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Theorem List for Metamath Proof Explorer - 32601-32700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgt-lt 32601 Simple relationship between  < and  >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V )  ->  ( A  >  B  <->  B  <  A ) )
 
Theoremgte-lteh 32602 Relationship between  <_ and  >_ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  >_  B  <->  B  <_  A )
 
Theoremgt-lth 32603 Relationship between  < and  > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  >  B  <->  B  <  A )
 
Theoremex-gt 32604 Simple example of  >, in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)
 |-  -.  0  >  0
 
Theoremex-gte 32605 Simple example of  >_, in this case, 0 is greater than or equal to 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)
 |-  0  >_  0
 
21.26.3  Hyperbolic trigonometric functions

It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as  ( cos `  ( _i  x.  x ) ). However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved.

 
Syntaxcsinh 32606 Extend class notation to include the hyperbolic sine function, see df-sinh 32609.
 class sinh
 
Syntaxccosh 32607 Extend class notation to include the hyperbolic cosine function. see df-cosh 32610.
 class cosh
 
Syntaxctanh 32608 Extend class notation to include the hyperbolic tangent function, see df-tanh 32611.
 class tanh
 
Definitiondf-sinh 32609 Define the hyperbolic sine function (sinh). We define it this way for cmpt 4511, which requires the form  (
x  e.  A  |->  B ). See sinhval-named 32612 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 32615 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  e.  CC  |->  ( ( sin `  ( _i  x.  x ) ) 
 /  _i ) )
 
Definitiondf-cosh 32610 Define the hyperbolic cosine function (cosh). We define it this way for cmpt 4511, which requires the form  (
x  e.  A  |->  B ). (Contributed by David A. Wheeler, 10-May-2015.)
 |- cosh  =  ( x  e.  CC  |->  ( cos `  ( _i  x.  x ) ) )
 
Definitiondf-tanh 32611 Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4511, which requires the form  (
x  e.  A  |->  B ). (Contributed by David A. Wheeler, 10-May-2015.)
 |- tanh  =  ( x  e.  ( `'cosh " ( CC  \  { 0 } )
 )  |->  ( ( tan `  ( _i  x.  x ) )  /  _i ) )
 
Theoremsinhval-named 32612 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 32609. See sinhval 13767 for a theorem to convert this further. See sinh-conventional 32615 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  e.  CC  ->  (sinh `  A )  =  ( ( sin `  ( _i  x.  A ) ) 
 /  _i ) )
 
Theoremcoshval-named 32613 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 32610. See coshval 13768 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  CC  ->  (cosh `  A )  =  ( cos `  ( _i  x.  A ) ) )
 
Theoremtanhval-named 32614 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 32611. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  ( `'cosh " ( CC  \  {
 0 } ) ) 
 ->  (tanh `  A )  =  ( ( tan `  ( _i  x.  A ) ) 
 /  _i ) )
 
Theoremsinh-conventional 32615 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  e.  CC  ->  (sinh `  A )  =  (
 -u _i  x.  ( sin `  ( _i  x.  A ) ) ) )
 
Theoremsinhpcosh 32616 Prove that  (sinh `  A
)  +  (cosh `  A )  =  ( exp `  A ) using the conventional hyperbolic trigonometric functions. (Contributed by David A. Wheeler, 27-May-2015.)
 |-  ( A  e.  CC  ->  ( (sinh `  A )  +  (cosh `  A )
 )  =  ( exp `  A ) )
 
21.26.4  Reciprocal trigonometric functions (sec, csc, cot)

Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them.

