let K be Field; :: thesis: for V being VectSp of K
for v being Vector of V
for X being Subspace of V
for y being Vector of (X + (Lin {v}))
for W being Subspace of X + (Lin {v}) st v = y & X = W & not v in X holds
for w being Vector of (X + (Lin {v}))
for x being Vector of X
for t, r being Element of K st w |-- (W,(Lin {y})) = [x,(r * v)] holds
(t * w) |-- (W,(Lin {y})) = [(t * x),((t * r) * v)]
let V be VectSp of K; :: thesis: for v being Vector of V
for X being Subspace of V
for y being Vector of (X + (Lin {v}))
for W being Subspace of X + (Lin {v}) st v = y & X = W & not v in X holds
for w being Vector of (X + (Lin {v}))
for x being Vector of X
for t, r being Element of K st w |-- (W,(Lin {y})) = [x,(r * v)] holds
(t * w) |-- (W,(Lin {y})) = [(t * x),((t * r) * v)]
let v be Vector of V; :: thesis: for X being Subspace of V
for y being Vector of (X + (Lin {v}))
for W being Subspace of X + (Lin {v}) st v = y & X = W & not v in X holds
for w being Vector of (X + (Lin {v}))
for x being Vector of X
for t, r being Element of K st w |-- (W,(Lin {y})) = [x,(r * v)] holds
(t * w) |-- (W,(Lin {y})) = [(t * x),((t * r) * v)]
let X be Subspace of V; :: thesis: for y being Vector of (X + (Lin {v}))
for W being Subspace of X + (Lin {v}) st v = y & X = W & not v in X holds
for w being Vector of (X + (Lin {v}))
for x being Vector of X
for t, r being Element of K st w |-- (W,(Lin {y})) = [x,(r * v)] holds
(t * w) |-- (W,(Lin {y})) = [(t * x),((t * r) * v)]
let y be Vector of (X + (Lin {v})); :: thesis: for W being Subspace of X + (Lin {v}) st v = y & X = W & not v in X holds
for w being Vector of (X + (Lin {v}))
for x being Vector of X
for t, r being Element of K st w |-- (W,(Lin {y})) = [x,(r * v)] holds
(t * w) |-- (W,(Lin {y})) = [(t * x),((t * r) * v)]
let W be Subspace of X + (Lin {v}); :: thesis: ( v = y & X = W & not v in X implies for w being Vector of (X + (Lin {v}))
for x being Vector of X
for t, r being Element of K st w |-- (W,(Lin {y})) = [x,(r * v)] holds
(t * w) |-- (W,(Lin {y})) = [(t * x),((t * r) * v)] )
assume that
A1: v = y and
A2: X = W and
A3: not v in X ; :: thesis: for w being Vector of (X + (Lin {v}))
for x being Vector of X
for t, r being Element of K st w |-- (W,(Lin {y})) = [x,(r * v)] holds
(t * w) |-- (W,(Lin {y})) = [(t * x),((t * r) * v)]
A4: X + (Lin {v}) is_the_direct_sum_of W, Lin {y} by A1, A2, A3, Th14;
let w be Vector of (X + (Lin {v})); :: thesis: for x being Vector of X
for t, r being Element of K st w |-- (W,(Lin {y})) = [x,(r * v)] holds
(t * w) |-- (W,(Lin {y})) = [(t * x),((t * r) * v)]
let x be Vector of X; :: thesis: for t, r being Element of K st w |-- (W,(Lin {y})) = [x,(r * v)] holds
(t * w) |-- (W,(Lin {y})) = [(t * x),((t * r) * v)]
let t, r be Element of K; :: thesis: ( w |-- (W,(Lin {y})) = [x,(r * v)] implies (t * w) |-- (W,(Lin {y})) = [(t * x),((t * r) * v)] )
assume A5: w |-- (W,(Lin {y})) = [x,(r * v)] ; :: thesis: (t * w) |-- (W,(Lin {y})) = [(t * x),((t * r) * v)]
reconsider z = x as Vector of (X + (Lin {v})) by A2, VECTSP_4:10;
r * y = r * v by A1, VECTSP_4:14;
then A6: t * w = t * (z + (r * y)) by A4, A5, Th6
.= (t * z) + (t * (r * y)) by VECTSP_1:def 14
.= (t * z) + ((t * r) * y) by VECTSP_1:def 16 ;
reconsider u = x as Vector of V by VECTSP_4:10;
A7: (t * r) * y in Lin {y} by Th3;
A8: (t * r) * y = (t * r) * v by A1, VECTSP_4:14;
A9: t * z = t * u by VECTSP_4:14
.= t * x by VECTSP_4:14 ;
then t * z in W by A2;
hence (t * w) |-- (W,(Lin {y})) = [(t * x),((t * r) * v)] by A4, A9, A8, A7, A6, Th5; :: thesis: verum