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Theorem List for Metamath Proof Explorer - 42301-42400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdignn0ehalf 42301 The digits of the half of an even nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 3-Jun-2010.)
(((𝐴 / 2) ∈ ℕ0𝐴 ∈ ℕ0𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(𝐴 / 2)))
 
Theoremdignn0flhalf 42302 The digits of the rounded half of a nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 7-Jun-2010.)
((𝐴 ∈ (ℤ‘2) ∧ 𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(⌊‘(𝐴 / 2))))
 
Theoremnn0sumshdiglemA 42303* Lemma for nn0sumshdig 42307 (induction step, even multiplier). (Contributed by AV, 3-Jun-2020.)
(((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b𝑥) = 𝑦𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
 
Theoremnn0sumshdiglemB 42304* Lemma for nn0sumshdig 42307 (induction step, odd multiplier). (Contributed by AV, 7-Jun-2020.)
(((𝑎 ∈ ℕ ∧ ((𝑎 − 1) / 2) ∈ ℕ0) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b𝑥) = 𝑦𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
 
Theoremnn0sumshdiglem1 42305* Lemma 1 for nn0sumshdig 42307 (induction step). (Contributed by AV, 7-Jun-2020.)
(𝑦 ∈ ℕ → (∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝑦𝑎 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘))) → ∀𝑎 ∈ ℕ0 ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
 
Theoremnn0sumshdiglem2 42306* Lemma 2 for nn0sumshdig 42307. (Contributed by AV, 7-Jun-2020.)
(𝐿 ∈ ℕ → ∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝐿𝑎 = Σ𝑘 ∈ (0..^𝐿)((𝑘(digit‘2)𝑎) · (2↑𝑘))))
 
Theoremnn0sumshdig 42307* A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.)
(𝐴 ∈ ℕ0𝐴 = Σ𝑘 ∈ (0..^(#b𝐴))((𝑘(digit‘2)𝐴) · (2↑𝑘)))
 
20.34.15.12  Algorithms for the multiplication of nonnegative integers
 
Theoremnn0mulfsum 42308* Trivial algorithm to calculate the product of two nonnegative integers 𝑎 and 𝑏 by adding up 𝑏 𝑎 times. (Contributed by AV, 17-May-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (1...𝐴)𝐵)
 
Theoremnn0mullong 42309* Standard algorithm (also known as "long multiplication" or "grade-school multiplication") to calculate the product of two nonnegative integers 𝑎 and 𝑏 by multiplying the multiplicand 𝑏 by each digit of the multiplier 𝑎 and then add up all the properly shifted results. Here, the binary representation of the multiplier 𝑎 is used, i.e. the above mentioned "digits" are 0 or 1. This is a similar result as provided by smumul 14924. (Contributed by AV, 7-Jun-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵))
 
20.35  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/.

 
20.35.1  Natural deduction
 
Theorem19.8ad 42310 If a wff is true, it is true for at least one instance. Deductive form of 19.8a 1988. (Contributed by DAW, 13-Feb-2017.)
(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)
 
Theoremsbidd 42311 An identity theorem for substitution. See sbid 2133. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
(𝜑 → [𝑥 / 𝑥]𝜓)       (𝜑𝜓)
 
Theoremsbidd-misc 42312 An identity theorem for substitution. See sbid 2133. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
((𝜑 → [𝑥 / 𝑥]𝜓) ↔ (𝜑𝜓))
 
20.35.2  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.

 
Syntaxcge-real 42313 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 42315.
class
 
Syntaxcgt 42314 Extend wff notation to include the 'greater than' relation, see df-gt 42316.
class >
 
Definitiondf-gte 42315 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 9834.

We do not write this as (𝑥𝑦𝑦𝑥), and similarly we do not write ` > ` as (𝑥 > 𝑦𝑦 < 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*) ∧ 𝑦𝑥)} but these are very complicated. This definition of , and the similar one for > (df-gt 42316), 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 42317 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.)

