Article HOLDER_1, MML version 4.99.1005

:: HOLDER_1:funcreg 1
registration
  let a1 be real set;
  cluster right_closed_halfline a1 -> non empty;
end;

:: HOLDER_1:th 1
theorem
for b1, b2 being Element of REAL
   st 0 < b1 & 0 < b2
for b3 being Element of REAL
      st 0 <= b3
   holds (b3 to_power b1) * (b3 to_power b2) = b3 to_power (b1 + b2);

:: HOLDER_1:th 2
theorem
for b1, b2 being Element of REAL
   st 0 < b1 & 0 < b2
for b3 being Element of REAL
      st 0 <= b3
   holds (b3 to_power b1) to_power b2 = b3 to_power (b1 * b2);

:: HOLDER_1:th 3
theorem
for b1 being Element of REAL
   st 0 < b1
for b2, b3 being Element of REAL
      st 0 <= b2 & b2 <= b3
   holds b2 to_power b1 <= b3 to_power b1;

:: HOLDER_1:th 4
theorem
for b1, b2, b3, b4 being Element of REAL
      st 1 < b1 & (1 / b1) + (1 / b2) = 1 & 0 < b3 & 0 < b4
   holds b3 * b4 <= ((b3 #R b1) / b1) + ((b4 #R b2) / b2) &
    (b3 * b4 = ((b3 #R b1) / b1) + ((b4 #R b2) / b2) implies b3 #R b1 = b4 #R b2) &
    (b3 #R b1 = b4 #R b2 implies b3 * b4 = ((b3 #R b1) / b1) + ((b4 #R b2) / b2));

:: HOLDER_1:th 5
theorem
for b1, b2, b3, b4 being Element of REAL
      st 1 < b1 & (1 / b1) + (1 / b2) = 1 & 0 <= b3 & 0 <= b4
   holds b3 * b4 <= ((b3 to_power b1) / b1) + ((b4 to_power b2) / b2) &
    (b3 * b4 = ((b3 to_power b1) / b1) + ((b4 to_power b2) / b2) implies b3 to_power b1 = b4 to_power b2) &
    (b3 to_power b1 = b4 to_power b2 implies b3 * b4 = ((b3 to_power b1) / b1) + ((b4 to_power b2) / b2));

:: HOLDER_1:th 6
theorem
for b1, b2 being Element of REAL
   st 1 < b1 & (1 / b1) + (1 / b2) = 1
for b3, b4, b5, b6, b7 being Function-like quasi_total Relation of NAT,REAL
   st for b8 being Element of NAT holds
        b5 . b8 = (abs (b3 . b8)) to_power b1 &
         b6 . b8 = (abs (b4 . b8)) to_power b2 &
         b7 . b8 = abs ((b3 . b8) * (b4 . b8))
for b8 being Element of NAT holds
   (Partial_Sums b7) . b8 <= (((Partial_Sums b5) . b8) to_power (1 / b1)) * (((Partial_Sums b6) . b8) to_power (1 / b2));

:: HOLDER_1:th 7
theorem
for b1 being Element of REAL
   st 1 < b1
for b2, b3, b4, b5, b6 being Function-like quasi_total Relation of NAT,REAL
   st for b7 being Element of NAT holds
        b4 . b7 = (abs (b2 . b7)) to_power b1 &
         b5 . b7 = (abs (b3 . b7)) to_power b1 &
         b6 . b7 = (abs ((b2 . b7) + (b3 . b7))) to_power b1
for b7 being Element of NAT holds
   ((Partial_Sums b6) . b7) to_power (1 / b1) <= (((Partial_Sums b4) . b7) to_power (1 / b1)) + (((Partial_Sums b5) . b7) to_power (1 / b1));

:: HOLDER_1:th 8
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st (for b3 being Element of NAT holds
            b1 . b3 <= b2 . b3) &
         b2 is convergent &
         b1 is non-decreasing
   holds b1 is convergent & lim b1 <= lim b2;

:: HOLDER_1:th 9
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st (for b4 being Element of NAT holds
            b1 . b4 <= (b2 . b4) + (b3 . b4)) &
         b2 is convergent &
         b3 is convergent &
         b1 is non-decreasing
   holds b1 is convergent & lim b1 <= (lim b2) + lim b3;

:: HOLDER_1:th 10
theorem
for b1 being Element of REAL
   st 0 < b1
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st b2 is convergent &
         (for b4 being Element of NAT holds
            0 <= b2 . b4) &
         (for b4 being Element of NAT holds
            b3 . b4 = (b2 . b4) to_power b1)
   holds b3 is convergent & lim b3 = (lim b2) to_power b1;

:: HOLDER_1:th 11
theorem
for b1 being Element of REAL
   st 0 < b1
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st b2 is summable &
         (for b4 being Element of NAT holds
            0 <= b2 . b4) &
         (for b4 being Element of NAT holds
            b3 . b4 = ((Partial_Sums b2) . b4) to_power b1)
   holds b3 is convergent &
    lim b3 = (Sum b2) to_power b1 &
    b3 is non-decreasing &
    (for b4 being Element of NAT holds
       b3 . b4 <= (Sum b2) to_power b1);

:: HOLDER_1:th 12
theorem
for b1, b2 being Element of REAL
   st 1 < b1 & (1 / b1) + (1 / b2) = 1
for b3, b4, b5, b6, b7 being Function-like quasi_total Relation of NAT,REAL
      st (for b8 being Element of NAT holds
            b5 . b8 = (abs (b3 . b8)) to_power b1 &
             b6 . b8 = (abs (b4 . b8)) to_power b2 &
             b7 . b8 = abs ((b3 . b8) * (b4 . b8))) &
         b5 is summable &
         b6 is summable
   holds b7 is summable &
    Sum b7 <= ((Sum b5) to_power (1 / b1)) * ((Sum b6) to_power (1 / b2));

:: HOLDER_1:th 13
theorem
for b1 being Element of REAL
   st 1 < b1
for b2, b3, b4, b5, b6 being Function-like quasi_total Relation of NAT,REAL
      st (for b7 being Element of NAT holds
            b4 . b7 = (abs (b2 . b7)) to_power b1 &
             b5 . b7 = (abs (b3 . b7)) to_power b1 &
             b6 . b7 = (abs ((b2 . b7) + (b3 . b7))) to_power b1) &
         b4 is summable &
         b5 is summable
   holds b6 is summable &
    (Sum b6) to_power (1 / b1) <= ((Sum b4) to_power (1 / b1)) + ((Sum b5) to_power (1 / b1));