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Type | Label | Description |
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Statement | ||
Theorem | ex-gte 43001 | Simple example of ≥, in this case, 0 is greater than or equal to 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.) |
⊢ 0 ≥ 0 | ||
It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as (cos‘(i · 𝑥)). However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved. | ||
Syntax | csinh 43002 | Extend class notation to include the hyperbolic sine function, see df-sinh 43005. |
class sinh | ||
Syntax | ccosh 43003 | Extend class notation to include the hyperbolic cosine function. see df-cosh 43006. |
class cosh | ||
Syntax | ctanh 43004 | Extend class notation to include the hyperbolic tangent function, see df-tanh 43007. |
class tanh | ||
Definition | df-sinh 43005 | Define the hyperbolic sine function (sinh). We define it this way for cmpt 4863, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 43008 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 43011 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 43006 | Define the hyperbolic cosine function (cosh). We define it this way for cmpt 4863, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (Contributed by David A. Wheeler, 10-May-2015.) |
⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥))) | ||
Definition | df-tanh 43007 | Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4863, 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 43008 | 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 43005. See sinhval 15090 for a theorem to convert this further. See sinh-conventional 43011 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 43009 | 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 43006. See coshval 15091 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 43010 | 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 43007. (Contributed by David A. Wheeler, 10-May-2015.) |
⊢ (𝐴 ∈ (^{◡}cosh “ (ℂ ∖ {0})) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i)) | ||
Theorem | sinh-conventional 43011 | 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)))) | ||
Theorem | sinhpcosh 43012 | Prove that (sinh‘𝐴) + (cosh‘𝐴) = (exp‘𝐴) using the conventional hyperbolic trigonometric functions. (Contributed by David A. Wheeler, 27-May-2015.) |
⊢ (𝐴 ∈ ℂ → ((sinh‘𝐴) + (cosh‘𝐴)) = (exp‘𝐴)) | ||
Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them. | ||
Syntax | csec 43013 | Extend class notation to include the secant function, see df-sec 43016. |
class sec | ||
Syntax | ccsc 43014 | Extend class notation to include the cosecant function, see df-csc 43017. |
class csc | ||
Syntax | ccot 43015 | Extend class notation to include the cotangent function, see df-cot 43018. |
class cot | ||
Definition | df-sec 43016* | Define the secant function. We define it this way for cmpt 4863, 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 (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥))) | ||
Definition | df-csc 43017* | Define the cosecant function. We define it this way for cmpt 4863, 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 (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥))) | ||
Definition | df-cot 43018* | Define the cotangent function. We define it this way for cmpt 4863, 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 (Contributed by David A. Wheeler, 14-Mar-2014.) |
⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥))) | ||
Theorem | secval 43019 | 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 43020 | 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 43021 | 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 43022 | 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 43023 | 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 43024 | 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 43025 | 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 43026 | 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 43027 | 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 43028 | 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 43029 | 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 43030 | 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 43031 | 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 43032 | The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.) |
⊢ (sec‘0) = 1 | ||
