:: The Inner Product of Finite Sequences and of Points of $n$-dimensional Topological Space
:: by Kanchun and Yatsuka Nakamura
::
:: Received February 3, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users

theorem Th1: :: EUCLID_2:1

for i, n being Nat
for v being Element of n -tuples_on REAL holds (mlt (v,(0* n))) . i = 0

proof end;

theorem Th2: :: EUCLID_2:2

for n being Nat
for v being Element of n -tuples_on REAL holds mlt (v,(0* n)) = 0* n

proof end;

theorem :: EUCLID_2:3

for n being Nat
for y1, y2 being real-valued FinSequence
for x1, x2 being Element of REAL n st x1 = y1 & x2 = y2 holds
|(y1,y2)| = (1 / 4) * ((|.(x1 + x2).| ^2) - (|.(x1 - x2).| ^2))

proof end;

Lm1: now :: thesis: for x being real-valued FinSequence holds x is FinSequence of REAL

let x be real-valued FinSequence; :: thesis: x is FinSequence of REAL
rng x c= REAL ;
hence x is FinSequence of REAL by FINSEQ_1:def 4; :: thesis: verum

end;

theorem Th4: :: EUCLID_2:4

for x being real-valued FinSequence holds |.x.| ^2 = |(x,x)|

proof end;

theorem Th5: :: EUCLID_2:5

for x being real-valued FinSequence holds |.x.| = sqrt |(x,x)|

proof end;

theorem :: EUCLID_2:6

canceled;

::$CT

theorem Th6: :: EUCLID_2:7

for x being real-valued FinSequence holds
( |(x,x)| = 0 iff x = 0* (len x) )

proof end;

theorem :: EUCLID_2:8

for x being real-valued FinSequence holds
( |(x,x)| = 0 iff |.x.| = 0 )

proof end;

theorem Th8: :: EUCLID_2:9

for x being real-valued FinSequence holds |(x,(0* (len x)))| = 0

proof end;

theorem :: EUCLID_2:10

for x being real-valued FinSequence holds |((0* (len x)),x)| = 0 by Th8;

theorem Th10: :: EUCLID_2:11

for x, y being real-valued FinSequence st len x = len y holds
|.(x + y).| ^2 = ((|.x.| ^2) + (2 * |(y,x)|)) + (|.y.| ^2)

proof end;

theorem Th11: :: EUCLID_2:12

for x, y being real-valued FinSequence st len x = len y holds
|.(x - y).| ^2 = ((|.x.| ^2) - (2 * |(y,x)|)) + (|.y.| ^2)

proof end;

theorem :: EUCLID_2:13

for x, y being real-valued FinSequence st len x = len y holds
(|.(x + y).| ^2) + (|.(x - y).| ^2) = 2 * ((|.x.| ^2) + (|.y.| ^2))

proof end;

theorem :: EUCLID_2:14

for x, y being real-valued FinSequence st len x = len y holds
(|.(x + y).| ^2) - (|.(x - y).| ^2) = 4 * |(x,y)|

proof end;

theorem Th14: :: EUCLID_2:15

for x, y being real-valued FinSequence st len x = len y holds
|.|(x,y)|.| <= |.x.| * |.y.|

proof end;

theorem Th15: :: EUCLID_2:16

for x, y being real-valued FinSequence st len x = len y holds
|.(x + y).| <= |.x.| + |.y.|

proof end;

theorem :: EUCLID_2:17

canceled;

