Article IRRAT_1, MML version 4.99.1005

:: IRRAT_1:attrnot 1 => RAT_1:attr 1
notation
  let a1 be real set;
  antonym irrational for rational;
end;

:: IRRAT_1:funcnot 1 => POWER:func 3
notation
  let a1, a2 be real set;
  synonym a1 ^ a2 for a1 to_power a2;
end;

:: IRRAT_1:th 1
theorem
for b1 being Element of NAT
      st b1 is prime
   holds sqrt b1 is not rational;

:: IRRAT_1:th 2
theorem
ex b1, b2 being real set st
   b1 is not rational & b2 is not rational & b1 to_power b2 is rational;

:: IRRAT_1:sch 1
scheme IRRAT_1:sch 1
{F1 -> real set}:
(ex b1 being Function-like quasi_total Relation of NAT,REAL st
    for b2 being Element of NAT holds
       b1 . b2 = F1(b2)) &
 (for b1, b2 being Function-like quasi_total Relation of NAT,REAL
       st (for b3 being Element of NAT holds
             b1 . b3 = F1(b3)) &
          (for b3 being Element of NAT holds
             b2 . b3 = F1(b3))
    holds b1 = b2)


:: IRRAT_1:funcnot 2 => IRRAT_1:func 1
definition
  let a1 be natural set;
  func aseq A1 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = (b1 - a1) / b1;
end;

:: IRRAT_1:def 1
theorem
for b1 being natural set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b2 = aseq b1
   iff
      for b3 being Element of NAT holds
         b2 . b3 = (b3 - b1) / b3;

:: IRRAT_1:funcnot 3 => IRRAT_1:func 2
definition
  let a1 be natural set;
  func bseq A1 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = (b1 choose a1) * (b1 to_power - a1);
end;

:: IRRAT_1:def 2
theorem
for b1 being natural set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b2 = bseq b1
   iff
      for b3 being Element of NAT holds
         b2 . b3 = (b3 choose b1) * (b3 to_power - b1);

:: IRRAT_1:funcnot 4 => IRRAT_1:func 3
definition
  let a1 be natural set;
  func cseq A1 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = (a1 choose b1) * (a1 to_power - b1);
end;

:: IRRAT_1:def 3
theorem
for b1 being natural set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b2 = cseq b1
   iff
      for b3 being Element of NAT holds
         b2 . b3 = (b1 choose b3) * (b1 to_power - b3);

:: IRRAT_1:th 3
theorem
for b1, b2 being Element of NAT holds
(cseq b1) . b2 = (bseq b2) . b1;

:: IRRAT_1:funcnot 5 => IRRAT_1:func 4
definition
  func dseq -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = (1 + (1 / b1)) to_power b1;
end;

:: IRRAT_1:def 4
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 = dseq
   iff
      for b2 being Element of NAT holds
         b1 . b2 = (1 + (1 / b2)) to_power b2;

:: IRRAT_1:funcnot 6 => IRRAT_1:func 5
definition
  func eseq -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = 1 / (b1 !);
end;

:: IRRAT_1:def 5
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 = eseq
   iff
      for b2 being Element of NAT holds
         b1 . b2 = 1 / (b2 !);

:: IRRAT_1:th 4
theorem
for b1, b2 being Element of NAT
      st 0 < b1
   holds b1 to_power - (b2 + 1) = (b1 to_power - b2) / b1;

:: IRRAT_1:th 6
theorem
for b1, b2 being Element of NAT holds
b1 choose (b2 + 1) = ((b1 - b2) / (b2 + 1)) * (b1 choose b2);

:: IRRAT_1:th 7
theorem
for b1, b2 being Element of NAT
      st 0 < b1
   holds (bseq (b2 + 1)) . b1 = ((1 / (b2 + 1)) * ((bseq b2) . b1)) * ((aseq b2) . b1);

:: IRRAT_1:th 8
theorem
for b1, b2 being Element of NAT
      st 0 < b1
   holds (aseq b2) . b1 = 1 - (b2 / b1);

:: IRRAT_1:th 9
theorem
for b1 being Element of NAT holds
   aseq b1 is convergent & lim aseq b1 = 1;

:: IRRAT_1:th 10
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           b2 . b3 = b1
   holds b2 is convergent & lim b2 = b1;

:: IRRAT_1:th 11
theorem
for b1 being Element of NAT holds
   (bseq 0) . b1 = 1;

:: IRRAT_1:th 12
theorem
for b1 being Element of NAT holds
   (1 / (b1 + 1)) * (1 / (b1 !)) = 1 / ((b1 + 1) !);

:: IRRAT_1:th 13
theorem
for b1 being Element of NAT holds
   bseq b1 is convergent & lim bseq b1 = 1 / (b1 !) & lim bseq b1 = eseq . b1;

:: IRRAT_1:th 14
theorem
for b1, b2 being Element of NAT
      st b1 < b2
   holds 0 < (aseq b1) . b2 & (aseq b1) . b2 <= 1;

:: IRRAT_1:th 15
theorem
for b1, b2 being Element of NAT
      st 0 < b1
   holds 0 <= (bseq b2) . b1 &
    (bseq b2) . b1 <= 1 / (b2 !) &
    (bseq b2) . b1 <= eseq . b2 &
    0 <= (cseq b1) . b2 &
    (cseq b1) . b2 <= 1 / (b2 !) &
    (cseq b1) . b2 <= eseq . b2;

:: IRRAT_1:th 16
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 ^\ 1 is summable
   holds b1 is summable &
    Sum b1 = (b1 . 0) + Sum (b1 ^\ 1);

