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Theorem List for Metamath Proof Explorer - 36501-36600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremint-eqineqd 36501 EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  B  <_  A )
 
Theoremint-ineqmvtd 36502 IneqMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  B  <_  A )   &    |-  ( ph  ->  A  =  ( C  +  D ) )   =>    |-  ( ph  ->  ( B  -  D )  <_  C )
 
Theoremint-ineq1stprincd 36503 FirstPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  B  <_  A )   &    |-  ( ph  ->  D 
 <_  C )   =>    |-  ( ph  ->  ( B  +  D )  <_  ( A  +  C ) )
 
Theoremint-ineq2ndprincd 36504 SecondPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <_  A )   &    |-  ( ph  ->  0  <_  C )   =>    |-  ( ph  ->  ( B  x.  C )  <_  ( A  x.  C ) )
 
Theoremint-ineqtransd 36505 InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <_  A )   &    |-  ( ph  ->  C  <_  B )   =>    |-  ( ph  ->  C  <_  A )
 
21.26.3  N-Digit Addition Proof Generator

This section formalizes theorems used in an n-digit addition proof generator.

Other theorems required: deccl 11067 addcomli 9827 00id 9810 addid1i 9822 addid2i 9823 eqid 2423 dec0h 11069 decadd 11094 decaddc 11095.

 
Theoremunitadd 36506 Theorem used in conjunction with decaddc 11095 to absorb carry when generating n-digit addition synthetic proofs. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( A  +  B )  =  F   &    |-  ( C  +  1 )  =  B   &    |-  A  e.  NN0   &    |-  C  e.  NN0   =>    |-  ( ( A  +  C )  +  1
 )  =  F
 
Theorem5p5e10b 36507 5 + 5 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  (
 5  +  5 )  = ; 1 0
 
Theorem6p4e10b 36508 6 + 4 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  (
 6  +  4 )  = ; 1 0
 
Theorem7p3e10b 36509 7 + 3 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  (
 7  +  3 )  = ; 1 0
 
Theorem8p2e10b 36510 8 + 2 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  (
 8  +  2 )  = ; 1 0
 
Theorem9p1e10b 36511 9 + 1 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  (
 9  +  1 )  = ; 1 0
 
21.26.4  AM-GM (for k = 2,3,4)
 
Theoremgsumws3 36512 Valuation of a length 3 word in a monoid. (Contributed by Stanislas Polu, 9-Sep-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( G  e.  Mnd  /\  ( S  e.  B  /\  ( T  e.  B  /\  U  e.  B ) ) )  ->  ( G  gsumg 
 <" S T U "> )  =  ( S  .+  ( T 
 .+  U ) ) )
 
Theoremgsumws4 36513 Valuation of a length 4 word in a monoid (Contributed by Stanislas Polu, 10-Sep-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( G  e.  Mnd  /\  ( S  e.  B  /\  ( T  e.  B  /\  ( U  e.  B  /\  V  e.  B ) ) ) )  ->  ( G  gsumg 
 <" S T U V "> )  =  ( S  .+  ( T  .+  ( U  .+  V ) ) ) )
 
Theoremamgm2d 36514 Arithmetic-geometric mean inequality for  n  =  2, derived from amgmlem 23907. (Contributed by Stanislas Polu, 8-Sep-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  B )  ^c  ( 1 
 /  2 ) ) 
 <_  ( ( A  +  B )  /  2
 ) )
 
Theoremamgm3d 36515 Arithmetic-geometric mean inequality for  n  =  3. (Contributed by Stanislas Polu, 11-Sep-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  ( B  x.  C ) ) 
 ^c  ( 1 
 /  3 ) ) 
 <_  ( ( A  +  ( B  +  C ) )  /  3
 ) )
 
Theoremamgm4d 36516 Arithmetic-geometric mean inequality for  n  =  4. (Contributed by Stanislas Polu, 11-Sep-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  D  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  ( B  x.  ( C  x.  D ) ) ) 
 ^c  ( 1 
 /  4 ) ) 
 <_  ( ( A  +  ( B  +  ( C  +  D )
 ) )  /  4
 ) )
 
