Step
*
1
1
1
1
1
1
of Lemma
chain-rule_0
1. I : Interval
2. J : Interval
3. f : I ⟶ℝ
4. f' : I ⟶ℝ
5. g : J ⟶ℝ
6. g' : J ⟶ℝ
7. f[x] (proper)continuous for x ∈ I
8. f'[x] (proper)continuous for x ∈ I
9. g'[x] (proper)continuous for x ∈ J
10. d(f[x])/dx = λx.f'[x] on I
11. d(g[x])/dx = λx.g'[x] on J
12. k : ℕ+
13. n : {n:ℕ+| icompact(i-approx(I;n)) ∧ iproper(i-approx(I;n))}
14. m : {m:ℕ+| icompact(i-approx(J;m)) ∧ iproper(i-approx(J;m))}
15. ∀x:{x:ℝ| x ∈ i-approx(I;n)} . (f[x] ∈ i-approx(J;m))
16. mc1 : f'[x] continuous for x ∈ i-approx(I;n)
17. ∃a:ℝ. ∀[x:{r:ℝ| r ∈ i-approx(I;n)} ]. (|f'[x]| ≤ a)
18. mc2 : g'[x] continuous for x ∈ i-approx(J;m)
19. ∃b:ℝ. ∀[x:{r:ℝ| r ∈ i-approx(J;m)} ]. (|g'[x]| ≤ b)
20. f' ∈ i-approx(I;n) ⟶ℝ
21. A : ℕ+
22. ∀r:ℝ. ((r ∈ i-approx(I;n)) ⇒ (|f'[r]| ≤ (r(A)/r(2 * k))))
⊢ ∃del:ℝ [((r0 < del)
∧ (∀x,y:ℝ.
((x ∈ i-approx(I;n))
⇒ (y ∈ i-approx(I;n))
⇒ (|y - x| ≤ del)
⇒ (|g[f[y]] - g[f[x]] - (g'[f[x]] * f'[x]) * (y - x)| ≤ ((r1/r(k)) * |y - x|)))))]
BY
{ TACTIC:((((Assert g ∈ i-approx(J;m) ⟶ℝ BY Auto) THEN (Assert g' ∈ i-approx(J;m) ⟶ℝ BY Auto))
THEN (Assert f ∈ i-approx(I;n) ⟶ℝ BY
Auto)
)
THEN Assert ⌜∃d:ℝ
((r0 < d)
∧ (∀x,y:ℝ.
((x ∈ i-approx(I;n))
⇒ (y ∈ i-approx(I;n))
⇒ (|y - x| ≤ d)
⇒ (|g[f[y]] - g[f[x]] - g'[f[x]] * (f[y] - f[x])| ≤ ((r1/r(A)) * |f[y] - f[x]|)))))⌝⋅
) }
1
.....assertion.....
1. I : Interval
2. J : Interval
3. f : I ⟶ℝ
4. f' : I ⟶ℝ
5. g : J ⟶ℝ
6. g' : J ⟶ℝ
7. f[x] (proper)continuous for x ∈ I
8. f'[x] (proper)continuous for x ∈ I
9. g'[x] (proper)continuous for x ∈ J
10. d(f[x])/dx = λx.f'[x] on I
11. d(g[x])/dx = λx.g'[x] on J
12. k : ℕ+
13. n : {n:ℕ+| icompact(i-approx(I;n)) ∧ iproper(i-approx(I;n))}
14. m : {m:ℕ+| icompact(i-approx(J;m)) ∧ iproper(i-approx(J;m))}
15. ∀x:{x:ℝ| x ∈ i-approx(I;n)} . (f[x] ∈ i-approx(J;m))
16. mc1 : f'[x] continuous for x ∈ i-approx(I;n)
17. ∃a:ℝ. ∀[x:{r:ℝ| r ∈ i-approx(I;n)} ]. (|f'[x]| ≤ a)
18. mc2 : g'[x] continuous for x ∈ i-approx(J;m)
19. ∃b:ℝ. ∀[x:{r:ℝ| r ∈ i-approx(J;m)} ]. (|g'[x]| ≤ b)
20. f' ∈ i-approx(I;n) ⟶ℝ
21. A : ℕ+
22. ∀r:ℝ. ((r ∈ i-approx(I;n)) ⇒ (|f'[r]| ≤ (r(A)/r(2 * k))))
23. g ∈ i-approx(J;m) ⟶ℝ
24. g' ∈ i-approx(J;m) ⟶ℝ
25. f ∈ i-approx(I;n) ⟶ℝ
⊢ ∃d:ℝ
((r0 < d)
∧ (∀x,y:ℝ.
