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Theorem List for Metamath Proof Explorer - 32901-33000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdpfrac1 32901 Prove a simple equivalence involving the decimal point. See df-dp 32897 and dpcl 32900. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  =  (; A B  /  10 ) )
 
21.26.8  Logarithms generalized to arbitrary base using ` logb `
 
Theoremene0 32902  _e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  0
 
Theoremene1 32903  _e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  1
 
Theoremelogb 32904 Using  _e as the base is the same as  log. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  ( A  e.  ( CC  \  { 0 } )  ->  ( _elogb A )  =  ( log `  A ) )
 
21.26.9  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 ) ` ;; 1 0 0 )  =  2

This form is less convenient to work with inside metamath as compared to the  ( Blogb X
) form defined separately.

 
Syntaxclog_ 32905 Extend class notation to include the logarithm generalized to an arbitrary base.
 class log_
 
Definitiondf-log_ 32906* 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.)
 |- log_  =  ( b  e.  ( CC  \  { 0 ,  1 } )  |->  ( x  e.  ( CC  \  { 0 } )  |->  ( ( log `  x )  /  ( log `  b
 ) ) ) )
 
21.26.10  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  e. 
_V  |->  { r  |  r  C_  ( x  X.  x ) } ), ReflBinRel =  ( x  e.  _V  |->  { r  e.  ( BinRel  `  x )  |  (Diag `  x )  C_  r } ), and IrreflBinRel =  ( x  e.  _V  |->  { r  e.  ( BinRel  `  x )  |  ( r  i^i  (Diag `  x )
)  =  (/) } ).

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

 
Syntaxwreflexive 32907 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 32908. (Contributed by David A. Wheeler, 1-Dec-2019.)
 wff  RReflexive A
 
Definitiondf-reflexive 32908* Define relexive relationship; relation R is reflexive over the set A iff  A. x  e.  A x R x. (Contributed by David A. Wheeler, 1-Dec-2019.)
 |-  ( RReflexive A  <->  ( R  C_  ( A  X.  A ) 
 /\  A. x  e.  A  x R x ) )
 
Syntaxwirreflexive 32909 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 32910. (Contributed by David A. Wheeler, 1-Dec-2019.)
 wff  RIrreflexive A
 
Definitiondf-irreflexive 32910* Define irrelexive relationship; relation R is irreflexive over the set A iff  A. x  e.  A -.  x R x. Note that a relationship can be neither reflexive nor irreflexive. (Contributed by David A. Wheeler, 1-Dec-2019.)
 |-  ( RIrreflexive A  <->  ( R  C_  ( A  X.  A ) 
 /\  A. x  e.  A  -.  x R x ) )
 
21.26.11  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 9775. We might want to switch to a naming convention like addcomli 9775.

 
Theoremcomraddd 32911 Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =  ( B  +  C ) )   =>    |-  ( ph  ->  A  =  ( C  +  B ) )
 
Theoremcomraddi 32912 Commute RHS addition. See addcomli 9775 to commute addition on LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  B  e.  CC   &    |-  C  e.  CC   &    |-  A  =  ( B  +  C )   =>    |-  A  =  ( C  +  B )
 
Theoremmvlladdd 32913 Move LHS left addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  C )   =>    |-  ( ph  ->  B  =  ( C  -  A ) )
 
Theoremmvlraddd 32914 Move LHS right addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  C )   =>    |-  ( ph  ->  A  =  ( C  -  B ) )
 
Theoremmvlraddi 32915 Move LHS right addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  ( A  +  B )  =  C   =>    |-  A  =  ( C  -  B )
 
Theoremmvrladdd 32916 Move RHS left addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =  ( B  +  C ) )   =>    |-  ( ph  ->  ( A  -  B )  =  C )
 
Theoremmvrladdi 32917 Move RHS left addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  B  e.  CC   &    |-  C  e.  CC   &    |-  A  =  ( B  +  C )   =>    |-  ( A  -  B )  =  C
 
Theoremmvrraddd 32918 Move RHS right addition to LHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =  ( B  +  C ) )   =>    |-  ( ph  ->  ( A  -  C )  =  B )
 
Theoremmvrraddi 32919 Move RHS right addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  B  e.  CC   &    |-  C  e.  CC   &    |-  A  =  ( B  +  C )   =>    |-  ( A  -  C )  =  B
 
