:: Lines in $n$-Dimensional Euclidean Spaces
:: by Akihiro Kubo
::
:: Received August 8, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users

theorem Th1: :: EUCLID_4:1

for n being Nat
for x being Element of REAL n holds
( (0 * x) + x = x & x + (0* n) = x )

proof end;

theorem Th2: :: EUCLID_4:2

for a being Real
for n being Nat holds a * (0* n) = 0* n

proof end;

theorem Th3: :: EUCLID_4:3

for n being Nat
for x being Element of REAL n holds
( 1 * x = x & 0 * x = 0* n )

proof end;

theorem :: EUCLID_4:4

for a, b being Real
for n being Nat
for x being Element of REAL n holds (a * b) * x = a * (b * x) by RVSUM_1:49;

theorem :: EUCLID_4:5

for a being Real
for n being Nat
for x being Element of REAL n holds
( not a * x = 0* n or a = 0 or x = 0* n )

proof end;

theorem :: EUCLID_4:6

for a being Real
for n being Nat
for x1, x2 being Element of REAL n holds a * (x1 + x2) = (a * x1) + (a * x2) by RVSUM_1:51;

theorem :: EUCLID_4:7

for a, b being Real
for n being Nat
for x being Element of REAL n holds (a + b) * x = (a * x) + (b * x) by RVSUM_1:50;

theorem :: EUCLID_4:8

for a being Real
for n being Nat
for x1, x2 being Element of REAL n holds
( not a * x1 = a * x2 or a = 0 or x1 = x2 )

proof end;

definition

let n be Nat;
let x1, x2 be Element of REAL n;

func Line (x1,x2) -> Subset of (REAL n) equals :: EUCLID_4:def 1
{ (((1 - lambda) * x1) + (lambda * x2)) where lambda is Real : verum } ;

coherence
{ (((1 - lambda) * x1) + (lambda * x2)) where lambda is Real : verum } is Subset of (REAL n)

proof end;

end;

:: deftheorem defines Line EUCLID_4:def 1 :
for n being Nat
for x1, x2 being Element of REAL n holds Line (x1,x2) = { (((1 - lambda) * x1) + (lambda * x2)) where lambda is Real : verum } ;

registration

let n be Nat;
let x1, x2 be Element of REAL n;

cluster Line (x1,x2) -> non empty ;

coherence
not Line (x1,x2) is empty

proof end;

end;

Lm1: for n being Nat
for x1, x2 being Element of REAL n holds Line (x1,x2) c= Line (x2,x1)

proof end;

definition

let n be Nat;
let x1, x2 be Element of REAL n;
:: original: Line
redefine func Line (x1,x2) -> Subset of (REAL n);
commutativity
for x1, x2 being Element of REAL n holds Line (x1,x2) = Line (x2,x1) by Lm1;

end;

theorem Th9: :: EUCLID_4:9

for n being Nat
for x1, x2 being Element of REAL n holds
( x1 in Line (x1,x2) & x2 in Line (x1,x2) )

proof end;

Lm2: for n being Nat
for x1, x2, x3 being Element of REAL n holds x1 + (x2 + x3) = (x1 + x2) + x3
by FINSEQOP:28;

theorem Th10: :: EUCLID_4:10

for n being Nat
for x1, x2, y1, y2 being Element of REAL n st y1 in Line (x1,x2) & y2 in Line (x1,x2) holds
Line (y1,y2) c= Line (x1,x2)

proof end;

theorem Th11: :: EUCLID_4:11

for n being Nat
for x1, x2, y1, y2 being Element of REAL n st y1 in Line (x1,x2) & y2 in Line (x1,x2) & y1 <> y2 holds
Line (x1,x2) c= Line (y1,y2)

proof end;

definition

let n be Nat;
let A be Subset of (REAL n);

attr A is being_line means :: EUCLID_4:def 2
ex x1, x2 being Element of REAL n st
( x1 <> x2 & A = Line (x1,x2) );

end;

:: deftheorem defines being_line EUCLID_4:def 2 :
for n being Nat
for A being Subset of (REAL n) holds
( A is being_line iff ex x1, x2 being Element of REAL n st
( x1 <> x2 & A = Line (x1,x2) ) );