21.27  Mathbox for Steve Rodriguez
 
21.27.1  Miscellanea
 
Theoremnanorxor 36517 'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020.)
 |-  (
 ( ph  -/\  ps )  <->  ( ( ph  \/  ps ) 
 <->  ( ph  \/_  ps ) ) )
 
Theoremundisjrab 36518 Union of two disjoint restricted class abstractions; compare unrab 3745. (Contributed by Steve Rodriguez, 28-Feb-2020.)
 |-  (
 ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  ps } )  =  (/)  <->  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  ps } )  =  { x  e.  A  |  ( ph  \/_  ps ) } )
 
Theoremiso0 36519 The empty set is an  R ,  S isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
 |-  (/)  Isom  R ,  S  ( (/) ,  (/) )
 
Theoremssrecnpr 36520  RR is a subset of both  RR and  CC. (Contributed by Steve Rodriguez, 22-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  RR  C_  S )
 
Theoremseff 36521 Let set  S be the real or complex numbers. Then the exponential function restricted to  S is a mapping from  S to  S. (Contributed by Steve Rodriguez, 6-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   =>    |-  ( ph  ->  ( exp  |`  S ) : S --> S )
 
Theoremsblpnf 36522 The infinity ball in the absolute value metric is just the whole space.  S analog of blpnf 21404. (Contributed by Steve Rodriguez, 8-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  D  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )   =>    |-  ( ( ph  /\  P  e.  S ) 
 ->  ( P ( ball `  D ) +oo )  =  S )
 
Theoremprmunb2 36523* The primes are unbounded. This generalizes prmunb 14851 to real  A with arch 10868 and lttrd 9798: every real is less than some positive integer, itself less than some prime. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( A  e.  RR  ->  E. p  e.  Prime  A  <  p )
 
Theoremisprm7 36524* One need only check prime divisors of  P up to  sqr P in order to ensure primality. This version of isprm5 14644 combines the primality and bound on  z into a finite interval of prime numbers. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  (
 ( 2 ... ( |_ `  ( sqr `  P ) ) )  i^i 
 Prime )  -.  z  ||  P ) )
 
21.27.2  Ratio test for infinite series convergence and divergence
 
Theoremdvgrat 36525* Ratio test for divergence of a complex infinite series. See e.g. remark "if  ( abs `  (
( a `  (
n  +  1 ) )  /  ( a `
 n ) ) )  >_  1 for all large n..." in https://en.wikipedia.org/wiki/Ratio_test#The_test. (Contributed by Steve Rodriguez, 28-Feb-2020.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  ( ZZ>= `  N )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  W )  ->  ( F `  k )  =/=  0 )   &    |-  ( ( ph  /\  k  e.  W ) 
 ->  ( abs `  ( F `  k ) ) 
 <_  ( abs `  ( F `  ( k  +  1 ) ) ) )   =>    |-  ( ph  ->  seq M (  +  ,  F )  e/  dom  ~~>  )
 
Theoremcvgdvgrat 36526* Ratio test for convergence and divergence of a complex infinite series. If the ratio  R of the absolute values of successive terms in an infinite sequence  F converges to less than one, then the infinite sum of the terms of  F converges to a complex number; and if  R converges greater then the sum diverges. This combined form of cvgrat 13932 and dvgrat 36525 directly uses the limit of the ratio.

(It also demonstrates how to use climi2 13568 and absltd 13485 to transform a limit to an inequality cf. https://math.stackexchange.com/q/2215191, and how to use r19.29a 2971 in a similar fashion to Mario Carneiro's proof sketch with rexlimdva 2918 at https://groups.google.com/forum/#!topic/metamath/2RPikOiXLMo.) (Contributed by Steve Rodriguez, 28-Feb-2020.)