Theorem | onetansqsecsq 43033 | 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 43034 | Prove the tangent squared cosecant squared identity (1 + ((cot A ) ^ 2 ) ) = ( ( csc 𝐴)↑2)). (Contributed by David A. Wheeler, 27-May-2015.) |
⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (1 + ((cot‘𝐴)↑2)) = ((csc‘𝐴)↑2)) | ||
Utility theorems for "if". | ||
Theorem | ifnmfalse 43035 | 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 4234 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (𝐴 ∉ 𝐵 → if(𝐴 ∈ 𝐵, 𝐶, 𝐷) = 𝐷) | ||
Most of this subsection was moved to main set.mm, section "Logarithms to an arbitrary base". | ||
Theorem | logb2aval 43036 | Define the value of the log_{b} function, the logarithm generalized to an arbitrary base, when used in the 2-argument form log_{b} ⟨𝐵, 𝑋⟩ (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.) |
⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → ( log_{b} ‘⟨𝐵, 𝑋⟩) = ((log‘𝑋) / (log‘𝐵))) | ||
Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear. This supports the notational form ((log_‘𝐵)‘𝑋); that looks a little more like traditional notation, but is different from other 2-parameter functions. E.g., ((log_‘;10)‘;;100) = 2. This form is less convenient to work with inside metamath as compared to the (𝐵 log_{b} 𝑋) form defined separately. | ||
Syntax | clog- 43037 | Extend class notation to include the logarithm generalized to an arbitrary base. |
class log_ | ||
Definition | df-logbALT 43038* | Define the log_ operator. This is the logarithm generalized to an arbitrary base. It can be used as ((log_‘𝐵)‘𝑋) for "log base B of X". This formulation suggested by Mario Carneiro. (Contributed by David A. Wheeler, 14-Jul-2017.) (New usage is discouraged.) |
⊢ log_ = (𝑏 ∈ (ℂ ∖ {0, 1}) ↦ (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝑏)))) | ||
EXPERIMENTAL. Several terms are used in comments but not directly defined in set.mm. For example, there are proofs that a number of specific relations are reflexive, but there is no formal definition of what being reflexive actually *means*. Stating the relationships directly, instead of defining a broader property such as being reflexive, can reduce proof size (because the definition of that property does not need to be expanded later). A disadvantage, however, is that there are several terms that are widely used in comments but do not have a clear formal definition. Here we define wffs that formally define some of these key terms. The intent isn't to use these directly, but to instead provide a clear formal definition of widely-used mathematical terminology (we even use this terminology within the comments of set.mm itself). We could define these using extensible structures, but doing so appears overly restrictive. These definitions don't require the use of extensible structures; requiring something to be in an extensible structure to use them is too restrictive. Even if an extensible structure is already in use, it may in use for other things. For example, in geometry, there is a "less-than" relation, but while the geometry itself is an extensible structure, we would have to build a new structure to state "the geometric less-than relation is transitive" (which is more work than it's probably worth). By creating definitions that aren't tied to extensible structures we create definitions that can be applied to anything, including extensible structures, in whatever way we'd like. Benoit 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 Benoit's approach is very much worth considering. Examples would be: BinRel = (𝑥 ∈ V ↦ {𝑟 ∣ 𝑟 ⊆ (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 | ||
Syntax | wreflexive 43039 | 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 43040. (Contributed by David A. Wheeler, 1-Dec-2019.) |
wff 𝑅Reflexive𝐴 | ||
Definition | df-reflexive 43040* | Define reflexive relation; relation 𝑅 is reflexive over the set 𝐴 iff ∀𝑥 ∈ 𝐴𝑥𝑅𝑥. (Contributed by David A. Wheeler, 1-Dec-2019.) |
⊢ (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)) | ||
Syntax | wirreflexive 43041 | 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 43042. (Contributed by David A. Wheeler, 1-Dec-2019.) |
wff 𝑅Irreflexive𝐴 | ||
Definition | df-irreflexive 43042* | Define irreflexive relation; relation 𝑅 is irreflexive over the set 𝐴 iff ∀𝑥 ∈ 𝐴¬ 𝑥𝑅𝑥. Note that a relation can be neither reflexive nor irreflexive. (Contributed by David A. Wheeler, 1-Dec-2019.) |
⊢ (𝑅Irreflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)) | ||
