P. 330 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32901-33000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dpfrac1 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 `
Theorem	ene0 32902	_e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
|- _e =/= 0
Theorem	ene1 32903	_e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
|- _e =/= 1
Theorem	elogb 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.
Syntax	clog_ 32905	Extend class notation to include the logarithm generalized to an arbitrary base.
class log_
Definition	df-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
Syntax	wreflexive 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
Definition	df-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 ) )
Syntax	wirreflexive 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
Definition	df-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.
Theorem	comraddd 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 ) )
Theorem	comraddi 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 )
Theorem	mvlladdd 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 ) )
Theorem	mvlraddd 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 ) )
Theorem	mvlraddi 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 )
Theorem	mvrladdd 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 )
Theorem	mvrladdi 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
Theorem	mvrraddd 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 )
Theorem	mvrraddi 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
Theorem	assraddsubd 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 ) ) )
Theorem	assraddsubi 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 ) )
Theorem	joinlmuladdmuld 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 )
Theorem	joinlmuladdmuli 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
Theorem	joinlmulsubmuld 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 )
Theorem	joinlmulsubmuli 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
Theorem	mvlrmuld 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 ) )
Theorem	mvlrmuli 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.
Theorem	i2linesi 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 ) )
Theorem	i2linesd 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.
Theorem	alimp-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 ) )
Theorem	alimp-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 )
Theorem	empty-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
Theorem	empty-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.
Theorem	eximp-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 ) )
Theorem	eximp-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.
Syntax	walsi 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 )
Syntax	walsc 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
Definition	df-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 ) )
Definition	df-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 ) )
Theorem	alsconv 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 )
Theorem	alsi1d 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 ) )
Theorem	alsi2d 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 )
Theorem	alsc1d 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 )
Theorem	alsc2d 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 )
Theorem	alscn0d 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 =/= (/) )
Theorem	alsi-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.
Theorem	5m4e1 32947	Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)
|- ( 5 - 4 ) = 1
Theorem	2p2ne5 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
Theorem	resolution 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 ) )
Theorem	testable 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
Theorem	idiALT 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
Theorem	exbir 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 ) ) ) )
Theorem	3impexp 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 ) ) ) )
Theorem	3impexpbicom 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 ) ) ) ) )
Theorem	3impexpbicomi 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 ) ) ) )
Theorem	ee02 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
Theorem	ad4ant13 32957	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ( ph /\ th ) /\ ps ) /\ ta ) -> ch )
Theorem	ad4ant14 32958	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ( ph /\ th ) /\ ta ) /\ ps ) -> ch )
Theorem	ad4ant123 32959	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ph /\ ps ) /\ ch ) /\ ta ) -> th )
Theorem	ad4ant124 32960	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ph /\ ps ) /\ ta ) /\ ch ) -> th )
Theorem	ad4ant134 32961	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ph /\ ta ) /\ ps ) /\ ch ) -> th )
Theorem	ad4ant23 32962	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ( th /\ ph ) /\ ps ) /\ ta ) -> ch )
Theorem	ad4ant24 32963	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ( th /\ ph ) /\ ta ) /\ ps ) -> ch )
Theorem	ad4ant234 32964	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ta /\ ph ) /\ ps ) /\ ch ) -> th )
Theorem	ad5ant12 32965	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ( ( ph /\ ps ) /\ th ) /\ ta ) /\ et ) -> ch )
Theorem	ad5ant13 32966	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ( ( ph /\ th ) /\ ps ) /\ ta ) /\ et ) -> ch )
Theorem	ad5ant14 32967	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ( ( ph /\ th ) /\ ta ) /\ ps ) /\ et ) -> ch )
Theorem	ad5ant15 32968	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ( ( ph /\ th ) /\ ta ) /\ et ) /\ ps ) -> ch )
Theorem	ad5ant23 32969	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ( ( th /\ ph ) /\ ps ) /\ ta ) /\ et ) -> ch )
Theorem	ad5ant24 32970	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ( ( th /\ ph ) /\ ta ) /\ ps ) /\ et ) -> ch )
Theorem	ad5ant25 32971	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ( ( th /\ ph ) /\ ta ) /\ et ) /\ ps ) -> ch )
Theorem	ad5ant245 32972	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ( ta /\ ph ) /\ et ) /\ ps ) /\ ch ) -> th )
Theorem	ad5ant234 32973	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ( ta /\ ph ) /\ ps ) /\ ch ) /\ et ) -> th )
Theorem	ad5ant235 32974	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ( ta /\ ph ) /\ ps ) /\ et ) /\ ch ) -> th )
Theorem	ad5ant123 32975	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ( ph /\ ps ) /\ ch ) /\ ta ) /\ et ) -> th )
Theorem	ad5ant124 32976	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ( ph /\ ps ) /\ ta ) /\ ch ) /\ et ) -> th )
Theorem	ad5ant125 32977	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ( ph /\ ps ) /\ ta ) /\ et ) /\ ch ) -> th )
Theorem	ad5ant134 32978	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ( ph /\ ta ) /\ ps ) /\ ch ) /\ et ) -> th )
Theorem	ad5ant135 32979	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ( ph /\ ta ) /\ ps ) /\ et ) /\ ch ) -> th )
Theorem	ad5ant145 32980	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ( ( ( ph /\ ta ) /\ et ) /\ ps ) /\ ch ) -> th )
Theorem	ad5ant1345 32981	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
|- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta )   =>    |- ( ( ( ( ( ph /\ et ) /\ ps ) /\ ch ) /\ th ) -> ta )
Theorem	ad5ant2345 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
Theorem	biimp 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 )
Theorem	bi2imp 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 )
Theorem	bi3impb 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 )
Theorem	bi3impa 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 )
Theorem	bi23impib 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 )
Theorem	bi13impib 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 )
Theorem	bi123impib 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 )
Theorem	bi13impia 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 )
Theorem	bi123impia 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 )
Theorem	bi33imp12 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 )
Theorem	bi23imp13 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 )
Theorem	bi13imp23 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 )
Theorem	bi13imp2 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 )
Theorem	bi12imp3 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 )
Theorem	bi23imp1 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 )
Theorem	bi123imp0 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 )
Theorem	4animp1 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 )
Theorem	4an31 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 )