 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  ( ZZ>= `  N )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  W )  ->  ( F `  k )  =/=  0 )   &    |-  R  =  ( k  e.  W  |->  ( abs `  ( ( F `  ( k  +  1 ) )  /  ( F `  k ) ) ) )   &    |-  ( ph  ->  R  ~~>  L )   &    |-  ( ph  ->  L  =/=  1
 )   =>    |-  ( ph  ->  ( L  <  1  <->  seq M (  +  ,  F )  e.  dom  ~~>  ) )
 
Theoremradcnvrat 36527* Let  L be the limit, if one exists, of the ratio  ( abs `  (
( A `  (
k  +  1 ) )  /  ( A `
 k ) ) ) (as in the ratio test cvgdvgrat 36526) as  k increases. Then the radius of convergence of power series  sum_ n  e.  NN0 ( ( A `  n )  x.  (
x ^ n ) ) is  ( 1  /  L ) if  L is nonzero. Proof "The limit involved in the ratio test..." in https://en.wikipedia.org/wiki/Radius_of_convergence —a few lines that evidently hide quite an involved process to confirm. (Contributed by Steve Rodriguez, 8-Mar-2020.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0
 (  +  ,  ( G `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  D  =  ( k  e.  NN0  |->  ( abs `  (
 ( A `  (
 k  +  1 ) )  /  ( A `
  k ) ) ) )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( A `  k )  =/=  0 )   &    |-  ( ph  ->  D  ~~>  L )   &    |-  ( ph  ->  L  =/=  0 )   =>    |-  ( ph  ->  R  =  ( 1  /  L ) )
 
21.27.3  Multiples
 
Theoremreldvds 36528 The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  Rel  ||
 
Theoremnznngen 36529 All positive integers in the set of multiples of n, nℤ, are the absolute value of n or greater. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  (
 (  ||  " { N } )  i^i  NN )  C_  ( ZZ>= `  ( abs `  N ) ) )
 
Theoremnzss 36530 The set of multiples of m, mℤ, is a subset of those of n, nℤ, iff n divides m. Lemma 2.1(a) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5, with mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  V )   =>    |-  ( ph  ->  (
 (  ||  " { M } )  C_  (  ||  " { N } )  <->  N 
 ||  M ) )
 
Theoremnzin 36531 The intersection of the set of multiples of m, mℤ, and those of n, nℤ, is the set of multiples of their least common multiple. Roughly Lemma 2.1(c) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5 and Problem 1(b) of https://people.math.binghamton.edu/mazur/teach/40107/40107h16sol.pdf p. 1, with mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  (
 (  ||  " { M } )  i^i  (  ||  " { N } )
 )  =  (  ||  " { ( M lcm  N ) } ) )
 
Theoremnzprmdif 36532 Subtract one prime's multiples from an unequal prime's. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ph  ->  M  e.  Prime )   &    |-  ( ph  ->  N  e.  Prime )   &    |-  ( ph  ->  M  =/=  N )   =>    |-  ( ph  ->  ( (  ||  " { M } )  \  (  ||  " { N } )
 )  =  ( ( 
 ||  " { M }
 )  \  (  ||  " {
 ( M  x.  N ) } ) ) )
 
Theoremhashnzfz 36533 Special case of hashdvds 14716: the count of multiples in nℤ restricted to an interval. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  J  e.  ZZ )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  ( J  -  1 ) ) )   =>    |-  ( ph  ->  ( # `
  ( (  ||  " { N } )  i^i  ( J ... K ) ) )  =  ( ( |_ `  ( K  /  N ) )  -  ( |_ `  (
 ( J  -  1
 )  /  N )
 ) ) )
 
Theoremhashnzfz2 36534 Special case of hashnzfz 36533: the count of multiples in nℤ, n greater than one, restricted to an interval starting at two. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  K  e.  NN )   =>    |-  ( ph  ->  ( # `
  ( (  ||  " { N } )  i^i  ( 2 ... K ) ) )  =  ( |_ `  ( K  /  N ) ) )
 
Theoremhashnzfzclim 36535* As the upper bound  K of the constraint interval  ( J ... K
) in hashnzfz 36533 increases, the resulting count of multiples tends to  ( K  /  M ) —that is, there are approximately  ( K  /  M
) multiples of  M in a finite interval of integers. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  J  e.  ZZ )   =>    |-  ( ph  ->  (
 k  e.  ( ZZ>= `  ( J  -  1
 ) )  |->  ( ( # `  ( (  ||  " { M } )  i^i  ( J ... k
 ) ) )  /  k ) )  ~~>  ( 1  /  M ) )
 
21.27.4  Function operations
 
Theoremcaofcan 36536* Transfer a cancellation law like mulcan 10251 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> T )   &    |-  ( ph  ->  G : A --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x R y )  =  ( x R z )  <->  y  =  z
 ) )   =>    |-  ( ph  ->  (
 ( F  oF R G )  =  ( F  oF R H )  <->  G  =  H ) )
 