This is an experimental approach to make it clearer (and easier) to do basic algebra in set.mm. These little theorems support basic algebra on equations at a slightly higher conceptual level. Instead of always having to "build up" equivalent expressions for one side of an equation, these theorems allow you to directly manipulate an equality. These higher-level steps lead to easier to understand proofs when they can be used, as well as proofs that are slightly shorter (when measured in steps). There are disadvantages. In particular, this approach requires many theorems (for many permutations to provide all of the operations). It can also only handle certain cases; more complex approaches must still be approached by "building up" equalities as is done today. However, I expect that we can create enough theorems to make it worth doing. I'm trying this out to see if this is helpful and if the number of permutations is manageable. To commute LHS for addition, use addcomli 10430. We might want to switch to a naming convention like addcomli 10430. | ||
Theorem | comraddi 43043 | Commute RHS addition. See addcomli 10430 to commute addition on LHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐴 = (𝐵 + 𝐶) ⇒ ⊢ 𝐴 = (𝐶 + 𝐵) | ||
Theorem | mvlladdd 43044 | Move LHS left addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶) ⇒ ⊢ (𝜑 → 𝐵 = (𝐶 − 𝐴)) | ||
Theorem | mvlraddi 43045 | Move LHS right addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 + 𝐵) = 𝐶 ⇒ ⊢ 𝐴 = (𝐶 − 𝐵) | ||
Theorem | mvrladdd 43046 | Move RHS left addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶) | ||
Theorem | mvrladdi 43047 | Move RHS left addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐴 = (𝐵 + 𝐶) ⇒ ⊢ (𝐴 − 𝐵) = 𝐶 | ||
Theorem | assraddsubd 43048 | Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 15-Oct-2018.) |
⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = ((𝐵 + 𝐶) − 𝐷)) ⇒ ⊢ (𝜑 → 𝐴 = (𝐵 + (𝐶 − 𝐷))) | ||
Theorem | assraddsubi 43049 | 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 43050 | 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 43051 | 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 43052 | 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 43053 | Move LHS right multiplication to RHS. (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 43054 | 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) | ||
Examples using the algebra helpers. | ||
Theorem | i2linesi 43055 | 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 43056 | Solve for the intersection of two lines expressed in Y = MX+B form (note that the lines cannot be vertical). Here we use deduction form. We just solve for X, since Y can be trivially found by using X. This is an example of how to use the algebra helpers. Notice that because this proof uses algebra helpers, the main steps of the proof are higher level and easier to follow by a human reader. (Contributed by David A. Wheeler, 15-Oct-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑌 = ((𝐴 · 𝑋) + 𝐵)) & ⊢ (𝜑 → 𝑌 = ((𝐶 · 𝑋) + 𝐷)) & ⊢ (𝜑 → (𝐴 − 𝐶) ≠ 0) ⇒ ⊢ (𝜑 → 𝑋 = ((𝐷 − 𝐵) / (𝐴 − 𝐶))) | ||
Prove that some formal expressions using classical logic have meanings that might not be obvious to some lay readers. I find these are common mistakes and are worth pointing out to new people. In particular we prove alimp-surprise 43057, empty-surprise 43059, and eximp-surprise 43061. | ||
Theorem | alimp-surprise 43057 |
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 840). 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 43058) or using the allsome quantifier (df-alsi 43065) . For other "surprises" for new users of classical logic, see empty-surprise 43059 and eximp-surprise 43061. (Contributed by David A. Wheeler, 17-Oct-2018.) |
⊢ ¬ ∃𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓)) | ||
Theorem | alimp-no-surprise 43058 | 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 43057. The allsome quantifier also counters this problem, see df-alsi 43065. (Contributed by David A. Wheeler, 27-Oct-2018.) |
⊢ ¬ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑) | ||
Theorem | empty-surprise 43059 |
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 3066). Thus, if 𝐴 is the empty set, this expression is always true regardless of the value of 𝜑 (see alimp-surprise 43057). 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 4078, 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 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 2716, celaront 2717, and cesaro 2722. For another "surprise" for new users of classical logic, see alimp-surprise 43057 and eximp-surprise 43061. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ ¬ ∃𝑥 𝑥 ∈ 𝐴 ⇒ ⊢ ∀𝑥 ∈ 𝐴 𝜑 | ||
