### Nuprl Lemma : derivative-continuous

`∀I:Interval. ∀f,g:I ⟶ℝ.`
`  ((∀x,y:{x:ℝ| x ∈ I} .  (g[x] ≠ g[y] `` x ≠ y)) `` λx.g[x] = d(f[x])/dx on I `` g[x] continuous for x ∈ I)`

Proof

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

Latex:
\mforall{}I:Interval.  \mforall{}f,g:I  {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{x:\mBbbR{}|  x  \mmember{}  I\}  .    (g[x]  \mneq{}  g[y]  {}\mRightarrow{}  x  \mneq{}  y))
{}\mRightarrow{}  \mlambda{}x.g[x]  =  d(f[x])/dx  on  I
{}\mRightarrow{}  g[x]  continuous  for  x  \mmember{}  I)

Date html generated: 2016_05_18-AM-10_00_54
Last ObjectModification: 2016_01_17-AM-00_44_47

Theory : reals

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