Theoremofsubid 36537 Function analog of subid 9895. (Contributed by Steve Rodriguez, 5-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC )  ->  ( F  oF  -  F )  =  ( A  X.  { 0 } ) )
 
Theoremofmul12 36538 Function analog of mul12 9801. (Contributed by Steve Rodriguez, 13-Nov-2015.)
 |-  (
 ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A --> CC  /\  H : A --> CC )
 )  ->  ( F  oF  x.  ( G  oF  x.  H ) )  =  ( G  oF  x.  ( F  oF  x.  H ) ) )
 
Theoremofdivrec 36539 Function analog of divrec 10288, a division analog of ofnegsub 10609. (Contributed by Steve Rodriguez, 3-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC  \  { 0 } )
 )  ->  ( F  oF  x.  (
 ( A  X.  {
 1 } )  oF  /  G ) )  =  ( F  oF  /  G ) )
 
Theoremofdivcan4 36540 Function analog of divcan4 10297. (Contributed by Steve Rodriguez, 4-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC  \  { 0 } )
 )  ->  ( ( F  oF  x.  G )  oF  /  G )  =  F )
 
Theoremofdivdiv2 36541 Function analog of divdiv2 10321. (Contributed by Steve Rodriguez, 23-Nov-2015.)
 |-  (
 ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A --> ( CC  \  { 0 } )  /\  H : A --> ( CC  \  { 0 } )
 ) )  ->  ( F  oF  /  ( G  oF  /  H ) )  =  (
 ( F  oF  x.  H )  oF  /  G ) )
 
21.27.5  Calculus
 
Theoremlhe4.4ex1a 36542 Example of the Fundamental Theorem of Calculus, part two (ftc2 22988):  S. ( 1 (,) 2 ) ( ( x ^ 2 )  -  3 )  _d x  =  -u ( 2  /  3
). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 22988 as simply the "Fundamental Theorem of Calculus", then ftc1 22986 as the "Second Fundamental Theorem of Calculus".) (Contributed by Steve Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.)
 |-  S. ( 1 (,) 2
 ) ( ( x ^ 2 )  -  3 )  _d x  =  -u ( 2  / 
 3 )
 
Theoremdvsconst 36543 Derivative of a constant function on the real or complex numbers. The function may return a complex 
A even if  S is  RR. (Contributed by Steve Rodriguez, 11-Nov-2015.)
 |-  (
 ( S  e.  { RR ,  CC }  /\  A  e.  CC )  ->  ( S  _D  ( S  X.  { A }
 ) )  =  ( S  X.  { 0 } ) )
 
Theoremdvsid 36544 Derivative of the identity function on the real or complex numbers. (Contributed by Steve Rodriguez, 11-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  (  _I  |`  S ) )  =  ( S  X.  { 1 } ) )
 
Theoremdvsef 36545 Derivative of the exponential function on the real or complex numbers. (Contributed by Steve Rodriguez, 12-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( exp  |`  S ) )  =  ( exp  |`  S ) )
 
Theoremexpgrowthi 36546* Exponential growth and decay model. See expgrowth 36548 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  K  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  Y  =  ( t  e.  S  |->  ( C  x.  ( exp `  ( K  x.  t
 ) ) ) )   =>    |-  ( ph  ->  ( S  _D  Y )  =  ( ( S  X.  { K } )  oF  x.  Y ) )
 
Theoremdvconstbi 36547* The derivative of a function on  S is zero iff it is a constant function. Roughly a biconditional  S analog of dvconst 22863 and dveq0 22944. Corresponds to integration formula " S. 0  _d x  =  C " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  Y : S --> CC )   &    |-  ( ph  ->  dom  ( S  _D  Y )  =  S )   =>    |-  ( ph  ->  ( ( S  _D  Y )  =  ( S  X.  {
 0 } )  <->  E. c  e.  CC  Y  =  ( S  X.  { c } )
 ) )
 
Theoremexpgrowth 36548* Exponential growth and decay model. The derivative of a function y of variable t equals a constant k times y itself, iff y equals some constant C times the exponential of kt. This theorem and expgrowthi 36546 illustrate one of the simplest and most crucial classes of differential equations, equations that relate functions to their derivatives.