 
Syntaxcsec 32617 Extend class notation to include the secant function, see df-sec 32620.
 class  sec
 
Syntaxccsc 32618 Extend class notation to include the cosecant function, see df-csc 32621.
 class  csc
 
Syntaxccot 32619 Extend class notation to include the cotangent function, see df-cot 32622.
 class  cot
 
Definitiondf-sec 32620* Define the secant function. We define it this way for cmpt 4511, which requires the form  ( x  e.  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  e.  {
 y  e.  CC  |  ( cos `  y )  =/=  0 }  |->  ( 1 
 /  ( cos `  x ) ) )
 
Definitiondf-csc 32621* Define the cosecant function. We define it this way for cmpt 4511, which requires the form  ( x  e.  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  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1 
 /  ( sin `  x ) ) )
 
Definitiondf-cot 32622* Define the cotangent function. We define it this way for cmpt 4511, which requires the form  ( x  e.  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  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x )  /  ( sin `  x ) ) )
 
Theoremsecval 32623 Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  =  ( 1  /  ( cos `  A ) ) )
 
Theoremcscval 32624 Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  =  ( 1  /  ( sin `  A ) ) )
 
Theoremcotval 32625 Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  =  ( ( cos `  A )  /  ( sin `  A ) ) )
 
Theoremseccl 32626 The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  CC )
 
Theoremcsccl 32627 The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  CC )
 
Theoremcotcl 32628 The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  CC )
 
Theoremreseccl 32629 The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  RR )
 
Theoremrecsccl 32630 The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  RR )
 
Theoremrecotcl 32631 The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  RR )
 
Theoremrecsec 32632 The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( cos `  A )  =  ( 1  /  ( sec `  A ) ) )
 
Theoremreccsc 32633 The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( sin `  A )  =  ( 1  /  ( csc `  A ) ) )
 
Theoremreccot 32634 The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  =  ( 1  /  ( cot `  A ) ) )
 
Theoremrectan 32635 The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( cot `  A )  =  ( 1  /  ( tan `  A ) ) )
 
Theoremsec0 32636 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( sec `  0 )  =  1
 
Theoremonetansqsecsq 32637 Prove the tangent squared secant squared identity  ( 1  +  ( ( tan A ) ^ 2 ) ) = ( ( sec  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 25-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( 1  +  (
 ( tan `  A ) ^ 2 ) )  =  ( ( sec `  A ) ^ 2
 ) )
 
Theoremcotsqcscsq 32638 Prove the tangent squared cosecant squared identity  ( 1  +  ( ( cot A ) ^ 2 ) ) = ( ( csc  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 27-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( 1  +  (
 ( cot `  A ) ^ 2 ) )  =  ( ( csc `  A ) ^ 2
 ) )
 
21.26.5  Identities for "if"

Utility theorems for "if".

 
Theoremifnmfalse 32639 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 3954 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
 
21.26.6  Not-member-of
 
TheoremAnelBC 32640 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using 
e/. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  A  =/=  B   &    |-  A  =/=  C   =>    |-  A  e/  { B ,  C }
 
21.26.7  Decimal point

Define the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 32644 and df-dp2 32643 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 10989.

TODO: Fix non-existent label dfpval.

 
Syntaxcdp2 32641 Constant used for decimal fraction constructor. See df-dp2 32643.
 class _ A B
 
Syntaxcdp 32642 Decimal point operator. See df-dp 32644.
 class  period
 
Definitiondf-dp2 32643 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 10989. (Contributed by David A. Wheeler, 15-May-2015.)
 |- _ A B  =  ( A  +  ( B 
 /  10 ) )
 
Definitiondf-dp 32644* Define the  period (decimal point) operator. For example,  ( 1 period 5 )  =  ( 3  /  2 ), and  -u (; 3 2 period_ 7_ 1 8 )  = 
-u (;;;; 3 2 7 1 8  / ;;; 1 0 0 0 ) Unary minus, if applied, should normally be applied in front of the parentheses.

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

The RHS is  RR, not  QQ; this should simplify some proofs. The LHS is  NN0, since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression  -u ( A period B ) is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.)

 |-  period  =  ( x  e.  NN0 ,  y  e.  RR  |-> _ x y )
 
Theoremdp2cl 32645 Define the closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  -> _ A B  e.  RR )
 
Theoremdpval 32646 Define the value of the decimal point operator. See df-dp 32644. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
 
Theoremdpcl 32647 Prove that the closure of the decimal point is  RR as we have defined it. See df-dp 32644. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  e.  RR )
 
Theoremdpfrac1 32648 Prove a simple equivalence involving the decimal point. See df-dp 32644 and dpcl 32647. (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 32649  _e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  0
 
