Nuprl Lemma : derivative-rexp
d(e^x)/dx = λx.e^x on (-∞, ∞)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞),
rexp: e^x
Definitions unfolded in proof :
member: t ∈ T,
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_apply: x[s],
all: ∀x:A. B[x],
nequal: a ≠ b ∈ T ,
guard: {T},
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rneq: x ≠ y,
squash: ↓T,
sq_stable: SqStable(P),
int_upper: {i...},
true: True,
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
eq_int: (i =z j),
pointwise-req: x[k] = y[k] for k ∈ [n,m]
Lemmas referenced :
fun-converges-to-rexp,
riiint_wf,
iproper-riiint,
rsum_wf,
int-rdiv_wf,
fact_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_sets,
less_than_wf,
nequal_wf,
nat_plus_properties,
int_seg_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
rnexp_wf,
int_seg_wf,
real_wf,
i-member_wf,
nat_wf,
subtract_wf,
subtract-add-cancel,
rexp_wf,
req_functionality,
rsum_functionality2,
int-rdiv_functionality,
le_wf,
rnexp_functionality,
nat_plus_wf,
req_weakening,
req_wf,
set_wf,
fun-converges-to-derivative,
icompact_wf,
rless_wf,
rless-int,
int-to-real_wf,
rdiv_wf,
rsub_wf,
rabs_wf,
rleq_wf,
all_wf,
int_upper_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermAdd_wf,
itermSubtract_wf,
intformle_wf,
intformnot_wf,
decidable__le,
i-approx_wf,
sq_stable__icompact,
int_upper_properties,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
not-lt-2,
decidable__lt,
derivative-rsum,
subtype_rel_self,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
true_wf,
derivative-const,
fact0_redex_lemma,
rnexp_zero_lemma,
derivative-rnexp,
derivative-const-mul,
minus-zero,
not-equal-2,
rmul_wf,
derivative_functionality,
rmul-one-both,
rmul-rdiv-cancel2,
rmul-rdiv-cancel,
rmul-ac,
rmul_comm,
rmul_functionality,
rmul-assoc,
req_inversion,
uiff_transitivity,
int-rdiv-req,
rmul_preserves_req,
rneq-int,
fact-non-zero,
set_subtype_base,
fact_unroll_1,
rmul-int,
sq_stable__less_than,
rinv_wf2,
req_transitivity,
real_term_polynomial,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul-rinv3,
rinv-mul-as-rdiv,
rdiv_functionality,
rsum-split-shift,
radd_wf,
radd-zero-both,
rsum-single,
radd_functionality,
rsum_functionality,
decidable__equal_int
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
hypothesis,
lambdaEquality,
sqequalRule,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesisEquality,
applyEquality,
addEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
intEquality,
setEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
productElimination,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
baseClosed,
independent_functionElimination,
dependent_set_memberEquality,
inrFormation,
imageElimination,
imageMemberEquality,
minusEquality,
unionElimination,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
addLevel,
multiplyEquality
Latex:
d(e\^{}x)/dx = \mlambda{}x.e\^{}x on (-\minfty{}, \minfty{})
Date html generated:
2017_10_04-PM-10_17_35
Last ObjectModification:
2017_07_28-AM-08_47_57
Theory : reals_2
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