Section 6.3 of [Strang] p. 242 calls y' = ky "the most important differential equation in applied mathematics". In the field of population ecology it is known as the Malthusian growth model or exponential law, and C, k, and t correspond to initial population size, growth rate, and time respectively (https://en.wikipedia.org/wiki/Malthusian_growth_model); and in finance, the model appears in a similar role in continuous compounding with C as the initial amount of money. In exponential decay models, k is often expressed as the negative of a positive constant λ.

Here y' is given as  ( S  _D  Y
), C as  c, and ky as  ( ( S  X.  { K }
)  oF  x.  Y ).  ( S  X.  { K }
) is the constant function that maps any real or complex input to k and  oF  x. is multiplication as a function operation.

The leftward direction of the biconditional is as given in http://www.saylor.org/site/wp-content/uploads/2011/06/MA221-2.1.1.pdf pp. 1-2, which also notes the reverse direction ("While we will not prove this here, it turns out that these are the only functions that satisfy this equation."). The rightward direction is Theorem 5.1 of [LarsonHostetlerEdwards] p. 375 (which notes " C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0."); its proof here closely follows the proof of y' = y in https://proofwiki.org/wiki/Exponential_Growth_Equation/Special_Case.

Statements for this and expgrowthi 36546 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)

 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  K  e.  CC )   &    |-  ( ph  ->  Y : S --> CC )   &    |-  ( ph  ->  dom  ( S  _D  Y )  =  S )   =>    |-  ( ph  ->  (
 ( S  _D  Y )  =  ( ( S  X.  { K }
 )  oF  x.  Y )  <->  E. c  e.  CC  Y  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t ) ) ) ) ) )
 
21.27.6  The generalized binomial coefficient operation
 
Syntaxcbcc 36549 Extend class notation to include the generalized binomial coefficient operation.
 class C𝑐
 
Definitiondf-bcc 36550* Define a generalized binomial coefficient operation, which unlike df-bc 12489 allows complex numbers for the first argument. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |- C𝑐  =  ( c  e.  CC ,  k  e. 
 NN0  |->  ( ( c FallFac  k )  /  ( ! `  k ) ) )
 
Theorembccval 36551 Value of the generalized binomial coefficient,  C choose  K. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ph  ->  ( CC𝑐 K )  =  (
 ( C FallFac  K )  /  ( ! `  K ) ) )
 
Theorembcccl 36552 Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ph  ->  ( CC𝑐 K )  e.  CC )
 
Theorembcc0 36553 The generalized binomial coefficient  C choose  K is zero iff  C is an integer between zero and  ( K  - 
1 ) inclusive. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ph  ->  (
 ( CC𝑐 K )  =  0  <->  C  e.  ( 0 ... ( K  -  1
 ) ) ) )
 
Theorembccp1k 36554 Generalized binomial coefficient: 
C choose  ( K  +  1 ). (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ph  ->  ( CC𝑐 ( K  +  1
 ) )  =  ( ( CC𝑐 K )  x.  (
 ( C  -  K )  /  ( K  +  1 ) ) ) )
 
Theorembccm1k 36555 Generalized binomial coefficient: 
C choose  ( K  -  1 ), when  C is not  ( K  -  1 ). (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  ( CC  \  { ( K  -  1 ) }
 ) )   &    |-  ( ph  ->  K  e.  NN )   =>    |-  ( ph  ->  ( CC𝑐 ( K  -  1
 ) )  =  ( ( CC𝑐 K )  /  (
 ( C  -  ( K  -  1 ) ) 
 /  K ) ) )
 
Theorembccn0 36556 Generalized binomial coefficient: 
C choose  0. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( CC𝑐 0 )  =  1
 )
 
Theorembccn1 36557 Generalized binomial coefficient: 
C choose  1. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( CC𝑐 1 )  =  C )
 
Theorembccbc 36558 The binomial coefficient and generalized binomial coefficient are equal when their arguments are nonnegative integers. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ph  ->  ( NC𝑐 K )  =  ( N  _C  K ) )
 
