Step
*
1
1
of Lemma
derivative-rinv
.....assertion.....
1. I : Interval
2. f : I ⟶ℝ
3. g : I ⟶ℝ
4. ∀x,y:{t:ℝ| t ∈ I} . ((x = y)
⇒ (g[x] = g[y]))
5. d(f[x])/dx = λx.g[x] on I
6. k : ℕ+
7. n : {n:ℕ+| icompact(i-approx(I;n)) ∧ iproper(i-approx(I;n))}
8. ∀a,b:{x:ℝ| x ∈ i-approx(I;n)} . ((a = b)
⇒ (f[a] = f[b]))
9. i-approx(I;n) ⊆ I
10. ∀x:ℝ. ((x ∈ I)
⇒ f[x] ≠ r0)
11. ∃c:ℝ [((r0 < c) ∧ (∀x:ℝ. ((x ∈ i-approx(I;n))
⇒ (c ≤ |f[x]|))))]
12. (r1/f[x]) continuous for x ∈ i-approx(I;n)
13. g[x] continuous for x ∈ i-approx(I;n)
14. ∀x:ℝ. ((x ∈ i-approx(I;n))
⇒ f[x] ≠ r0)
⊢ ∃M:ℕ+. ∀x:ℝ. ((x ∈ i-approx(I;n))
⇒ ((|g[x]| ≤ r(M)) ∧ (|(r1/f[x])| ≤ r(M))))
BY
{ (RenameVar `mcg' (-2)⋅
THEN (Assert ∃a:ℝ. ∀[x:{r:ℝ| r ∈ i-approx(I;n)} ]. (|g[x]| ≤ a) BY
((InstLemma `Inorm-bound` [⌜i-approx(I;n)⌝;⌜g⌝;⌜mcg⌝]⋅ THENA Auto)
THEN (InstLemma `Inorm_wf` [⌜i-approx(I;n)⌝;⌜g⌝;⌜mcg⌝]⋅ THENA Auto)
THEN With ⌜||g[x]||_i-approx(I;n)⌝ (D 0)⋅
THEN Auto))
) }
1
1. I : Interval
2. f : I ⟶ℝ
3. g : I ⟶ℝ
4. ∀x,y:{t:ℝ| t ∈ I} . ((x = y)
⇒ (g[x] = g[y]))
5. d(f[x])/dx = λx.g[x] on I
6. k : ℕ+
7. n : {n:ℕ+| icompact(i-approx(I;n)) ∧ iproper(i-approx(I;n))}
8. ∀a,b:{x:ℝ| x ∈ i-approx(I;n)} . ((a = b)
⇒ (f[a] = f[b]))
9. i-approx(I;n) ⊆ I
10. ∀x:ℝ. ((x ∈ I)
⇒ f[x] ≠ r0)
11. ∃c:ℝ [((r0 < c) ∧ (∀x:ℝ. ((x ∈ i-approx(I;n))
⇒ (c ≤ |f[x]|))))]
12. (r1/f[x]) continuous for x ∈ i-approx(I;n)
13. mcg : g[x] continuous for x ∈ i-approx(I;n)
14. ∀x:ℝ. ((x ∈ i-approx(I;n))
⇒ f[x] ≠ r0)
15. ∃a:ℝ. ∀[x:{r:ℝ| r ∈ i-approx(I;n)} ]. (|g[x]| ≤ a)
⊢ ∃M:ℕ+. ∀x:ℝ. ((x ∈ i-approx(I;n))
⇒ ((|g[x]| ≤ r(M)) ∧ (|(r1/f[x])| ≤ r(M))))
Latex:
Latex:
.....assertion.....
1. I : Interval
2. f : I {}\mrightarrow{}\mBbbR{}
3. g : I {}\mrightarrow{}\mBbbR{}
4. \mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} (g[x] = g[y]))
5. d(f[x])/dx = \mlambda{}x.g[x] on I
6. k : \mBbbN{}\msupplus{}
7. n : \{n:\mBbbN{}\msupplus{}| icompact(i-approx(I;n)) \mwedge{} iproper(i-approx(I;n))\}
8. \mforall{}a,b:\{x:\mBbbR{}| x \mmember{} i-approx(I;n)\} . ((a = b) {}\mRightarrow{} (f[a] = f[b]))
9. i-approx(I;n) \msubseteq{} I
10. \mforall{}x:\mBbbR{}. ((x \mmember{} I) {}\mRightarrow{} f[x] \mneq{} r0)
11. \mexists{}c:\mBbbR{} [((r0 < c) \mwedge{} (\mforall{}x:\mBbbR{}. ((x \mmember{} i-approx(I;n)) {}\mRightarrow{} (c \mleq{} |f[x]|))))]
12. (r1/f[x]) continuous for x \mmember{} i-approx(I;n)
13. g[x] continuous for x \mmember{} i-approx(I;n)
14. \mforall{}x:\mBbbR{}. ((x \mmember{} i-approx(I;n)) {}\mRightarrow{} f[x] \mneq{} r0)
\mvdash{} \mexists{}M:\mBbbN{}\msupplus{}. \mforall{}x:\mBbbR{}. ((x \mmember{} i-approx(I;n)) {}\mRightarrow{} ((|g[x]| \mleq{} r(M)) \mwedge{} (|(r1/f[x])| \mleq{} r(M))))
By
Latex:
(RenameVar `mcg' (-2)\mcdot{}
THEN (Assert \mexists{}a:\mBbbR{}. \mforall{}[x:\{r:\mBbbR{}| r \mmember{} i-approx(I;n)\} ]. (|g[x]| \mleq{} a) BY
((InstLemma `Inorm-bound` [\mkleeneopen{}i-approx(I;n)\mkleeneclose{};\mkleeneopen{}g\mkleeneclose{};\mkleeneopen{}mcg\mkleeneclose{}]\mcdot{} THENA Auto)
THEN (InstLemma `Inorm\_wf` [\mkleeneopen{}i-approx(I;n)\mkleeneclose{};\mkleeneopen{}g\mkleeneclose{};\mkleeneopen{}mcg\mkleeneclose{}]\mcdot{} THENA Auto)
THEN With \mkleeneopen{}||g[x]||\_i-approx(I;n)\mkleeneclose{} (D 0)\mcdot{}
THEN Auto))
)
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