Theorem | empty-surprise2 43060 |
"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 43059. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1667); 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 43066. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ ¬ ∃𝑥 𝑥 ∈ 𝐴 ⇒ ⊢ ∀𝑥 ∈ 𝐴 ⊥ | ||
Theorem | eximp-surprise 43061 |
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 840, 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 43062. See also alimp-surprise 43057 and empty-surprise 43059. (Contributed by David A. Wheeler, 17-Oct-2018.) |
⊢ (∃𝑥(𝜑 → 𝜓) ↔ ∃𝑥(¬ 𝜑 ∨ 𝜓)) | ||
Theorem | eximp-surprise2 43062 |
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 43061, which shows what implication really expands to. See also empty-surprise 43059. (Contributed by David A. Wheeler, 18-Oct-2018.) |
⊢ ∃𝑥 ¬ 𝜑 ⇒ ⊢ ∃𝑥(𝜑 → 𝜓) | ||
These are definitions and proofs involving an experimental "allsome" quantifier (aka "all some"). In informal language, statements like "All Martians are green" imply that there is at least one Martian. But it's easy to mistranslate informal language into formal notations because similar statements like ∀𝑥𝜑 → 𝜓 do not imply that 𝜑 is ever true, leading to vacuous truths. See alimp-surprise 43057 and empty-surprise 43059 as examples of the problem. Some systems include a mechanism to counter this, e.g., PVS allows types to be appended with "+" to declare that they are nonempty. This section presents a different solution to the same problem. The "allsome" quantifier expressly includes the notion of both "all" and "there exists at least one" (aka some), and is defined to make it easier to more directly express both notions. The hope is that if a quantifier more directly expresses this concept, it will be used instead and reduce the risk of creating formal expressions that look okay but in fact are mistranslations. The term "allsome" was chosen because it's short, easy to say, and clearly hints at the two concepts it combines. I do not expect this to be used much in metamath, because in metamath there's a general policy of avoiding the use of new definitions unless there are very strong reasons to do so. Instead, my goal is to rigorously define this quantifier and demonstrate a few basic properties of it. The syntax allows two forms that look like they would be problematic, but they are fine. When applied to a top-level implication we allow ∀!𝑥(𝜑 → 𝜓), and when restricted (applied to a class) we allow ∀!𝑥 ∈ 𝐴𝜑. The first symbol after the setvar variable must always be ∈ if it is the form applied to a class, and since ∈ cannot begin a wff, it is unambiguous. The → looks like it would be a problem because 𝜑 or 𝜓 might include implications, but any implication arrow → within any wff must be surrounded by parentheses, so only the implication arrow of ∀! can follow the wff. The implication syntax would work fine without the parentheses, but I added the parentheses because it makes things clearer inside larger complex expressions, and it's also more consistent with the rest of the syntax. For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html I hope that others will eventually agree that allsome is awesome. | ||
Syntax | walsi 43063 | Extend wff definition to include "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true, and there is at least least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) |
wff ∀!𝑥(𝜑 → 𝜓) | ||
Syntax | walsc 43064 | Extend wff definition to include "all some" applied to a class, which means 𝜑 is true for all 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.) |
wff ∀!𝑥 ∈ 𝐴𝜑 | ||
Definition | df-alsi 43065 | Define "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true and there is at least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (∀!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑)) | ||
Definition | df-alsc 43066 | Define "all some" applied to a class, which means 𝜑 is true for all 𝑥 in 𝐴 and there is at least one 𝑥 in 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (∀!𝑥 ∈ 𝐴𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴)) | ||
Theorem | alsconv 43067 | There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.) |