Lm3: for n being Nat
for A being Subset of (REAL n)
for x1, x2 being Element of REAL n st A is being_line & x1 in A & x2 in A & x1 <> x2 holds
A = Line (x1,x2)
by Th10, Th11;

theorem :: EUCLID_4:12

for n being Nat
for A, C being Subset of (REAL n)
for x1, x2 being Element of REAL n st A is being_line & C is being_line & x1 in A & x2 in A & x1 in C & x2 in C & not x1 = x2 holds
A = C

proof end;

theorem Th13: :: EUCLID_4:13

for n being Nat
for A being Subset of (REAL n) st A is being_line holds
ex x1, x2 being Element of REAL n st
( x1 in A & x2 in A & x1 <> x2 )

proof end;

theorem :: EUCLID_4:14

for n being Nat
for x1 being Element of REAL n
for A being Subset of (REAL n) st A is being_line holds
ex x2 being Element of REAL n st
( x1 <> x2 & x2 in A )

proof end;

theorem :: EUCLID_4:15

for n being Nat
for x being Element of REAL n holds |.x.| = sqrt |(x,x)| ;

theorem Th16: :: EUCLID_4:16

for n being Nat
for x being Element of REAL n holds
( |(x,x)| = 0 iff |.x.| = 0 )

proof end;

theorem Th17: :: EUCLID_4:17

for n being Nat
for x being Element of REAL n holds
( |(x,x)| = 0 iff x = 0* n )

proof end;

theorem Th18: :: EUCLID_4:18

for n being Nat
for x being Element of REAL n holds |(x,(0* n))| = 0

proof end;

theorem :: EUCLID_4:19

for n being Nat
for x being Element of REAL n holds |((0* n),x)| = 0 by Th18;

theorem Th20: :: EUCLID_4:20

for n being Nat
for x1, x2, x3 being Element of REAL n holds |((x1 + x2),x3)| = |(x1,x3)| + |(x2,x3)|

proof end;

theorem Th21: :: EUCLID_4:21

for n being Nat
for x1, x2 being Element of REAL n
for a being Real holds |((a * x1),x2)| = a * |(x1,x2)|

proof end;

theorem :: EUCLID_4:22

for n being Nat
for x1, x2 being Element of REAL n
for a being Real holds |(x1,(a * x2))| = a * |(x1,x2)| by Th21;

theorem Th23: :: EUCLID_4:23

for n being Nat
for x1, x2 being Element of REAL n holds |((- x1),x2)| = - |(x1,x2)|

proof end;

theorem :: EUCLID_4:24

for n being Nat
for x1, x2 being Element of REAL n holds |(x1,(- x2))| = - |(x1,x2)| by Th23;

theorem :: EUCLID_4:25

for n being Nat
for x1, x2 being Element of REAL n holds |((- x1),(- x2))| = |(x1,x2)|

proof end;

theorem Th26: :: EUCLID_4:26

for n being Nat
for x1, x2, x3 being Element of REAL n holds |((x1 - x2),x3)| = |(x1,x3)| - |(x2,x3)|

proof end;

theorem :: EUCLID_4:27

for n being Nat
for a, b being Real
for x1, x2, x3 being Element of REAL n holds |(((a * x1) + (b * x2)),x3)| = (a * |(x1,x3)|) + (b * |(x2,x3)|)

proof end;

theorem :: EUCLID_4:28

for n being Nat
for x1, y1, y2 being Element of REAL n holds |(x1,(y1 + y2))| = |(x1,y1)| + |(x1,y2)| by Th20;

theorem :: EUCLID_4:29

for n being Nat
for x1, y1, y2 being Element of REAL n holds |(x1,(y1 - y2))| = |(x1,y1)| - |(x1,y2)| by Th26;

theorem Th30: :: EUCLID_4:30

for n being Nat
for x1, x2, y1, y2 being Element of REAL n holds |((x1 + x2),(y1 + y2))| = ((|(x1,y1)| + |(x1,y2)|) + |(x2,y1)|) + |(x2,y2)|

proof end;

theorem Th31: :: EUCLID_4:31

for n being Nat
for x1, x2, y1, y2 being Element of REAL n holds |((x1 - x2),(y1 - y2))| = ((|(x1,y1)| - |(x1,y2)|) - |(x2,y1)|) + |(x2,y2)|

proof end;

theorem Th32: :: EUCLID_4:32

for n being Nat
for x, y being Element of REAL n holds |((x + y),(x + y))| = (|(x,x)| + (2 * |(x,y)|)) + |(y,y)|

proof end;

theorem Th33: :: EUCLID_4:33

for n being Nat
for x, y being Element of REAL n holds |((x - y),(x - y))| = (|(x,x)| - (2 * |(x,y)|)) + |(y,y)|

proof end;

theorem :: EUCLID_4:34

for n being Nat
for x, y being Element of REAL n holds |.(x + y).| ^2 = ((|.x.| ^2) + (2 * |(x,y)|)) + (|.y.| ^2)

proof end;

theorem :: EUCLID_4:35

for n being Nat
for x, y being Element of REAL n holds |.(x - y).| ^2 = ((|.x.| ^2) - (2 * |(x,y)|)) + (|.y.| ^2)

proof end;

theorem :: EUCLID_4:36

for n being Nat
for x, y being Element of REAL n holds (|.(x + y).| ^2) + (|.(x - y).| ^2) = 2 * ((|.x.| ^2) + (|.y.| ^2))

proof end;

theorem :: EUCLID_4:37

for n being Nat
for x, y being Element of REAL n holds (|.(x + y).| ^2) - (|.(x - y).| ^2) = 4 * |(x,y)|

proof end;

theorem :: EUCLID_4:38

for n being Nat
for x, y being Element of REAL n holds |.|(x,y)|.| <= |.x.| * |.y.|

proof end;