≥ =
 
Definitiondf-gt 42316 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 9703. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 42315 for a discussion on why this approach is used for the definition. See gt-lt 42318 and gt-lth 42320 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.)

> = <
 
Theoremgte-lte 42317 Simple relationship between and . (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵𝐵𝐴))
 
Theoremgt-lt 42318 Simple relationship between < and >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵𝐵 < 𝐴))
 
Theoremgte-lteh 42319 Relationship between and using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵𝐵𝐴)
 
Theoremgt-lth 42320 Relationship between < and > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 > 𝐵𝐵 < 𝐴)
 
Theoremex-gt 42321 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
 
Theoremex-gte 42322 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
 
20.35.3  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.

 
Syntaxcsinh 42323 Extend class notation to include the hyperbolic sine function, see df-sinh 42326.
class sinh
 
Syntaxccosh 42324 Extend class notation to include the hyperbolic cosine function. see df-cosh 42327.
class cosh
 
Syntaxctanh 42325 Extend class notation to include the hyperbolic tangent function, see df-tanh 42328.
class tanh
 
Definitiondf-sinh 42326 Define the hyperbolic sine function (sinh). We define it this way for cmpt 4541, which requires the form (𝑥𝐴𝐵). See sinhval-named 42329 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 metamath). See sinh-conventional 42332 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))
 
Definitiondf-cosh 42327 Define the hyperbolic cosine function (cosh). We define it this way for cmpt 4541, which requires the form (𝑥𝐴𝐵). (Contributed by David A. Wheeler, 10-May-2015.)
cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
 
Definitiondf-tanh 42328 Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4541, which requires the form (𝑥𝐴𝐵). (Contributed by David A. Wheeler, 10-May-2015.)
tanh = (𝑥 ∈ (cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
 
Theoremsinhval-named 42329 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 42326. See sinhval 14590 for a theorem to convert this further. See sinh-conventional 42332 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))
 
Theoremcoshval-named 42330 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 42327. See coshval 14591 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
(𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴)))
 
Theoremtanhval-named 42331 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 42328. (Contributed by David A. Wheeler, 10-May-2015.)
(𝐴 ∈ (cosh “ (ℂ ∖ {0})) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i))
 
Theoremsinh-conventional 42332 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 metamath. (Contributed by David A. Wheeler, 10-May-2015.)
(𝐴 ∈ ℂ → (sinh‘𝐴) = (-i · (sin‘(i · 𝐴))))
 
Theoremsinhpcosh 42333 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))
 
20.35.4  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.

 
Syntaxcsec 42334 Extend class notation to include the secant function, see df-sec 42337.
class sec
 
Syntaxccsc 42335 Extend class notation to include the cosecant function, see df-csc 42338.
class csc
 
Syntaxccot 42336 Extend class notation to include the cotangent function, see df-cot 42339.
class cot
 
Definitiondf-sec 42337* Define the secant function. We define it this way for cmpt 4541, which requires the form (𝑥𝐴𝐵). 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 (Contributed by David A. Wheeler, 14-Mar-2014.)
sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
 
Definitiondf-csc 42338* Define the cosecant function. We define it this way for cmpt 4541, which requires the form (𝑥𝐴𝐵). 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 (Contributed by David A. Wheeler, 14-Mar-2014.)
csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
 
Definitiondf-cot 42339* Define the cotangent function. We define it this way for cmpt 4541, which requires the form (𝑥𝐴𝐵). 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 (Contributed by David A. Wheeler, 14-Mar-2014.)
cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
 
Theoremsecval 42340 Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) = (1 / (cos‘𝐴)))
 
Theoremcscval 42341 Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴)))
 
Theoremcotval 42342 Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))
 
Theoremseccl 42343 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)
 
Theoremcsccl 42344 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)
 
Theoremcotcl 42345 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)
 
Theoremreseccl 42346 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)
 
Theoremrecsccl 42347 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)
 
Theoremrecotcl 42348 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)
 
Theoremrecsec 42349 The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (cos‘𝐴) = (1 / (sec‘𝐴)))
 
Theoremreccsc 42350 The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (sin‘𝐴) = (1 / (csc‘𝐴)))
 
Theoremreccot 42351 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)))
 
Theoremrectan 42352 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)))
 
Theoremsec0 42353 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
(sec‘0) = 1
 
Theoremonetansqsecsq 42354 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))
 
Theoremcotsqcscsq 42355 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))
 
20.35.5  Identities for "if"

Utility theorems for "if".