Theoremassraddsubd 32920 Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  A  =  ( ( B  +  C )  -  D ) )   =>    |-  ( ph  ->  A  =  ( B  +  ( C  -  D ) ) )
 
Theoremassraddsubi 32921 Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   &    |-  A  =  ( ( B  +  C )  -  D )   =>    |-  A  =  ( B  +  ( C  -  D ) )
 
Theoremjoinlmuladdmuld 32922 Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( ( A  x.  B )  +  ( C  x.  B ) )  =  D )   =>    |-  ( ph  ->  (
 ( A  +  C )  x.  B )  =  D )
 
Theoremjoinlmuladdmuli 32923 Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  ( ( A  x.  B )  +  ( C  x.  B ) )  =  D   =>    |-  (
 ( A  +  C )  x.  B )  =  D
 
Theoremjoinlmulsubmuld 32924 Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( ( A  x.  B )  -  ( C  x.  B ) )  =  D )   =>    |-  ( ph  ->  (
 ( A  -  C )  x.  B )  =  D )
 
Theoremjoinlmulsubmuli 32925 Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  ( ( A  x.  B )  -  ( C  x.  B ) )  =  D   =>    |-  (
 ( A  -  C )  x.  B )  =  D
 
Theoremmvlrmuld 32926 Move LHS right multiplication to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  ( A  x.  B )  =  C )   =>    |-  ( ph  ->  A  =  ( C  /  B ) )
 
Theoremmvlrmuli 32927 Move LHS right multiplication to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B  =/=  0   &    |-  ( A  x.  B )  =  C   =>    |-  A  =  ( C  /  B )
 
21.26.12  Algebra helper examples

Examples using the algebra helpers.

 
Theoremi2linesi 32928 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  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   &    |-  X  e.  CC   &    |-  Y  =  ( ( A  x.  X )  +  B )   &    |-  Y  =  ( ( C  x.  X )  +  D )   &    |-  ( A  -  C )  =/=  0   =>    |-  X  =  ( ( D  -  B ) 
 /  ( A  -  C ) )
 
Theoremi2linesd 32929 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.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  Y  =  ( ( A  x.  X )  +  B ) )   &    |-  ( ph  ->  Y  =  ( ( C  x.  X )  +  D )
 )   &    |-  ( ph  ->  ( A  -  C )  =/=  0 )   =>    |-  ( ph  ->  X  =  ( ( D  -  B )  /  ( A  -  C ) ) )
 
21.26.13  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 32930, empty-surprise 32932, and eximp-surprise 32934.

 
Theoremalimp-surprise 32930 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  e.  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  A. x ( ph  ->  ps ). In this expression  ph is true iff  x is a Martian and  ps is true iff  x is green. Similarly, "All Martians are not green" would typically be represented as  A. x (
ph  ->  -.  ps ). However, if there are no Martians ( -.  E. x ph), 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  ( ph  ->  ps ) is equivalent to  -.  ph  \/  ps (as proven in imor 412). When  ph is always false,  -.  ph is always true, and an or with true is always true.

Here are a few technical notes. In this notation,  ph and  ps 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  ph,  ps, and  x be distinct (so  x can be part of  ph or  ps).

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 32931) or using the allsome quantifier (df-alsi 32938) .

For other "surprises" for new users of classical logic, see empty-surprise 32932 and eximp-surprise 32934. (Contributed by David A. Wheeler, 17-Oct-2018.)

 |-  -.  E. x ph   =>    |-  ( A. x (
 ph  ->  ps )  /\  A. x ( ph  ->  -. 
 ps ) )
 
Theoremalimp-no-surprise 32931 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 32930. The allsome quantifier also counters this problem, see df-alsi 32938. (Contributed by David A. Wheeler, 27-Oct-2018.)
 |-  -.  ( A. x ( ph  ->  ps )  /\  A. x ( ph  ->  -. 
 ps )  /\  E. x ph )
 
Theoremempty-surprise 32932 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 
A. x  e.  A ph is simply an abbreviation for  A. x ( x  e.  A  ->  ph ) (per df-ral 2798). Thus, if  A is the empty set, this expression is always true regardless of the value of  ph (see alimp-surprise 32930).