⊢ (∀!𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀!𝑥 ∈ 𝐴𝜑) | ||
Theorem | alsi1d 43068 | Deduction rule: Given "all some" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) | ||
Theorem | alsi2d 43069 | Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | alsc1d 43070 | Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) | ||
Theorem | alsc2d 43071 | Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓) ⇒ ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) | ||
Theorem | alscn0d 43072* | Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018.) |
⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓) ⇒ ⊢ (𝜑 → 𝐴 ≠ ∅) | ||
Theorem | alsi-no-surprise 43073 | Demonstrate that there is never a "surprise" when using the allsome quantifier, that is, it is never possible for the consequent to be both always true and always false. This uses the definition of df-alsi 43065; the proof itself builds on alimp-no-surprise 43058. For a contrast, see alimp-surprise 43057. (Contributed by David A. Wheeler, 27-Oct-2018.) |
⊢ ¬ (∀!𝑥(𝜑 → 𝜓) ∧ ∀!𝑥(𝜑 → ¬ 𝜓)) | ||
Miscellaneous proofs. | ||
Theorem | 5m4e1 43074 | Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.) |
⊢ (5 − 4) = 1 | ||
Theorem | 2p2ne5 43075 | Prove that 2 + 2 ≠ 5. In George Orwell's "1984", Part One, Chapter Seven, the protagonist Winston notes that, "In the end the Party would announce that two and two made five, and you would have to believe it." http://www.sparknotes.com/lit/1984/section4.rhtml. More generally, the phrase 2 + 2 = 5 has come to represent an obviously false dogma one may be required to believe. See the Wikipedia article for more about this: https://en.wikipedia.org/wiki/2_%2B_2_%3D_5. Unsurprisingly, we can easily prove that this claim is false. (Contributed by David A. Wheeler, 31-Jan-2017.) |
⊢ (2 + 2) ≠ 5 | ||
Theorem | resolution 43076 | Resolution rule. This is the primary inference rule in some automated theorem provers such as prover9. The resolution rule can be traced back to Davis and Putnam (1960). (Contributed by David A. Wheeler, 9-Feb-2017.) |
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) → (𝜓 ∨ 𝜒)) | ||
Theorem | testable 43077 | In classical logic all wffs are testable, that is, it is always true that (¬ 𝜑 ∨ ¬ ¬ 𝜑). This is not necessarily true in intuitionistic logic. In intuitionistic logic, if this statement is true for some 𝜑, then 𝜑 is testable. The proof is trivial because it's simply a special case of the law of the excluded middle, which is true in classical logic but not necessarily true in intuitionisic logic. (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑) | ||
Theorem | aacllem 43078* | Lemma for other theorems about 𝔸. (Contributed by Brendan Leahy, 3-Jan-2020.) (Revised by Alexander van der Vekens and David A. Wheeler, 25-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ_{0}) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑𝑘) = Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝔸) | ||
Theorem | amgmwlem 43079 | Weighted version of amgmlem 24937. (Contributed by Kunhao Zheng, 19-Jun-2021.) |
⊢ 𝑀 = (mulGrp‘ℂ_{fld}) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ^{+}) & ⊢ (𝜑 → 𝑊:𝐴⟶ℝ^{+}) & ⊢ (𝜑 → (ℂ_{fld} Σ_{g} 𝑊) = 1) ⇒ ⊢ (𝜑 → (𝑀 Σ_{g} (𝐹 ∘_{𝑓} ↑_{𝑐}𝑊)) ≤ (ℂ_{fld} Σ_{g} (𝐹 ∘_{𝑓} · 𝑊))) | ||
Theorem | amgmlemALT 43080 | Alternate proof of amgmlem 24937 using amgmwlem 43079. (Contributed by Kunhao Zheng, 20-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑀 = (mulGrp‘ℂ_{fld}) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ^{+}) ⇒ ⊢ (𝜑 → ((𝑀 Σ_{g} 𝐹)↑_{𝑐}(1 / (♯‘𝐴))) ≤ ((ℂ_{fld} Σ_{g} 𝐹) / (♯‘𝐴))) | ||
Theorem | amgmw2d 43081 | Weighted arithmetic-geometric mean inequality for 𝑛 = 2 (compare amgm2d 39027). (Contributed by Kunhao Zheng, 20-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ^{+}) & ⊢ (𝜑 → 𝑃 ∈ ℝ^{+}) & ⊢ (𝜑 → 𝐵 ∈ ℝ^{+}) & ⊢ (𝜑 → 𝑄 ∈ ℝ^{+}) & ⊢ (𝜑 → (𝑃 + 𝑄) = 1) ⇒ ⊢ (𝜑 → ((𝐴↑_{𝑐}𝑃) · (𝐵↑_{𝑐}𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄))) | ||
Theorem | young2d 43082 | Young's inequality for 𝑛 = 2, a direct application of amgmw2d 43081. (Contributed by Kunhao Zheng, 6-Jul-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ^{+}) & ⊢ (𝜑 → 𝑃 ∈ ℝ^{+}) & ⊢ (𝜑 → 𝐵 ∈ ℝ^{+}) & ⊢ (𝜑 → 𝑄 ∈ ℝ^{+}) & ⊢ (𝜑 → ((1 / 𝑃) + (1 / 𝑄)) = 1) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ≤ (((𝐴↑_{𝑐}𝑃) / 𝑃) + ((𝐵↑_{𝑐}𝑄) / 𝑄))) |
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