((x ∈ i-approx(I;n))
⇒ (y ∈ i-approx(I;n))
⇒ (|y - x| ≤ d)
⇒ (|g[f[y]] - g[f[x]] - g'[f[x]] * (f[y] - f[x])| ≤ ((r1/r(A)) * |f[y] - f[x]|)))))
2
1. I : Interval
2. J : Interval
3. f : I ⟶ℝ
4. f' : I ⟶ℝ
5. g : J ⟶ℝ
6. g' : J ⟶ℝ
7. f[x] (proper)continuous for x ∈ I
8. f'[x] (proper)continuous for x ∈ I
9. g'[x] (proper)continuous for x ∈ J
10. d(f[x])/dx = λx.f'[x] on I
11. d(g[x])/dx = λx.g'[x] on J
12. k : ℕ+
13. n : {n:ℕ+| icompact(i-approx(I;n)) ∧ iproper(i-approx(I;n))}
14. m : {m:ℕ+| icompact(i-approx(J;m)) ∧ iproper(i-approx(J;m))}
15. ∀x:{x:ℝ| x ∈ i-approx(I;n)} . (f[x] ∈ i-approx(J;m))
16. mc1 : f'[x] continuous for x ∈ i-approx(I;n)
17. ∃a:ℝ. ∀[x:{r:ℝ| r ∈ i-approx(I;n)} ]. (|f'[x]| ≤ a)
18. mc2 : g'[x] continuous for x ∈ i-approx(J;m)
19. ∃b:ℝ. ∀[x:{r:ℝ| r ∈ i-approx(J;m)} ]. (|g'[x]| ≤ b)
20. f' ∈ i-approx(I;n) ⟶ℝ
21. A : ℕ+
22. ∀r:ℝ. ((r ∈ i-approx(I;n)) ⇒ (|f'[r]| ≤ (r(A)/r(2 * k))))
23. g ∈ i-approx(J;m) ⟶ℝ
24. g' ∈ i-approx(J;m) ⟶ℝ
25. f ∈ i-approx(I;n) ⟶ℝ
26. ∃d:ℝ
((r0 < d)
∧ (∀x,y:ℝ.
((x ∈ i-approx(I;n))
⇒ (y ∈ i-approx(I;n))
⇒ (|y - x| ≤ d)
⇒ (|g[f[y]] - g[f[x]] - g'[f[x]] * (f[y] - f[x])| ≤ ((r1/r(A)) * |f[y] - f[x]|)))))
⊢ ∃del:ℝ [((r0 < del)
∧ (∀x,y:ℝ.
((x ∈ i-approx(I;n))
⇒ (y ∈ i-approx(I;n))
⇒ (|y - x| ≤ del)
⇒ (|g[f[y]] - g[f[x]] - (g'[f[x]] * f'[x]) * (y - x)| ≤ ((r1/r(k)) * |y - x|)))))]
Latex:
Latex:
1. I : Interval
2. J : Interval
3. f : I {}\mrightarrow{}\mBbbR{}
4. f' : I {}\mrightarrow{}\mBbbR{}
5. g : J {}\mrightarrow{}\mBbbR{}
6. g' : J {}\mrightarrow{}\mBbbR{}
7. f[x] (proper)continuous for x \mmember{} I
8. f'[x] (proper)continuous for x \mmember{} I
9. g'[x] (proper)continuous for x \mmember{} J
10. d(f[x])/dx = \mlambda{}x.f'[x] on I
11. d(g[x])/dx = \mlambda{}x.g'[x] on J
12. k : \mBbbN{}\msupplus{}
13. n : \{n:\mBbbN{}\msupplus{}| icompact(i-approx(I;n)) \mwedge{} iproper(i-approx(I;n))\}
14. m : \{m:\mBbbN{}\msupplus{}| icompact(i-approx(J;m)) \mwedge{} iproper(i-approx(J;m))\}
15. \mforall{}x:\{x:\mBbbR{}| x \mmember{} i-approx(I;n)\} . (f[x] \mmember{} i-approx(J;m))
16. mc1 : f'[x] continuous for x \mmember{} i-approx(I;n)
17. \mexists{}a:\mBbbR{}. \mforall{}[x:\{r:\mBbbR{}| r \mmember{} i-approx(I;n)\} ]. (|f'[x]| \mleq{} a)
18. mc2 : g'[x] continuous for x \mmember{} i-approx(J;m)
19. \mexists{}b:\mBbbR{}. \mforall{}[x:\{r:\mBbbR{}| r \mmember{} i-approx(J;m)\} ]. (|g'[x]| \mleq{} b)
20. f' \mmember{} i-approx(I;n) {}\mrightarrow{}\mBbbR{}
21. A : \mBbbN{}\msupplus{}
22. \mforall{}r:\mBbbR{}. ((r \mmember{} i-approx(I;n)) {}\mRightarrow{} (|f'[r]| \mleq{} (r(A)/r(2 * k))))
\mvdash{} \mexists{}del:\mBbbR{} [((r0 < del)
\mwedge{} (\mforall{}x,y:\mBbbR{}.
((x \mmember{} i-approx(I;n))
{}\mRightarrow{} (y \mmember{} i-approx(I;n))
{}\mRightarrow{} (|y - x| \mleq{} del)
{}\mRightarrow{} (|g[f[y]] - g[f[x]] - (g'[f[x]] * f'[x]) * (y - x)| \mleq{} ((r1/r(k)) * |y - x|)))))]
By
Latex:
TACTIC:((((Assert g \mmember{} i-approx(J;m) {}\mrightarrow{}\mBbbR{} BY Auto) THEN (Assert g' \mmember{} i-approx(J;m) {}\mrightarrow{}\mBbbR{} BY Auto))
THEN (Assert f \mmember{} i-approx(I;n) {}\mrightarrow{}\mBbbR{} BY
Auto)
)
THEN Assert \mkleeneopen{}\mexists{}d:\mBbbR{}
((r0 < d)
\mwedge{} (\mforall{}x,y:\mBbbR{}.
((x \mmember{} i-approx(I;n))
{}\mRightarrow{} (y \mmember{} i-approx(I;n))
{}\mRightarrow{} (|y - x| \mleq{} d)
{}\mRightarrow{} (|g[f[y]] - g[f[x]] - g'[f[x]] * (f[y] - f[x])| \mleq{} ((r1/r(A))
* |f[y] - f[x]|)))))\mkleeneclose{}\mcdot{}
)
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