Nuprl Lemma : derivative_functionality

`∀[I:Interval]. ∀[f1,f2,g1,g2:I ⟶ℝ].`
`  (rfun-eq(I;f1;f2) `` rfun-eq(I;g1;g2) `` λx.g1[x] = d(f1[x])/dx on I `` λx.g2[x] = d(f2[x])/dx on I)`

Proof

Definitions occuring in Statement :  derivative: `λz.g[z] = d(f[x])/dx on I` rfun-eq: `rfun-eq(I;f;g)` rfun: `I ⟶ℝ` interval: `Interval` uall: `∀[x:A]. B[x]` so_apply: `x[s]` implies: `P `` Q`
Definitions unfolded in proof :  top: `Top` not: `¬A` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` decidable: `Dec(P)` sq_exists: `∃x:{A| B[x]}` rless: `x < y` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` or: `P ∨ Q` guard: `{T}` rneq: `x ≠ y` uimplies: `b supposing a` nat_plus: `ℕ+` all: `∀x:A. B[x]` and: `P ∧ Q` so_apply: `x[s]` rfun: `I ⟶ℝ` label: `...\$L... t` so_lambda: `λ2x.t[x]` prop: `ℙ` member: `t ∈ T` implies: `P `` Q` uall: `∀[x:A]. B[x]` derivative: `λz.g[z] = d(f[x])/dx on I` rev_uimplies: `rev_uimplies(P;Q)` rge: `x ≥ y` rfun-eq: `rfun-eq(I;f;g)` r-ap: `f(x)` uiff: `uiff(P;Q)`
Lemmas referenced :  rmul_functionality rsub_functionality rabs_functionality req_weakening rleq_functionality r-ap_wf rleq_weakening_equal rleq_functionality_wrt_implies icompact_wf nat_plus_wf less_than_wf all_wf derivative_wf real_wf i-member_wf rfun-eq_wf rfun_wf interval_wf rleq_wf rabs_wf rsub_wf i-approx_wf i-member-approx rmul_wf rdiv_wf int-to-real_wf rless-int nat_plus_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf rless_wf
Rules used in proof :  computeAll voidEquality voidElimination isect_memberEquality intEquality int_eqEquality dependent_pairFormation unionElimination inrFormation independent_isectElimination natural_numberEquality dependent_set_memberEquality rename setElimination independent_functionElimination dependent_functionElimination promote_hyp productElimination independent_pairFormation because_Cache hypothesis setEquality applyEquality lambdaEquality sqequalRule hypothesisEquality thin isectElimination sqequalHypSubstitution lemma_by_obid cut lambdaFormation isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution introduction productEquality functionEquality

Latex:
\mforall{}[I:Interval].  \mforall{}[f1,f2,g1,g2:I  {}\mrightarrow{}\mBbbR{}].
(rfun-eq(I;f1;f2)
{}\mRightarrow{}  rfun-eq(I;g1;g2)
{}\mRightarrow{}  \mlambda{}x.g1[x]  =  d(f1[x])/dx  on  I
{}\mRightarrow{}  \mlambda{}x.g2[x]  =  d(f2[x])/dx  on  I)

Date html generated: 2016_05_18-AM-09_59_27
Last ObjectModification: 2016_01_17-AM-00_42_25

Theory : reals

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