If you want the expression  A. x  e.  A ph to not be vacuously true, you need to ensure that set 
A is inhabited (e.g., 
E. x  e.  A). (Technical note: You can also assert that  A  =/=  (/); this is an equivalent claim in classical logic as proven in n0 3780, but in intuitionistic logic the statement  A  =/=  (/) is a weaker claim than  E. x  e.  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  E. x ph. Examples of proofs that do this include barbari 2386, celaront 2387, and cesaro 2392.

For another "surprise" for new users of classical logic, see alimp-surprise 32930 and eximp-surprise 32934. (Contributed by David A. Wheeler, 20-Oct-2018.)

 |-  -.  E. x  x  e.  A   =>    |-  A. x  e.  A  ph
 
Theoremempty-surprise2 32933 "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 32932. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1420); 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 32939. (Contributed by David A. Wheeler, 20-Oct-2018.)

 |-  -.  E. x  x  e.  A   =>    |-  A. x  e.  A F.
 
Theoremeximp-surprise 32934 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 412, 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 32935. See also alimp-surprise 32930 and empty-surprise 32932. (Contributed by David A. Wheeler, 17-Oct-2018.)

 |-  ( E. x ( ph  ->  ps )  <->  E. x ( -.  ph  \/  ps ) )
 
Theoremeximp-surprise2 32935 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  ph, then the expression  E. x (
ph  ->  ps ) as a whole is always true, no matter what  ps is ( ps could even be false, F.). New users of formal notation who use "there exists" with an implication should consider if they meant "and" instead of "implies". See eximp-surprise 32934, which shows what implication really expands to. See also empty-surprise 32932. (Contributed by David A. Wheeler, 18-Oct-2018.)

 |-  E. x  -.  ph   =>    |- 
 E. x ( ph  ->  ps )
 
21.26.14  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  A. x ph  ->  ps do not imply that  ph is ever true, leading to vacuous truths. See alimp-surprise 32930 and empty-surprise 32932 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  A.! x (
ph  ->  ps ), and when restricted (applied to a class) we allow  A.! x  e.  A ph. The first symbol after the setvar variable must always be  e. if it is the form applied to a class, and since  e. cannot begin a wff, it is unambiguous. The  -> looks like it would be a problem because  ph or  ps might include implications, but any implication arrow  -> within any wff must be surrounded by parentheses, so only the implication arrow of  A.! 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 32936 Extend wff definition to include "all some" applied to a top-level implication, which means  ps is true whenever 
ph is true, and there is at least least one  x where  ph is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
 wff  A.! x ( ph  ->  ps )
 
Syntaxwalsc 32937 Extend wff definition to include "all some" applied to a class, which means  ph 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  A.! x  e.  A ph
 
Definitiondf-alsi 32938 Define "all some" applied to a top-level implication, which means  ps is true whenever  ph is true and there is at least one  x where  ph is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
 |-  ( A.! x ( ph  ->  ps )  <->  ( A. x ( ph  ->  ps )  /\  E. x ph )
 )
 
Definitiondf-alsc 32939 Define "all some" applied to a class, which means  ph is true for all  x in  A and there is at least one  x in  A. (Contributed by David A. Wheeler, 20-Oct-2018.)
 |-  ( A.! x  e.  A ph  <->  (
 A. x  e.  A  ph 
 /\  E. x  x  e.  A ) )
 
Theoremalsconv 32940 There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.)
 |-  ( A.! x ( x  e.  A  ->  ph )  <->  A.! x  e.  A ph )
 
Theoremalsi1d 32941 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.)
 |-  ( ph  ->  A.! x ( ps 
 ->  ch ) )   =>    |-  ( ph  ->  A. x ( ps  ->  ch ) )
 
Theoremalsi2d 32942 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.)
 |-  ( ph  ->  A.! x ( ps 
 ->  ch ) )   =>    |-  ( ph  ->  E. x ps )
 
Theoremalsc1d 32943 Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
 |-  ( ph  ->  A.! x  e.  A ps )   =>    |-  ( ph  ->  A. x  e.  A  ps )
 
Theoremalsc2d 32944 Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
 |-  ( ph  ->  A.! x  e.  A ps )   =>    |-  ( ph  ->  E. x  x  e.  A )
 
Theoremalscn0d 32945* Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018.)
 |-  ( ph  ->  A.! x  e.  A ps )   =>    |-  ( ph  ->  A  =/= 
 (/) )
 
Theoremalsi-no-surprise 32946 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 32938; the proof itself builds on alimp-no-surprise 32931. For a contrast, see alimp-surprise 32930. (Contributed by David A. Wheeler, 27-Oct-2018.)
 |-  -.  ( A.! x ( ph  ->  ps )  /\  A.! x ( ph  ->  -. 
 ps ) )
 
21.26.15  Miscellaneous

Miscellaneous proofs.

