:: H\"older's Inequality and {M}inkowski's Inequality
:: by Yasumasa Suzuki
::
:: Copyright (c) 2004-2021 Association of Mizar Users

registration
let x be Real;
coherence
proof end;
end;

theorem :: HOLDER_1:1
for p, q being Real st 0 < p & 0 < q holds
for a being Real st 0 <= a holds
(a to_power p) * (a to_power q) = a to_power (p + q)
proof end;

theorem :: HOLDER_1:2
for p, q being Real st 0 < p & 0 < q holds
for a being Real st 0 <= a holds
(a to_power p) to_power q = a to_power (p * q)
proof end;

theorem Th3: :: HOLDER_1:3
for p being Real st 0 <= p holds
for a, b being Real st 0 <= a & a <= b holds
a to_power p <= b to_power p
proof end;

theorem Th4: :: HOLDER_1:4
for a, b, p, q being Real st 1 < p & (1 / p) + (1 / q) = 1 & 0 < a & 0 < b holds
( a * b <= ((a #R p) / p) + ((b #R q) / q) & ( a * b = ((a #R p) / p) + ((b #R q) / q) implies a #R p = b #R q ) & ( a #R p = b #R q implies a * b = ((a #R p) / p) + ((b #R q) / q) ) )
proof end;

theorem Th5: :: HOLDER_1:5
for a, b, p, q being Real st 1 < p & (1 / p) + (1 / q) = 1 & 0 <= a & 0 <= b holds
( a * b <= ((a to_power p) / p) + ((b to_power q) / q) & ( a * b = ((a to_power p) / p) + ((b to_power q) / q) implies a to_power p = b to_power q ) & ( a to_power p = b to_power q implies a * b = ((a to_power p) / p) + ((b to_power q) / q) ) )
proof end;

Lm1: for a being Real_Sequence st ( for n being Nat holds 0 <= a . n ) holds
for n being Nat holds a . n <= () . n

proof end;

Lm2: for a being Real_Sequence st ( for n being Nat holds 0 <= a . n ) holds
for n being Nat holds 0 <= () . n

proof end;

Lm3: for a being Real_Sequence st ( for n being Nat holds 0 <= a . n ) holds
for n being Nat st () . n = 0 holds
for k being Nat st k <= n holds
a . k = 0

proof end;

Lm4: for a being Real_Sequence
for n being Nat st ( for k being Nat st k <= n holds
a . k = 0 ) holds
() . n = 0

proof end;

theorem Th6: :: HOLDER_1:6
for p, q being Real st 1 < p & (1 / p) + (1 / q) = 1 holds
for a, b, ap, bq, ab being Real_Sequence st ( for n being Nat holds
( ap . n = |.(a . n).| to_power p & bq . n = |.(b . n).| to_power q & ab . n = |.((a . n) * (b . n)).| ) ) holds
for n being Nat holds () . n <= ((() . n) to_power (1 / p)) * ((() . n) to_power (1 / q))
proof end;

theorem Th7: :: HOLDER_1:7
for p being Real st 1 < p holds
for a, b, ap, bp, ab being Real_Sequence st ( for n being Nat holds
( ap . n = |.(a . n).| to_power p & bp . n = |.(b . n).| to_power p & ab . n = |.((a . n) + (b . n)).| to_power p ) ) holds
for n being Nat holds (() . n) to_power (1 / p) <= ((() . n) to_power (1 / p)) + ((() . n) to_power (1 / p))
proof end;

theorem Th8: :: HOLDER_1:8
for a, b being Real_Sequence st ( for n being Nat holds a . n <= b . n ) & b is convergent & a is V48() holds
( a is convergent & lim a <= lim b )
proof end;

theorem Th9: :: HOLDER_1:9
for a, b, c being Real_Sequence st ( for n being Nat holds a . n <= (b . n) + (c . n) ) & b is convergent & c is convergent & a is V48() holds
( a is convergent & lim a <= (lim b) + (lim c) )
proof end;

theorem Th10: :: HOLDER_1:10
for p being Real st 0 < p holds
for a, ap being Real_Sequence st a is convergent & ( for n being Nat holds 0 <= a . n ) & ( for n being Nat holds ap . n = (a . n) to_power p ) holds
( ap is convergent & lim ap = (lim a) to_power p )
proof end;

theorem :: HOLDER_1:11
for p being Real st 0 < p holds
for a, ap being Real_Sequence st a is summable & ( for n being Nat holds 0 <= a . n ) & ( for n being Nat holds ap . n = (() . n) to_power p ) holds
( ap is convergent & lim ap = (Sum a) to_power p & ap is V48() & ( for n being Nat holds ap . n <= (Sum a) to_power p ) )
proof end;

theorem :: HOLDER_1:12
for p, q being Real st 1 < p & (1 / p) + (1 / q) = 1 holds
for a, b, ap, bq, ab being Real_Sequence st ( for n being Nat holds
( ap . n = |.(a . n).| to_power p & bq . n = |.(b . n).| to_power q & ab . n = |.((a . n) * (b . n)).| ) ) & ap is summable & bq is summable holds
( ab is summable & Sum ab <= ((Sum ap) to_power (1 / p)) * ((Sum bq) to_power (1 / q)) )
proof end;

theorem :: HOLDER_1:13
for p being Real st 1 < p holds
for a, b, ap, bp, ab being Real_Sequence st ( for n being Nat holds
( ap . n = |.(a . n).| to_power p & bp . n = |.(b . n).| to_power p & ab . n = |.((a . n) + (b . n)).| to_power p ) ) & ap is summable & bp is summable holds
( ab is summable & (Sum ab) to_power (1 / p) <= ((Sum ap) to_power (1 / p)) + ((Sum bp) to_power (1 / p)) )
proof end;