21.27.7  Binomial series
 
Theoremuzmptshftfval 36559* When  F is a maps-to function on some set of upper integers  Z that returns a set  B,  ( F  shift  N ) is another maps-to function on the shifted set of upper integers  W. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  F  =  ( x  e.  Z  |->  B )   &    |-  B  e.  _V   &    |-  ( x  =  ( y  -  N )  ->  B  =  C )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  W  =  ( ZZ>= `  ( M  +  N ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( F  shift  N )  =  ( y  e.  W  |->  C ) )
 
Theoremdvradcnv2 36560* The radius of convergence of the (formal) derivative  H of the power series  G is (at least) as large as the radius of convergence of  G. This version of dvradcnv 23368 uses a shifted version of  H to match the sum form of  ( CC  _D  F
) in pserdv2 23377 (and shows how to use uzmptshftfval 36559 to shift a maps-to function on a set of upper integers). (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  H  =  ( n  e.  NN  |->  ( ( n  x.  ( A `  n ) )  x.  ( X ^ ( n  -  1 ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  R )   =>    |-  ( ph  ->  seq 1
 (  +  ,  H )  e.  dom  ~~>  )
 
Theorembinomcxplemwb 36561 Lemma for binomcxp 36570. The lemma in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  K  e.  NN )   =>    |-  ( ph  ->  (
 ( ( C  -  K )  x.  ( CC𝑐 K ) )  +  ( ( C  -  ( K  -  1
 ) )  x.  ( CC𝑐 ( K  -  1
 ) ) ) )  =  ( C  x.  ( CC𝑐 K ) ) )
 
Theorembinomcxplemnn0 36562* Lemma for binomcxp 36570. When  C is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 13881 equals this generalized infinite sum: the generalized binomial coefficient and exponentiation operators give exactly the same values in the standard index set  ( 0 ... C
), and when the index set is widened beyond  C the additional values are just zeroes. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ( ph  /\  C  e.  NN0 )  ->  ( ( A  +  B )  ^c  C )  =  sum_ k  e. 
 NN0  ( ( CC𝑐 k )  x.  ( ( A  ^c  ( C  -  k ) )  x.  ( B ^ k ) ) ) )
 
Theorembinomcxplemrat 36563* Lemma for binomcxp 36570. As  k increases, this ratio's absolute value converges to one. Part of equation "Since continuity of the absolute value..." in the Wikibooks proof (proven for the inverse ratio, which we later show is no problem). (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( k  e.  NN0  |->  ( abs `  ( ( C  -  k )  /  (
 k  +  1 ) ) ) )  ~~>  1 )
 
Theorembinomcxplemfrat 36564* Lemma for binomcxp 36570. binomcxplemrat 36563 implies that when  C is not a nonnegative integer, the absolute value of the ratio  ( ( F `
 ( k  +  1 ) )  / 
( F `  k
) ) converges to one. The rest of equation "Since continuity of the absolute value..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )   =>    |-  ( ( ph  /\ 
 -.  C  e.  NN0 )  ->  ( k  e. 
 NN0  |->  ( abs `  (
 ( F `  (
 k  +  1 ) )  /  ( F `
  k ) ) ) )  ~~>  1 )
 
Theorembinomcxplemradcnv 36565* Lemma for binomcxp 36570. By binomcxplemfrat 36564 and radcnvrat 36527 the radius of convergence of power series  sum_ k  e.  NN0 ( ( F `  k )  x.  (
b ^ k ) ) is one. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )   &    |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
 k ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( S `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   =>    |-  (
 ( ph  /\  -.  C  e.  NN0 )  ->  R  =  1 )
 
Theorembinomcxplemdvbinom 36566* Lemma for binomcxp 36570. By the power and chain rules, calculate the derivative of  ( ( 1  +  b )  ^c  -u C ), with respect to  b in the disk of convergence 
D. We later multiply the derivative in the later binomcxplemdvsum 36568 by this derivative to show that  ( ( 1  +  b )  ^c  C ) (with a non-negated  C) and the later sum, since both at  b  =  0 equal one, are the same. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )   &    |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
 k ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( S `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  E  =  ( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
 ) )  x.  (
 b ^ ( k  -  1 ) ) ) ) )   &    |-  D  =  ( `' abs " (
 0 [,) R ) )   =>    |-  ( ( ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  (
 b  e.  D  |->  ( ( 1  +  b
 )  ^c  -u C ) ) )  =  ( b  e.  D  |->  ( -u C  x.  ( ( 1  +  b )  ^c 
 ( -u C  -  1
 ) ) ) ) )
 