 
Theorem5m4e1 32947 Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)
 |-  (
 5  -  4 )  =  1
 
Theorem2p2ne5 32948 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
 
Theoremresolution 32949 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.)
 |-  (
 ( ( ph  /\  ps )  \/  ( -.  ph  /\ 
 ch ) )  ->  ( ps  \/  ch )
 )
 
Theoremtestable 32950 In classical logic all wffs are testable, that is, it is always true that  ( -.  ph  \/  -.  -.  ph ). This is not necessarily true in intuitionistic logic. In intuitionistic logic, if this statement is true for some  ph, then  ph 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.)
 |-  ( -.  ph  \/  -.  -.  ph )
 
21.27  Mathbox for Alan Sare

We are sad to report the passing of long-time contributor Alan Sare (Nov. 9, 1954 - Mar. 23, 2019).

Alan's first contribution to Metamath was a shorter proof for tfrlem8 7055 in 2008.

He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: http://us.metamath.org/other.html#completeusersproof. His virtual deduction method is explained in the comment for wvd1 33079.

Below are some excerpts from his first emails to NM in 2007:

...I have been interested in proving set theory theorems for many years for mental exercise. I enjoy it. I have used a book by Martin Zuckerman. It is informal. I am interested in completely and perfectly proving theorems. Mr. Zuckerman leaves out most of the steps of a proof, of course, like most authors do, as you have noted. A complete proof for higher theorems would require a volume of writing similar to the metamath documents. So I am frustrated when I am not capable of constructing a proof and Zuckerman leaves out steps I do not understand. I could search for the steps in other texts, but I don't do that too much. Metamath may be the answer for me....

...If we go beyond mathematics, I believe that it is possible to write down all human knowledge in a way similar to the way you have explicated large areas of mathematics. Of course, that would be a much, much more difficult job. For example, it is possible to take a hard science like physics construct axioms based on experimental results and to cast all of physics into a collection of axioms and theorems. Maybe his has already been attempted, although I am not familiar with it. When one then moves on to the soft sciences such as social science, this job gets much more difficult. The key is: All human thought consists of logical operations on abstract objects. Usually, these logical operations are done informally. There is no reason why one cannot take any subject and explicate it and take it down to the indivisible postulates in a formal rigorous way....

...When I read a math book or an engineering book I come across something I don't understand and I am compelled to understand it. But, often it is hopeless. I don't have the time. Or, I would have to read the same thing by multiple authors in the hope that different authors would give parts of the working proof that others have omitted. It is very inefficient. Because I have always been inclined to "get to the bottom" for a 100% fully understood proof....

 
21.27.1  Auxiliary theorems for the Virtual Deduction tool
 
TheoremidiALT 32951 Placeholder for idi 2. Though unnecessary, this theorem is sometimes used in proofs in this mathbox for pedagogical purposes. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   =>    |-  ph
 
Theoremexbir 32952 Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 33386. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  (
 ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->  ( ph  ->  ( ps  ->  ( th  ->  ch )
 ) ) )
 
Theorem3impexp 32953 Version of impexp 446 for a triple conjunction. Derived automatically from 3impexpVD 33389. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  (
 ( ( ph  /\  ps  /\ 
 ch )  ->  th )  <->  (
 ph  ->  ( ps  ->  ( ch  ->  th )
 ) ) )
 
Theorem3impexpbicom 32954 Version of 3impexp 32953 where in addition the consequent is commuted. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  (
 ( ( ph  /\  ps  /\ 
 ch )  ->  ( th 
 <->  ta ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
 
Theorem3impexpbicomi 32955 Inference associated with 3impexpbicom 32954. Derived automatically from 3impexpbicomiVD 33391. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )
 
Theoremee02 32956 Proof of e02 33216 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |-  ( ph  ->  ( th  ->  ta ) )   =>    |-  ( ps  ->  ( ch  ->  ta )
 )
 
21.27.2  Supplementary "adant" deductions
 
Theoremad4ant13 32957 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( (
 ph  /\  th )  /\  ps )  /\  ta )  ->  ch )
 