::$CT

theorem Th16: :: EUCLID_2:18

for n being Nat
for p1, p2, p3 being Point of (TOP-REAL n) holds |((p1 + p2),p3)| = |(p1,p3)| + |(p2,p3)|

proof end;

theorem Th17: :: EUCLID_2:19

for n being Nat
for p1, p2 being Point of (TOP-REAL n)
for x being Real holds |((x * p1),p2)| = x * |(p1,p2)|

proof end;

theorem :: EUCLID_2:20

for n being Nat
for p1, p2 being Point of (TOP-REAL n)
for x being Real holds |(p1,(x * p2))| = x * |(p1,p2)| by Th17;

theorem Th19: :: EUCLID_2:21

for n being Nat
for p1, p2 being Point of (TOP-REAL n) holds |((- p1),p2)| = - |(p1,p2)|

proof end;

theorem :: EUCLID_2:22

for n being Nat
for p1, p2 being Point of (TOP-REAL n) holds |(p1,(- p2))| = - |(p1,p2)| by Th19;

theorem :: EUCLID_2:23

for n being Nat
for p1, p2 being Point of (TOP-REAL n) holds |((- p1),(- p2))| = |(p1,p2)|

proof end;

theorem Th22: :: EUCLID_2:24

for n being Nat
for p1, p2, p3 being Point of (TOP-REAL n) holds |((p1 - p2),p3)| = |(p1,p3)| - |(p2,p3)|

proof end;

theorem :: EUCLID_2:25

for n being Nat
for x, y being Real
for p1, p2, p3 being Point of (TOP-REAL n) holds |(((x * p1) + (y * p2)),p3)| = (x * |(p1,p3)|) + (y * |(p2,p3)|)

proof end;

theorem :: EUCLID_2:26

for n being Nat
for p, q1, q2 being Point of (TOP-REAL n) holds |(p,(q1 + q2))| = |(p,q1)| + |(p,q2)| by Th16;

theorem :: EUCLID_2:27

for n being Nat
for p, q1, q2 being Point of (TOP-REAL n) holds |(p,(q1 - q2))| = |(p,q1)| - |(p,q2)| by Th22;

theorem Th26: :: EUCLID_2:28

for n being Nat
for p1, p2, q1, q2 being Point of (TOP-REAL n) holds |((p1 + p2),(q1 + q2))| = ((|(p1,q1)| + |(p1,q2)|) + |(p2,q1)|) + |(p2,q2)|

proof end;

theorem Th27: :: EUCLID_2:29

for n being Nat
for p1, p2, q1, q2 being Point of (TOP-REAL n) holds |((p1 - p2),(q1 - q2))| = ((|(p1,q1)| - |(p1,q2)|) - |(p2,q1)|) + |(p2,q2)|

proof end;

theorem Th28: :: EUCLID_2:30

for n being Nat
for p, q being Point of (TOP-REAL n) holds |((p + q),(p + q))| = (|(p,p)| + (2 * |(p,q)|)) + |(q,q)|

proof end;

theorem Th29: :: EUCLID_2:31

for n being Nat
for p, q being Point of (TOP-REAL n) holds |((p - q),(p - q))| = (|(p,p)| - (2 * |(p,q)|)) + |(q,q)|

proof end;

theorem Th30: :: EUCLID_2:32

for n being Nat
for p being Point of (TOP-REAL n) holds |(p,(0. (TOP-REAL n)))| = 0

proof end;

theorem :: EUCLID_2:33

for n being Nat
for p being Point of (TOP-REAL n) holds |((0. (TOP-REAL n)),p)| = 0 by Th30;

theorem :: EUCLID_2:34

for n being Nat holds |((0. (TOP-REAL n)),(0. (TOP-REAL n)))| = 0 by Th30;

theorem Th33: :: EUCLID_2:35

for n being Nat
for p being Point of (TOP-REAL n) holds |(p,p)| >= 0 by RVSUM_1:119;

theorem Th34: :: EUCLID_2:36

for n being Nat
for p being Point of (TOP-REAL n) holds |(p,p)| = |.p.| ^2 by Th4;

theorem Th35: :: EUCLID_2:37

for n being Nat
for p being Point of (TOP-REAL n) holds |.p.| = sqrt |(p,p)|

proof end;

theorem Th36: :: EUCLID_2:38

for n being Nat
for p being Point of (TOP-REAL n) holds 0 <= |.p.|

proof end;

theorem Th37: :: EUCLID_2:39

for n being Nat holds |.(0. (TOP-REAL n)).| = 0 by TOPRNS_1:23;