Lm4: for a being Real
for n being Nat
for x being Element of REAL n holds
( - (a * x) = (- a) * x & - (a * x) = a * (- x) )

proof end;

Lm5: for n being Nat
for x1, x2 being Element of REAL n st x1 <> x2 holds
|.(x1 - x2).| <> 0

proof end;

Lm6: for n being Nat
for y1, y2, x1, x2 being Element of REAL n st y1 in Line (x1,x2) & y2 in Line (x1,x2) holds
ex a being Real st y1 - y2 = a * (x1 - x2)

proof end;

Lm7: for n being Nat
for x1, x2, y1 being Element of REAL n ex y2 being Element of REAL n st
( y2 in Line (x1,x2) & x1 - x2,y1 - y2 are_orthogonal & ( for x being Element of REAL n st x in Line (x1,x2) holds
|.(y1 - y2).| <= |.(y1 - x).| ) )

proof end;

theorem :: EUCLID_4:39

canceled;

::$CT

theorem :: EUCLID_4:40

for n being Nat
for R being Subset of REAL
for x1, x2, y1 being Element of REAL n st R = { |.(y1 - x).| where x is Element of REAL n : x in Line (x1,x2) } holds
ex y2 being Element of REAL n st
( y2 in Line (x1,x2) & |.(y1 - y2).| = lower_bound R & x1 - x2,y1 - y2 are_orthogonal )

proof end;

definition

end;

:: deftheorem EUCLID_4:def 3 :
canceled;

Lm8: for n being Nat
for p1, p2 being Point of (TOP-REAL n) ex x1, x2 being Element of REAL n st
( p1 = x1 & p2 = x2 & Line (x1,x2) = Line (p1,p2) )

proof end;

theorem :: EUCLID_4:41

for n being Nat
for p1, p2 being Point of (TOP-REAL n) holds
( p1 in Line (p1,p2) & p2 in Line (p1,p2) ) by RLTOPSP1:72;

theorem :: EUCLID_4:42

for n being Nat
for p1, p2, q1, q2 being Point of (TOP-REAL n) st q1 in Line (p1,p2) & q2 in Line (p1,p2) holds
Line (q1,q2) c= Line (p1,p2) by RLTOPSP1:74;

theorem :: EUCLID_4:43

for n being Nat
for p1, p2, q1, q2 being Point of (TOP-REAL n) st q1 in Line (p1,p2) & q2 in Line (p1,p2) & q1 <> q2 holds
Line (p1,p2) c= Line (q1,q2) by RLTOPSP1:75;

definition

let n be Nat;
let A be Subset of (TOP-REAL n);

attr A is being_line means :: EUCLID_4:def 4
ex p1, p2 being Point of (TOP-REAL n) st
( p1 <> p2 & A = Line (p1,p2) );

end;

:: deftheorem defines being_line EUCLID_4:def 4 :
for n being Nat
for A being Subset of (TOP-REAL n) holds
( A is being_line iff ex p1, p2 being Point of (TOP-REAL n) st
( p1 <> p2 & A = Line (p1,p2) ) );

Lm9: for n being Nat
for A being Subset of (TOP-REAL n)
for p1, p2 being Point of (TOP-REAL n) st A is being_line & p1 in A & p2 in A & p1 <> p2 holds
A = Line (p1,p2)
by RLTOPSP1:75;

theorem :: EUCLID_4:44

for n being Nat
for p1, p2 being Point of (TOP-REAL n)
for A, C being Subset of (TOP-REAL n) st A is being_line & C is being_line & p1 in A & p2 in A & p1 in C & p2 in C & not p1 = p2 holds
A = C

proof end;

theorem Th44: :: EUCLID_4:45

for n being Nat
for A being Subset of (TOP-REAL n) st A is being_line holds
ex p1, p2 being Point of (TOP-REAL n) st
( p1 in A & p2 in A & p1 <> p2 )

proof end;

theorem :: EUCLID_4:46

for n being Nat
for p1 being Point of (TOP-REAL n)
for A being Subset of (TOP-REAL n) st A is being_line holds
ex p2 being Point of (TOP-REAL n) st
( p1 <> p2 & p2 in A )

proof end;

theorem :: EUCLID_4:47

for n being Nat
for R being Subset of REAL
for p1, p2, q1 being Point of (TOP-REAL n) st R = { |.(q1 - p).| where p is Point of (TOP-REAL n) : p in Line (p1,p2) } holds
ex q2 being Point of (TOP-REAL n) st
( q2 in Line (p1,p2) & |.(q1 - q2).| = lower_bound R & p1 - p2,q1 - q2 are_orthogonal )

proof end;