Theoremene1 32650  _e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  1
 
Theoremelogb 32651 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_ 32652 Extend class notation to include the logarithm generalized to an arbitrary base.
 class log_
 
Definitiondf-log_ 32653* 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 32654 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 32655. (Contributed by David A. Wheeler, 1-Dec-2019.)
 wff  RReflexive A
 
Definitiondf-reflexive 32655* 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 32656 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 32657. (Contributed by David A. Wheeler, 1-Dec-2019.)
 wff  RIrreflexive A
 
Definitiondf-irreflexive 32657* 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 9783. We might want to switch to a naming convention like addcomli 9783.

 
Theoremcomraddd 32658 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 32659 Commute RHS addition. See addcomli 9783 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 32660 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 32661 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 32662 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 32663 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 32664 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 32665 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 32666 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 32667 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 32668 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 32669 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 32670 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 32671 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 32672 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 32673 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 32674 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 32675 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 32676 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 32677, empty-surprise 32679, and eximp-surprise 32681.

 
Theoremalimp-surprise 32677 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 32678) or using the allsome quantifier (df-alsi 32685) .

For other "surprises" for new users of classical logic, see empty-surprise 32679 and eximp-surprise 32681. (Contributed by David A. Wheeler, 17-Oct-2018.)

 |-  -.  E. x ph   =>    |-  ( A. x (
 ph  ->  ps )  /\  A. x ( ph  ->  -. 
 ps ) )
 
Theoremalimp-no-surprise 32678 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 32677. The allsome quantifier also counters this problem, see df-alsi 32685. (Contributed by David A. Wheeler, 27-Oct-2018.)
 |-  -.  ( A. x ( ph  ->  ps )  /\  A. x ( ph  ->  -. 
 ps )  /\  E. x ph )
 
Theoremempty-surprise 32679 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 2822). Thus, if  A is the empty set, this expression is always true regardless of the value of  ph (see alimp-surprise 32677).

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 3799, 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 2410, celaront 2411, and cesaro 2416.

For another "surprise" for new users of classical logic, see alimp-surprise 32677 and eximp-surprise 32681. (Contributed by David A. Wheeler, 20-Oct-2018.)

 |-  -.  E. x  x  e.  A   =>    |-  A. x  e.  A  ph
 
Theoremempty-surprise2 32680 "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 32679. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1416); 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 32686. (Contributed by David A. Wheeler, 20-Oct-2018.)

 |-  -.  E. x  x  e.  A   =>    |-  A. x  e.  A F.
 
Theoremeximp-surprise 32681 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 32682. See also alimp-surprise 32677 and empty-surprise 32679. (Contributed by David A. Wheeler, 17-Oct-2018.)

 |-  ( E. x ( ph  ->  ps )  <->  E. x ( -.  ph  \/  ps ) )
 
Theoremeximp-surprise2 32682 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 32681, which shows what implication really expands to. See also empty-surprise 32679. (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 32677 and empty-surprise 32679 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 32683 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 32684 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 32685 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 32686 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 32687 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 32688 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 32689 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 32690 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 32691 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 32692* 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 32693 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 32685; the proof itself builds on alimp-no-surprise 32678. For a contrast, see alimp-surprise 32677. (Contributed by David A. Wheeler, 27-Oct-2018.)
 |-  -.  ( A.! x ( ph  ->  ps )  /\  A.! x ( ph  ->  -. 
 ps ) )
 
21.26.15  Miscellaneous

Miscellaneous proofs.

 
Theorem5m4e1 32694 Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)
 |-  (
 5  -  4 )  =  1
 
Theorem2p2ne5 32695 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 32696 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 32697 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 7065 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 32827.

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 32698 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 32699 Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 33134. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  (
 ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->  ( ph  ->  ( ps  ->  ( th  ->  ch )
 ) ) )
 
Theorem3impexp 32700 Version of impexp 446 for a triple conjunction. Derived automatically from 3impexpVD 33137. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  (
 ( ( ph  /\  ps  /\ 
 ch )  ->  th )  <->  (
 ph  ->  ( ps  ->  ( ch  ->  th )
 ) ) )
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