theorem Th38: :: EUCLID_2:40

for n being Nat
for p being Point of (TOP-REAL n) holds
( |(p,p)| = 0 iff |.p.| = 0 )

proof end;

theorem Th39: :: EUCLID_2:41

for n being Nat
for p being Point of (TOP-REAL n) holds
( |(p,p)| = 0 iff p = 0. (TOP-REAL n) )

proof end;

theorem :: EUCLID_2:42

for n being Nat
for p being Point of (TOP-REAL n) holds
( |.p.| = 0 iff p = 0. (TOP-REAL n) ) by Th37, TOPRNS_1:24;

theorem :: EUCLID_2:43

for n being Nat
for p being Point of (TOP-REAL n) holds
( p <> 0. (TOP-REAL n) iff |(p,p)| > 0 )

proof end;

theorem :: EUCLID_2:44

for n being Nat
for p being Point of (TOP-REAL n) holds
( p <> 0. (TOP-REAL n) iff |.p.| > 0 )

proof end;

theorem Th43: :: EUCLID_2:45

for n being Nat
for p, q being Point of (TOP-REAL n) holds |.(p + q).| ^2 = ((|.p.| ^2) + (2 * |(q,p)|)) + (|.q.| ^2)

proof end;

theorem Th44: :: EUCLID_2:46

for n being Nat
for p, q being Point of (TOP-REAL n) holds |.(p - q).| ^2 = ((|.p.| ^2) - (2 * |(q,p)|)) + (|.q.| ^2)

proof end;

theorem :: EUCLID_2:47

for n being Nat
for p, q being Point of (TOP-REAL n) holds (|.(p + q).| ^2) + (|.(p - q).| ^2) = 2 * ((|.p.| ^2) + (|.q.| ^2))

proof end;

theorem :: EUCLID_2:48

for n being Nat
for p, q being Point of (TOP-REAL n) holds (|.(p + q).| ^2) - (|.(p - q).| ^2) = 4 * |(p,q)|

proof end;

theorem :: EUCLID_2:49

for n being Nat
for p, q being Point of (TOP-REAL n) holds |(p,q)| = (1 / 4) * ((|.(p + q).| ^2) - (|.(p - q).| ^2))

proof end;

theorem :: EUCLID_2:50

for n being Nat
for p, q being Point of (TOP-REAL n) holds |(p,q)| <= |(p,p)| + |(q,q)|

proof end;

theorem :: EUCLID_2:51

for n being Nat
for p, q being Point of (TOP-REAL n) holds |.|(p,q)|.| <= |.p.| * |.q.|

proof end;

theorem :: EUCLID_2:52

for n being Nat
for p, q being Point of (TOP-REAL n) holds |.(p + q).| <= |.p.| + |.q.|

proof end;

theorem Th51: :: EUCLID_2:53

for n being Nat
for p being Point of (TOP-REAL n) holds p, 0. (TOP-REAL n) are_orthogonal by Th30;

theorem :: EUCLID_2:54

for n being Nat
for p being Point of (TOP-REAL n) holds 0. (TOP-REAL n),p are_orthogonal by Th51;

theorem Th53: :: EUCLID_2:55

for n being Nat
for p being Point of (TOP-REAL n) holds
( p,p are_orthogonal iff p = 0. (TOP-REAL n) ) by Th39;

theorem Th54: :: EUCLID_2:56

for n being Nat
for a being Real
for p, q being Point of (TOP-REAL n) st p,q are_orthogonal holds
a * p,q are_orthogonal

proof end;

theorem :: EUCLID_2:57

for n being Nat
for a being Real
for p, q being Point of (TOP-REAL n) st p,q are_orthogonal holds
p,a * q are_orthogonal by Th54;

theorem :: EUCLID_2:58

for n being Nat
for p being Point of (TOP-REAL n) st ( for q being Point of (TOP-REAL n) holds p,q are_orthogonal ) holds
p = 0. (TOP-REAL n) by Th53;