Theorembinomcxplemcvg 36567* Lemma for binomcxp 36570. The sum in binomcxplemnn0 36562 and its derivative (see the next theorem, binomcxplemdvsum 36568) converge, as long as their base  J is within the disk of convergence. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )   &    |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
 k ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( S `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  E  =  ( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
 ) )  x.  (
 b ^ ( k  -  1 ) ) ) ) )   &    |-  D  =  ( `' abs " (
 0 [,) R ) )   =>    |-  ( ( ph  /\  J  e.  D )  ->  (  seq 0 (  +  ,  ( S `  J ) )  e.  dom  ~~>  /\  seq 1
 (  +  ,  ( E `  J ) )  e.  dom  ~~>  ) )
 
Theorembinomcxplemdvsum 36568* Lemma for binomcxp 36570. The derivative of the generalized sum in binomcxplemnn0 36562. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )   &    |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
 k ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( S `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  E  =  ( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
 ) )  x.  (
 b ^ ( k  -  1 ) ) ) ) )   &    |-  D  =  ( `' abs " (
 0 [,) R ) )   &    |-  P  =  ( b  e.  D  |->  sum_ k  e.  NN0  ( ( S `  b ) `  k
 ) )   =>    |-  ( ph  ->  ( CC  _D  P )  =  ( b  e.  D  |->  sum_
 k  e.  NN  (
 ( E `  b
 ) `  k )
 ) )
 
Theorembinomcxplemnotnn0 36569* Lemma for binomcxp 36570. When  C is not a nonnegative integer, the generalized sum in binomcxplemnn0 36562 —which we will call  P —is a convergent power series: its base  b is always of smaller absolute value than the radius of convergence.

pserdv2 23377 gives the derivative of  P, which by dvradcnv 23368 also converges in that radius. When  A is fixed at one,  ( A  +  b ) times that derivative equals  ( C  x.  P
) and fraction  ( P  / 
( ( A  +  b )  ^c  C ) ) is always defined with derivative zero, so the fraction is a constant—specifically one, because  ( ( 1  +  0 )  ^c  C )  =  1. Thus  ( ( 1  +  b )  ^c  C )  =  ( P `  b ).

Finally, let  b be  ( B  /  A ), and multiply both the binomial  ( ( 1  +  ( B  /  A ) )  ^c  C ) and the sum  ( P `  ( B  /  A
) ) by  ( A  ^c  C ) to get the result. (Contributed by Steve Rodriguez, 22-Apr-2020.)

 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )   &    |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
 k ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( S `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  E  =  ( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
 ) )  x.  (
 b ^ ( k  -  1 ) ) ) ) )   &    |-  D  =  ( `' abs " (
 0 [,) R ) )   &    |-  P  =  ( b  e.  D  |->  sum_ k  e.  NN0  ( ( S `  b ) `  k
 ) )   =>    |-  ( ( ph  /\  -.  C  e.  NN0 )  ->  ( ( A  +  B )  ^c  C )  =  sum_ k  e. 
 NN0  ( ( CC𝑐 k )  x.  ( ( A  ^c  ( C  -  k ) )  x.  ( B ^ k ) ) ) )
 
Theorembinomcxp 36570* Generalize the binomial theorem binom 13881 to positive real summand  A, real summand  B, and complex exponent  C. Proof in https://en.wikibooks.org/wiki/Advanced_Calculus; see also https://en.wikipedia.org/wiki/Binomial_series, https://en.wikipedia.org/wiki/Binomial_theorem (sections "Newton's generalized binomial theorem" and "Future generalizations"), and proof "General Binomial Theorem" in https://proofwiki.org/wiki/Binomial_Theorem. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  B )  ^c  C )  =  sum_ k  e.  NN0  ( ( CC𝑐 k )  x.  ( ( A 
 ^c  ( C  -  k ) )  x.  ( B ^
 k ) ) ) )
 
