set f = minnorm ;
A1: for a, b being Element of [.0,1.] holds minnorm . (a,b) = minnorm . (b,a)
proof
let a, b be Element of [.0,1.]; :: thesis: minnorm . (a,b) = minnorm . (b,a)
minnorm . (a,b) = min (a,b) by MinDef
.= minnorm . (b,a) by MinDef ;
hence minnorm . (a,b) = minnorm . (b,a) ; :: thesis: verum
end;
AA: for a, b, c being Element of [.0,1.] holds minnorm . ((minnorm . (a,b)),c) = minnorm . (a,(minnorm . (b,c)))
proof
let a, b, c be Element of [.0,1.]; :: thesis: minnorm . ((minnorm . (a,b)),c) = minnorm . (a,(minnorm . (b,c)))
a2: ( min (a,b) = a or min (a,b) = b ) by XXREAL_0:15;
a3: ( min (b,c) = b or min (b,c) = c ) by XXREAL_0:15;
minnorm . ((minnorm . (a,b)),c) = minnorm . ((min (a,b)),c) by MinDef
.= min ((min (a,b)),c) by
.= min (a,(min (b,c))) by XXREAL_0:33
.= minnorm . (a,(min (b,c))) by
.= minnorm . (a,(minnorm . (b,c))) by MinDef ;
hence minnorm . ((minnorm . (a,b)),c) = minnorm . (a,(minnorm . (b,c))) ; :: thesis: verum
end;
D1: for a, b, c, d being Element of [.0,1.] st a <= c & b <= d holds
minnorm . (a,b) <= minnorm . (c,d)
proof
let a, b, c, d be Element of [.0,1.]; :: thesis: ( a <= c & b <= d implies minnorm . (a,b) <= minnorm . (c,d) )
assume ( a <= c & b <= d ) ; :: thesis: minnorm . (a,b) <= minnorm . (c,d)
then min (a,b) <= min (c,d) by XXREAL_0:18;
then min (a,b) <= minnorm . (c,d) by MinDef;
hence minnorm . (a,b) <= minnorm . (c,d) by MinDef; :: thesis: verum
end;
for a being Element of [.0,1.] holds minnorm . (a,1) = a
proof
let a be Element of [.0,1.]; :: thesis: minnorm . (a,1) = a
T1: 1 in [.0,1.] by XXREAL_1:1;
a <= 1 by XXREAL_1:1;
then min (a,1) = a by XXREAL_0:def 9;
hence minnorm . (a,1) = a by ; :: thesis: verum
end;
hence ( minnorm is commutative & minnorm is associative & minnorm is monotonic & minnorm is with-1-identity ) by ; :: thesis: verum