Theoremad4ant14 32958 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( (
 ph  /\  th )  /\  ta )  /\  ps )  ->  ch )
 
Theoremad4ant123 32959 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  ta )  ->  th )
 
Theoremad4ant124 32960 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ta )  /\  ch )  ->  th )
 
Theoremad4ant134 32961 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ta )  /\  ps )  /\  ch )  ->  th )
 
Theoremad4ant23 32962 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( th  /\  ph )  /\  ps )  /\  ta )  ->  ch )
 
Theoremad4ant24 32963 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( th  /\  ph )  /\  ta )  /\  ps )  ->  ch )
 
Theoremad4ant234 32964 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ta 
 /\  ph )  /\  ps )  /\  ch )  ->  th )
 
Theoremad5ant12 32965 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  ps )  /\  th )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant13 32966 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ps )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant14 32967 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ta )  /\  ps )  /\  et )  ->  ch )
 
Theoremad5ant15 32968 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ta )  /\  et )  /\  ps )  ->  ch )
 
Theoremad5ant23 32969 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ps )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant24 32970 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ta )  /\  ps )  /\  et )  ->  ch )
 
Theoremad5ant25 32971 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ta )  /\  et )  /\  ps )  ->  ch )
 
Theoremad5ant245 32972 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ( ta  /\  ph )  /\  et )  /\  ps )  /\  ch )  ->  th )
 
Theoremad5ant234 32973 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ( ta  /\  ph )  /\  ps )  /\  ch )  /\  et )  ->  th )
 
Theoremad5ant235 32974 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ( ta  /\  ph )  /\  ps )  /\  et )  /\  ch )  ->  th )
 
Theoremad5ant123 32975 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ps )  /\  ch )  /\  ta )  /\  et )  ->  th )
 
Theoremad5ant124 32976 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ps )  /\  ta )  /\  ch )  /\  et )  ->  th )
 
Theoremad5ant125 32977 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ps )  /\  ta )  /\  et )  /\  ch )  ->  th )
 
Theoremad5ant134 32978 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ta )  /\  ps )  /\  ch )  /\  et )  ->  th )
 
Theoremad5ant135 32979 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ta )  /\  ps )  /\  et )  /\  ch )  ->  th )
 
Theoremad5ant145 32980 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ta )  /\  et )  /\  ps )  /\  ch )  ->  th )
 
Theoremad5ant1345 32981 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ( ( ph  /\  et )  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
Theoremad5ant2345 32982 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ( ( et  /\  ph )  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
21.27.3  Supplementary unification deductions
 
Theorembiimp 32983 Importation inference similar to imp 429, except the outermost implication of the hypothesis is a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  ->  ch )
 )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theorembi2imp 32984 Importation inference similar to imp 429, except the both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  <->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theorembi3impb 32985 Similar to 3impb 1193 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ph  /\  ( ps 
 /\  ch ) )  <->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi3impa 32986 Similar to 3impa 1192 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  <->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi23impib 32987 3impib 1195 with the inner implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ( ps 
 /\  ch )  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi13impib 32988 3impib 1195 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ( ps  /\  ch )  ->  th )
 )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi123impib 32989 3impib 1195 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ( ps  /\  ch )  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi13impia 32990 3impia 1194 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ph  /\  ps )  <->  ( ch  ->  th )
 )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi123impia 32991 3impia 1194 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ph  /\  ps )  <->  ( ch  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi33imp12 32992 3imp 1191 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi23imp13 32993 3imp 1191 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ps  <->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi13imp23 32994 3imp 1191 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi13imp2 32995 Similar to 3imp 1191 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  ->  ( ch 
 <-> 
 th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi12imp3 32996 Similar to 3imp 1191 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  <->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi23imp1 32997 Similar to 3imp 1191 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ps  <->  ( ch  <->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi123imp0 32998 Similar to 3imp 1191 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  <->  ( ch  <->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorem4animp1 32999 A single hypothesis unification deduction with an assertion which is an implication with a 4-right-nested conjunction antecedent. (Contributed by Alan Sare, 30-May-2018.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th ) )   =>    |-  ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
Theorem4an31 33000 A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
 |-  (
 ( ( ( ch 
 /\  ps )  /\  ph )  /\  th )  ->  ta )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 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