(𝐴 + 𝐵)) ≠ 0)    ⇒   ⊢ (𝜑 → (tanh‘(𝐴 + 𝐵)) = (((tanh‘𝐴) + (tanh‘𝐵)) / (1 + ((tanh‘𝐴) · (tanh‘𝐵)))))
Theorem	coshsinhsq 51260	The fundamental identity for hyperbolic functions: cosh^2 - sinh^2 = 1. (Contributed by Jon Pennant, 9-Sep-2026.)
⊢ (𝐴 ∈ ℂ → (((cosh‘𝐴)↑2) − ((sinh‘𝐴)↑2)) = 1)
Theorem	ontanhsq 51261	Hyperbolic tangent/secant identity: 1 - tanh^2 = sech^2 (or 1 / cosh^2). (Contributed by Jon Pennant, 27-Sep-2026.)
⊢ ((𝐴 ∈ ℂ ∧ (cosh‘𝐴) ≠ 0) → (1 − ((tanh‘𝐴)↑2)) = (1 / ((cosh‘𝐴)↑2)))
21.55.4  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 51262	Extend class notation to include the secant function, see df-sec 51265.
class sec
Syntax	ccsc 51263	Extend class notation to include the cosecant function, see df-csc 51266.
class csc
Syntax	ccot 51264	Extend class notation to include the cotangent function, see df-cot 51267.
class cot
Definition	df-sec 51265*	Define the secant function. We define it this way for cmpt 5192, 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 5192. (Contributed by David A. Wheeler, 14-Mar-2014.)
⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
Definition	df-csc 51266*	Define the cosecant function. We define it this way for cmpt 5192, 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 5192. (Contributed by David A. Wheeler, 14-Mar-2014.)
⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
Definition	df-cot 51267*	Define the cotangent function. We define it this way for cmpt 5192, 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 5192. (Contributed by David A. Wheeler, 14-Mar-2014.)
⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
Theorem	secval 51268	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 51269	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 51270	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 51271	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 51272	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 51273	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 51274	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 51275	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 51276	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 51277	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 51278	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 51279	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 51280	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 51281	The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
⊢ (sec‘0) = 1
Theorem	onetansqsecsq 51282	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 51283	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 51284	Derivative of the secant function. (Contributed by Jon Pennant, 28-Aug-2026.)
⊢ (ℂ D sec) = (𝑥 ∈ dom sec ↦ ((sec‘𝑥) · (tan‘𝑥)))
Theorem	dvcsc 51285	Derivative of the cosecant function. (Contributed by Jon Pennant, 28-Aug-2026.)
⊢ (ℂ D csc) = (𝑥 ∈ dom csc ↦ ( -(csc‘𝑥) · (cot‘𝑥)))
Theorem	dvcot 51286	Derivative of the cotangent function. (Contributed by Jon Pennant, 28-Aug-2026.)
⊢ (ℂ D cot) = (𝑥 ∈ dom cot ↦ -((csc‘𝑥)↑2))
21.55.5  Identities for "if" Utility theorems for "if".
Theorem	ifnmfalse 51287	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 4494 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 ∉ 𝐵 → if(𝐴 ∈ 𝐵, 𝐶, 𝐷) = 𝐷)
21.55.6  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 51288	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)))
21.55.7  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- 51289	Extend class notation to include the logarithm generalized to an arbitrary base.
class log_
Definition	df-logbALT 51290*	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))))
21.55.8  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 51291	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 51292. (Contributed by David A. Wheeler, 1-Dec-2019.)
wff 𝑅 Reflexive 𝐴
Definition	df-reflexive 51292*	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 51293	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 51294. (Contributed by David A. Wheeler, 1-Dec-2019.)
wff 𝑅 Irreflexive 𝐴
Definition	df-irreflexive 51294*	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))
21.55.9  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 11541. We might want to switch to a naming convention like addcomli 11541.
Theorem	mvlraddi 51295	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 51296	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 51297	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 51298	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 51299	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 51300	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))