 
Theoremifnmfalse 42356 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 vs. applying iffalse 3948 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
 
20.35.6  Decimal point

Define the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 42361 and df-dp2 42359 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 11233.

TODO: Fix non-existent label dfpval.

 
Syntaxcdp2 42357 Constant used for decimal fraction constructor. See df-dp2 42359.
class 𝐴𝐵
 
Syntaxcdp 42358 Decimal point operator. See df-dp 42361.
class .
 
Definitiondf-dp2 42359 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 11233. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
𝐴𝐵 = (𝐴 + (𝐵 / 10))
 
Theoremdfdp2OLD 42360 Obsolete version of df-dp2 42359 as of 9-Sep-2021. (Contributed by David A. Wheeler, 15-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴𝐵 = (𝐴 + (𝐵 / 10))
 
Definitiondf-dp 42361* Define the . (decimal point) operator. For example, (1.5) = (3 / 2), and -(32.718) = -(32718 / 1000) Unary minus, if applied, should normally be applied in front of the parentheses.

Metamath intentionally does not have a built-in construct for numbers, so it can show that numbers are something you can build based on set theory. However, that means that metamath has no built-in way to handle decimal numbers as traditionally written, e.g., "2.54", and its parsing system intentionally does not include the complexities necessary to define such a parsing system. Here we create a system for modeling traditional decimal point notation; it is not syntactically identical, but it is sufficiently similar so it is a reasonable model of decimal point notation. It should also serve as a convenient way to write some fractional numbers.

The RHS is , not ; this should simplify some proofs. The LHS is 0, since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression -(𝐴.𝐵) is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.)

. = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ 𝑥𝑦)
 
Theoremdp2cl 42362 Define the closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴𝐵 ∈ ℝ)
 
Theoremdpval 42363 Define the value of the decimal point operator. See df-dp 42361. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = 𝐴𝐵)
 
Theoremdpcl 42364 Prove that the closure of the decimal point is as we have defined it. See df-dp 42361. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
 
Theoremdpfrac1 42365 Prove a simple equivalence involving the decimal point. See df-dp 42361 and dpcl 42364. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = (𝐴𝐵 / 10))
 
Theoremdpfrac1OLD 42366 Obsolete version of dpfrac1 42365 as of 9-Sep-2021. (Contributed by David A. Wheeler, 15-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = (𝐴𝐵 / 10))
 
20.35.7  Logarithms generalized to arbitrary base using ` logb `

Most of this subsection was moved to main set.mm, section "Logarithms to an arbitrary base".

 
Theoremlogb2aval 42367 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used in the 2-argument form logb𝐵, 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)))
 
20.35.8  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 metamath as compared to the (𝐵 logb 𝑋) form defined separately.

 
Syntaxclog- 42368 Extend class notation to include the logarithm generalized to an arbitrary base.
class log_
 
Definitiondf-logbALT 42369* 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))))
 
20.35.9  Formally define terms 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 relationships are reflexive, but there is no formal definition of what being reflexive actually *means*. Stating the relationships directly, instead of defining a broader test such as being reflexive, can reduce proof size (because the definition of 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 whatever way we'd like.

Benoit suggests that it might be better to define these as functions. There are many advantages to doing that, but then they it 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 Benoit's approach is very much worth considering. Examples would be: BinRel = (𝑥 ∈ V ↦ {𝑟𝑟 ⊆ (𝑥 × 𝑥)}), ReflBinRel = (𝑥 ∈ V ↦ {𝑟 ∈ ( BinRel 𝑥) ∣ (Diag‘𝑥) ⊆ 𝑟}), and IrreflBinRel = (𝑥 ∈ V ↦ {𝑟 ∈ ( BinRel 𝑥) ∣ (𝑟 ∩ (Diag‘𝑥)) = ∅}).

