let X, Y, Z be non empty MetrSpace; for x, y being Element of [: the carrier of X, the carrier of Y, the carrier of Z:] holds
( (dist_cart3S (X,Y,Z)) . (x,y) = 0 iff x = y )
let x, y be Element of [: the carrier of X, the carrier of Y, the carrier of Z:]; ( (dist_cart3S (X,Y,Z)) . (x,y) = 0 iff x = y )
reconsider x1 = x `1_3 , y1 = y `1_3 as Element of X ;
reconsider x2 = x `2_3 , y2 = y `2_3 as Element of Y ;
reconsider x3 = x `3_3 , y3 = y `3_3 as Element of Z ;
A1:
( x = [x1,x2,x3] & y = [y1,y2,y3] )
;
thus
( (dist_cart3S (X,Y,Z)) . (x,y) = 0 implies x = y )
( x = y implies (dist_cart3S (X,Y,Z)) . (x,y) = 0 )proof
set d3 =
dist (
x3,
y3);
set d2 =
dist (
x2,
y2);
set d1 =
dist (
x1,
y1);
A2:
(
0 <= (dist (x2,y2)) ^2 &
0 <= (dist (x3,y3)) ^2 )
by XREAL_1:63;
assume
(dist_cart3S (X,Y,Z)) . (
x,
y)
= 0
;
x = y
then
sqrt ((((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2)) + ((dist (x3,y3)) ^2)) = 0
by A1, Def13;
then A3:
sqrt (((dist (x1,y1)) ^2) + (((dist (x2,y2)) ^2) + ((dist (x3,y3)) ^2))) = 0
;
(
0 <= (dist (x2,y2)) ^2 &
0 <= (dist (x3,y3)) ^2 )
by XREAL_1:63;
then A4:
(
0 <= (dist (x1,y1)) ^2 &
0 + 0 <= ((dist (x2,y2)) ^2) + ((dist (x3,y3)) ^2) )
by XREAL_1:7, XREAL_1:63;
then
dist (
x1,
y1)
= 0
by A3, Lm1;
then A5:
x1 = y1
by METRIC_1:2;
A6:
((dist (x2,y2)) ^2) + ((dist (x3,y3)) ^2) = 0
by A3, A4, Lm1;
then
dist (
x2,
y2)
= 0
by A2, XREAL_1:27;
then A7:
x2 = y2
by METRIC_1:2;
dist (
x3,
y3)
= 0
by A6, A2, XREAL_1:27;
hence
x = y
by A1, A5, A7, METRIC_1:2;
verum
end;
assume A8:
x = y
; (dist_cart3S (X,Y,Z)) . (x,y) = 0
then A9:
( (dist (x1,y1)) ^2 = 0 ^2 & (dist (x2,y2)) ^2 = 0 ^2 )
by METRIC_1:1;
(dist_cart3S (X,Y,Z)) . (x,y) =
sqrt ((((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2)) + ((dist (x3,y3)) ^2))
by A1, Def13
.=
0 ^2
by A8, A9, METRIC_1:1, SQUARE_1:17
;
hence
(dist_cart3S (X,Y,Z)) . (x,y) = 0
; verum