### Nuprl Lemma : chain-rule_1

`∀I,J:Interval. ∀f,f':I ⟶ℝ. ∀g,g':J ⟶ℝ.`
`  (maps-compact(I;J;x.f[x])`
`  `` f[x] continuous for x ∈ I`
`  `` f'[x] continuous for x ∈ I`
`  `` g'[x] continuous for x ∈ J`
`  `` λx.f'[x] = d(f[x])/dx on I`
`  `` λx.g'[x] = d(g[x])/dx on J`
`  `` λx.g'[f[x]] * f'[x] = d(g[f[x]])/dx on I)`

Proof

Definitions occuring in Statement :  derivative: `λz.g[z] = d(f[x])/dx on I` maps-compact: `maps-compact(I;J;x.f[x])` continuous: `f[x] continuous for x ∈ I` rfun: `I ⟶ℝ` interval: `Interval` rmul: `a * b` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q`
Definitions unfolded in proof :  label: `...\$L... t` rfun: `I ⟶ℝ` so_apply: `x[s]` so_lambda: `λ2x.t[x]` uimplies: `b supposing a` subtype_rel: `A ⊆r B` prop: `ℙ` squash: `↓T` sq_stable: `SqStable(P)` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` member: `t ∈ T` maps-compact: `maps-compact(I;J;x.f[x])` derivative: `λz.g[z] = d(f[x])/dx on I` implies: `P `` Q` all: `∀x:A. B[x]` uiff: `uiff(P;Q)` top: `Top` not: `¬A` false: `False` satisfiable_int_formula: `satisfiable_int_formula(fmla)` decidable: `Dec(P)` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` or: `P ∨ Q` rneq: `x ≠ y` guard: `{T}` true: `True` less_than': `less_than'(a;b)` less_than: `a < b` nat_plus: `ℕ+` and: `P ∧ Q` rev_uimplies: `rev_uimplies(P;Q)` rge: `x ≥ y` rless: `x < y` le: `A ≤ B` rnonneg: `rnonneg(x)` rleq: `x ≤ y` cand: `A c∧ B` continuous: `f[x] continuous for x ∈ I` sq_exists: `∃x:{A| B[x]}` real: `ℝ` rsub: `x - y`
Rules used in proof :  lambdaEquality setEquality equalitySymmetry equalityTransitivity dependent_pairFormation independent_isectElimination because_Cache applyEquality dependent_set_memberEquality imageElimination baseClosed imageMemberEquality sqequalRule introduction independent_functionElimination hypothesis rename setElimination isectElimination lemma_by_obid cut productElimination hypothesisEquality thin dependent_functionElimination sqequalHypSubstitution lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution computeAll voidEquality voidElimination isect_memberEquality intEquality int_eqEquality unionElimination inrFormation multiplyEquality functionEquality independent_pairFormation natural_numberEquality productEquality axiomEquality minusEquality independent_pairEquality addEquality dependent_set_memberFormation equalityEquality addLevel impliesFunctionality isect_memberFormation universeEquality

Latex:
\mforall{}I,J:Interval.  \mforall{}f,f':I  {}\mrightarrow{}\mBbbR{}.  \mforall{}g,g':J  {}\mrightarrow{}\mBbbR{}.
(maps-compact(I;J;x.f[x])
{}\mRightarrow{}  f[x]  continuous  for  x  \mmember{}  I
{}\mRightarrow{}  f'[x]  continuous  for  x  \mmember{}  I
{}\mRightarrow{}  g'[x]  continuous  for  x  \mmember{}  J
{}\mRightarrow{}  \mlambda{}x.f'[x]  =  d(f[x])/dx  on  I
{}\mRightarrow{}  \mlambda{}x.g'[x]  =  d(g[x])/dx  on  J
{}\mRightarrow{}  \mlambda{}x.g'[f[x]]  *  f'[x]  =  d(g[f[x]])/dx  on  I)

Date html generated: 2016_05_18-AM-10_15_09
Last ObjectModification: 2016_01_17-AM-00_57_38

Theory : reals

Home Index