For more discussion see: https://github.com/metamath/set.mm/pull/1286

 
Syntaxwreflexive 42370 Extend wff definition to include "Reflexive" applied to a class, which is true iff class R is a reflexive relationship over the set A. See df-reflexive 42371. (Contributed by David A. Wheeler, 1-Dec-2019.)
wff 𝑅Reflexive𝐴
 
Definitiondf-reflexive 42371* Define relexive relationship; relation R is reflexive over the set A iff 𝑥𝐴𝑥𝑅𝑥. (Contributed by David A. Wheeler, 1-Dec-2019.)
(𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥𝐴 𝑥𝑅𝑥))
 
Syntaxwirreflexive 42372 Extend wff definition to include "Irreflexive" applied to a class, which is true iff class R is an irreflexive relationship over the set A. See df-irreflexive 42373. (Contributed by David A. Wheeler, 1-Dec-2019.)
wff 𝑅Irreflexive𝐴
 
Definitiondf-irreflexive 42373* Define irrelexive relationship; relation R is irreflexive over the set A iff 𝑥𝐴¬ 𝑥𝑅𝑥. Note that a relationship can be neither reflexive nor irreflexive. (Contributed by David A. Wheeler, 1-Dec-2019.)
(𝑅Irreflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥𝐴 ¬ 𝑥𝑅𝑥))
 
20.35.10  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 9978. We might want to switch to a naming convention like addcomli 9978.

 
Theoremcomraddi 42374 Commute RHS addition. See addcomli 9978 to commute addition on LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐴 = (𝐵 + 𝐶)       𝐴 = (𝐶 + 𝐵)
 
Theoremmvlladdd 42375 Move LHS left addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) = 𝐶)       (𝜑𝐵 = (𝐶𝐴))
 
Theoremmvlraddi 42376 Move LHS right addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 + 𝐵) = 𝐶       𝐴 = (𝐶𝐵)
 
Theoremmvrladdd 42377 Move RHS left addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 = (𝐵 + 𝐶))       (𝜑 → (𝐴𝐵) = 𝐶)
 
Theoremmvrladdi 42378 Move RHS left addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐴 = (𝐵 + 𝐶)       (𝐴𝐵) = 𝐶
 
Theoremassraddsubd 42379 Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 15-Oct-2018.)
(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐴 = ((𝐵 + 𝐶) − 𝐷))       (𝜑𝐴 = (𝐵 + (𝐶𝐷)))
 
Theoremassraddsubi 42380 Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐴 = ((𝐵 + 𝐶) − 𝐷)       𝐴 = (𝐵 + (𝐶𝐷))
 
Theoremjoinlmuladdmuli 42381 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
 
Theoremjoinlmulsubmuld 42382 Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → ((𝐴 · 𝐵) − (𝐶 · 𝐵)) = 𝐷)       (𝜑 → ((𝐴𝐶) · 𝐵) = 𝐷)
 
Theoremjoinlmulsubmuli 42383 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       ((𝐴𝐶) · 𝐵) = 𝐷
 
Theoremmvlrmuld 42384 Move LHS right multiplication to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑 → (𝐴 · 𝐵) = 𝐶)       (𝜑𝐴 = (𝐶 / 𝐵))
 
Theoremmvlrmuli 42385 Move LHS right multiplication to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 ≠ 0    &   (𝐴 · 𝐵) = 𝐶       𝐴 = (𝐶 / 𝐵)
 
20.35.11  Algebra helper examples

Examples using the algebra helpers.

 
Theoremi2linesi 42386 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)    &   (𝐴𝐶) ≠ 0       𝑋 = ((𝐷𝐵) / (𝐴𝐶))
 
Theoremi2linesd 42387 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑌 = ((𝐴 · 𝑋) + 𝐵))    &   (𝜑𝑌 = ((𝐶 · 𝑋) + 𝐷))    &   (𝜑 → (𝐴𝐶) ≠ 0)       (𝜑𝑋 = ((𝐷𝐵) / (𝐴𝐶)))
 
20.35.12  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 42388, empty-surprise 42390, and eximp-surprise 42392.