21.28  Mathbox for Andrew Salmon
 
21.28.1  Principia Mathematica * 10
 
Theorempm10.12 36571* Theorem *10.12 in [WhiteheadRussell] p. 146. In *10, this is treated as an axiom, and the proofs in *10 are based on this theorem. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x ( ph  \/  ps )  ->  ( ph  \/  A. x ps )
 )
 
Theorempm10.14 36572 Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  (
 ( A. x ph  /\  A. x ps )  ->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps )
 )
 
Theorempm10.251 36573 Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x  -.  ph  ->  -. 
 A. x ph )
 
Theorempm10.252 36574 Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.)
 |-  ( -.  E. x ph  <->  A. x  -.  ph )
 
Theorempm10.253 36575 Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( -.  A. x ph  <->  E. x  -.  ph )
 
Theoremalbitr 36576 Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ( ph  <->  ps )  /\  A. x ( ps  <->  ch ) )  ->  A. x ( ph  <->  ch ) )
 
Theorempm10.42 36577 Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  (
 ( E. x ph  \/  E. x ps )  <->  E. x ( ph  \/  ps ) )
 
Theorempm10.52 36578* Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x ph  ->  ( A. x ( ph  ->  ps )  <->  ps ) )
 
Theorempm10.53 36579 Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x ph  ->  A. x ( ph  ->  ps ) )
 
Theorempm10.541 36580* Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ch  \/  ps )
 ) 
 <->  ( ch  \/  A. x ( ph  ->  ps ) ) )
 
Theorempm10.542 36581* Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ch  ->  ps )
 ) 
 <->  ( ch  ->  A. x ( ph  ->  ps )
 ) )
 
Theorempm10.55 36582 Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( E. x (
 ph  /\  ps )  /\  A. x ( ph  ->  ps ) )  <->  ( E. x ph 
 /\  A. x ( ph  ->  ps ) ) )
 
Theorempm10.56 36583 Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ( ph  ->  ps )  /\  E. x ( ph  /\  ch ) )  ->  E. x ( ps  /\  ch )
 )
 
Theorempm10.57 36584 Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ps  \/  ch )
 )  ->  ( A. x ( ph  ->  ps )  \/  E. x ( ph  /\  ch )
 ) )
 
21.28.2  Principia Mathematica * 11
 
Theorem2alanimi 36585 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( A. x A. y ph  /\  A. x A. y ps )  ->  A. x A. y ch )
 
Theorem2al2imi 36586 Removes two universal qunatifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x A. y ph  ->  ( A. x A. y ps  ->  A. x A. y ch ) )
 
Theorempm11.11 36587 Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ph   =>    |- 
 A. z A. w [ z  /  x ] [ w  /  y ] ph
 
Theorempm11.12 36588* Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x A. y (
 ph  \/  ps )  ->  ( ph  \/  A. x A. y ps )
 )
 
Theorem19.21vv 36589* Compare theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1776. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ps  ->  ph )  <->  ( ps  ->  A. x A. y ph ) )
 
Theorem2alim 36590 Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  ->  ps )  ->  ( A. x A. y ph  ->  A. x A. y ps ) )
 
Theorem2albi 36591 Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph 
 <->  ps )  ->  ( A. x A. y ph  <->  A. x A. y ps )
 )
 
Theorem2exim 36592 Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  ->  ps )  ->  ( E. x E. y ph  ->  E. x E. y ps ) )
 
Theorem2exbi 36593 Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph 
 <->  ps )  ->  ( E. x E. y ph  <->  E. x E. y ps )
 )
 
Theoremspsbce-2 36594 Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  ->  E. x E. y ph )
 
Theorem19.33-2 36595 Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x A. y ph  \/  A. x A. y ps )  ->  A. x A. y ( ph  \/  ps ) )
 
Theorem19.36vv 36596* Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( E. x E. y (
 ph  ->  ps )  <->  ( A. x A. y ph  ->  ps )
 )
 
Theorem19.31vv 36597* Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  \/  ps )  <->  (
 A. x A. y ph  \/  ps ) )
 
Theorem19.37vv 36598* Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ( ps  ->  ph )  <->  ( ps  ->  E. x E. y ph ) )
 
Theorem19.28vv 36599* Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ps  /\  ph )  <->  ( ps  /\  A. x A. y ph ) )
 
Theorempm11.52 36600 Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y (
 ph  /\  ps )  <->  -. 
 A. x A. y
 ( ph  ->  -.  ps ) )
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