:: IRRAT_1:th 17
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being FinSequence of b2
      st 1 <= b1 & b1 < len b3
   holds (b3 /^ 1) . b1 = b3 . (b1 + 1);

:: IRRAT_1:th 18
theorem
for b1 being FinSequence of REAL
      st 0 < len b1
   holds Sum b1 = (b1 . 1) + Sum (b1 /^ 1);

:: IRRAT_1:th 19
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being FinSequence of REAL
      st len b3 = b1 &
         (for b4 being Element of NAT
               st b4 < b1
            holds b2 . b4 = b3 . (b4 + 1)) &
         (for b4 being Element of NAT
               st b1 <= b4
            holds b2 . b4 = 0)
   holds b2 is summable & Sum b2 = Sum b3;

:: IRRAT_1:th 20
theorem
for b1, b2 being Element of NAT
for b3, b4 being real set
      st b3 <> 0 & b4 <> 0 & b1 <= b2
   holds ((b3,b4)In_Power b2) . (b1 + 1) = ((b2 choose b1) * (b3 to_power (b2 - b1))) * (b4 to_power b1);

:: IRRAT_1:th 21
theorem
for b1, b2 being Element of NAT
      st 0 < b1 & b2 <= b1
   holds (cseq b1) . b2 = ((1,1 / b1)In_Power b1) . (b2 + 1);

:: IRRAT_1:th 22
theorem
for b1 being Element of NAT
      st 0 < b1
   holds cseq b1 is summable &
    Sum cseq b1 = (1 + (1 / b1)) to_power b1 &
    Sum cseq b1 = dseq . b1;

:: IRRAT_1:th 23
theorem
dseq is convergent & lim dseq = number_e;

:: IRRAT_1:th 24
theorem
eseq is summable & Sum eseq = exp_R 1;

:: IRRAT_1:th 25
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           b2 . b3 = (Partial_Sums cseq b3) . b1
   holds b2 is convergent & lim b2 = (Partial_Sums eseq) . b1;

:: IRRAT_1:th 26
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
   st b2 is convergent & lim b2 = b1
for b3 being real set
      st 0 < b3
   holds ex b4 being Element of NAT st
      for b5 being Element of NAT
            st b4 <= b5
         holds b1 - b3 < b2 . b5;

:: IRRAT_1:th 27
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st (for b3 being real set
               st 0 < b3
            holds ex b4 being Element of NAT st
               for b5 being Element of NAT
                     st b4 <= b5
                  holds b1 - b3 < b2 . b5) &
         (ex b3 being Element of NAT st
            for b4 being Element of NAT
                  st b3 <= b4
               holds b2 . b4 <= b1)
   holds b2 is convergent & lim b2 = b1;

:: IRRAT_1:th 28
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is summable
for b2 being real set
      st 0 < b2
   holds ex b3 being Element of NAT st
      (Sum b1) - b2 < (Partial_Sums b1) . b3;

:: IRRAT_1:th 29
theorem
for b1 being Element of NAT
      st 1 <= b1
   holds dseq . b1 <= Sum eseq;

:: IRRAT_1:th 30
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is summable &
         (for b3 being Element of NAT holds
            0 <= b2 . b3)
   holds (Partial_Sums b2) . b1 <= Sum b2;

:: IRRAT_1:th 31
theorem
dseq is convergent & lim dseq = Sum eseq;

:: IRRAT_1:funcnot 7 => POWER:func 7
definition
  func number_e -> real set equals
    Sum eseq;
end;

:: IRRAT_1:def 6
theorem
number_e = Sum eseq;

:: IRRAT_1:funcnot 8 => POWER:func 7
definition
  func number_e -> real set equals
    exp_R 1;
end;

:: IRRAT_1:def 7
theorem
number_e = exp_R 1;

:: IRRAT_1:th 32
theorem
for b1 being real set
      st b1 is rational
   holds ex b2 being Element of NAT st
      2 <= b2 & b2 ! * b1 is integer;

:: IRRAT_1:th 33
theorem
for b1, b2 being Element of NAT holds
b1 ! * (eseq . b2) = b1 ! / (b2 !);

:: IRRAT_1:th 34
theorem
for b1, b2 being Element of NAT holds
0 < b1 ! / (b2 !);

:: IRRAT_1:th 35
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is summable &
         (for b2 being Element of NAT holds
            0 < b1 . b2)
   holds 0 < Sum b1;

:: IRRAT_1:th 36
theorem
for b1 being Element of NAT holds
   0 < b1 ! * Sum (eseq ^\ (b1 + 1));

:: IRRAT_1:th 37
theorem
for b1, b2 being Element of NAT
      st b1 <= b2
   holds b2 ! / (b1 !) is integer;

:: IRRAT_1:th 38
theorem
for b1 being Element of NAT holds
   b1 ! * ((Partial_Sums eseq) . b1) is integer;

:: IRRAT_1:th 39
theorem
for b1, b2 being Element of NAT
for b3 being real set
      st b3 = 1 / (b1 + 1)
   holds b1 ! / (((b1 + b2) + 1) !) <= b3 to_power (b2 + 1);

:: IRRAT_1:th 40
theorem
for b1 being Element of NAT
for b2 being real set
      st 0 < b1 & b2 = 1 / (b1 + 1)
   holds b1 ! * Sum (eseq ^\ (b1 + 1)) <= b2 / (1 - b2);

:: IRRAT_1:th 41
theorem
for b1, b2 being real set
      st 2 <= b2 & b1 = 1 / (b2 + 1)
   holds b1 / (1 - b1) < 1;

:: IRRAT_1:th 42
theorem
number_e is not rational;