 
Theoremalimp-surprise 42388 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., 𝑥𝑀 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(𝜑𝜓). 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 (𝜑𝜓) is equivalent to ¬ 𝜑𝜓 (as proven in imor 426). 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 42389) or using the allsome quantifier (df-alsi 42396) .

For other "surprises" for new users of classical logic, see empty-surprise 42390 and eximp-surprise 42392. (Contributed by David A. Wheeler, 17-Oct-2018.)

¬ ∃𝑥𝜑       (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓))
 
Theoremalimp-no-surprise 42389 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 42388. The allsome quantifier also counters this problem, see df-alsi 42396. (Contributed by David A. Wheeler, 27-Oct-2018.)
¬ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑)
 
Theoremempty-surprise 42390 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 𝑥𝐴𝜑 is simply an abbreviation for 𝑥(𝑥𝐴𝜑) (per df-ral 2805). Thus, if 𝐴 is the empty set, this expression is always true regardless of the value of 𝜑 (see alimp-surprise 42388).

If you want the expression 𝑥𝐴𝜑 to not be vacuously true, you need to ensure that set 𝐴 is inhabited (e.g., 𝑥𝐴). (Technical note: You can also assert that 𝐴 ≠ ∅; this is an equivalent claim in classical logic as proven in n0 3793, but in intuitionistic logic the statement 𝐴 ≠ ∅ is a weaker claim than 𝑥𝐴.)

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 non-existent entities in his logic." (Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy, http://www.iep.utm.edu/aris-log/). 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 2459, celaront 2460, and cesaro 2465.

For another "surprise" for new users of classical logic, see alimp-surprise 42388 and eximp-surprise 42392. (Contributed by David A. Wheeler, 20-Oct-2018.)

¬ ∃𝑥 𝑥𝐴       𝑥𝐴 𝜑
 
Theoremempty-surprise2 42391 "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 42390. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1509); 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 42397. (Contributed by David A. Wheeler, 20-Oct-2018.)

¬ ∃𝑥 𝑥𝐴       𝑥𝐴
 
Theoremeximp-surprise 42392 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 426, 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 42393. See also alimp-surprise 42388 and empty-surprise 42390. (Contributed by David A. Wheeler, 17-Oct-2018.)

(∃𝑥(𝜑𝜓) ↔ ∃𝑥𝜑𝜓))
 
Theoremeximp-surprise2 42393 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(𝜑𝜓) 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 42392, which shows what implication really expands to. See also empty-surprise 42390. (Contributed by David A. Wheeler, 18-Oct-2018.)

𝑥 ¬ 𝜑       𝑥(𝜑𝜓)
 
20.35.13  Allsome quantifier

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 𝑥𝜑𝜓 do not imply that 𝜑 is ever true, leading to vacuous truths. See alimp-surprise 42388 and empty-surprise 42390 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(𝜑𝜓), and when restricted (applied to a class) we allow ∀!𝑥𝐴𝜑. 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 I hope that others will eventually agree that allsome is awesome.

 
Syntaxwalsi 42394 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(𝜑𝜓)
 
Syntaxwalsc 42395 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 ∀!𝑥𝐴𝜑
 
Definitiondf-alsi 42396 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(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∃𝑥𝜑))
 
Definitiondf-alsc 42397 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.)
(∀!𝑥𝐴𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∃𝑥 𝑥𝐴))
 
Theoremalsconv 42398 There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.)
(∀!𝑥(𝑥𝐴𝜑) ↔ ∀!𝑥𝐴𝜑)
 
Theoremalsi1d 42399 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(𝜓𝜒))       (𝜑 → ∀𝑥(𝜓𝜒))
 
Theoremalsi2d 42400 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(𝜓𝜒))       (𝜑 → ∃𝑥𝜓)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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