Nuprl Lemma : derivative-rinv
∀I:Interval. ∀f,g:I ⟶ℝ.
((∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (g[x] = g[y])))
⇒ f[x]≠r0 for x ∈ I
⇒ d(f[x])/dx = λx.g[x] on I
⇒ d((r1/f[x]))/dx = λx.(-(g[x])/f[x] * f[x]) on I)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
nonzero-on: f[x]≠r0 for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rdiv: (x/y),
req: x = y,
rmul: a * b,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
derivative: d(f[x])/dx = λz.g[z] on I,
member: t ∈ T,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
nonzero-on: f[x]≠r0 for x ∈ I,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
subinterval: I ⊆ J ,
rneq: x ≠ y,
label: ...$L... t,
guard: {T},
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
sq_exists: ∃x:A [B[x]],
continuous: f[x] continuous for x ∈ I,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
req_int_terms: t1 ≡ t2,
rdiv: (x/y)
Lemmas referenced :
i-approx-is-subinterval,
istype-less_than,
nonzero-on-implies,
icompact_wf,
i-approx_wf,
continuous-implies-functional,
rfun_subtype,
proper-continuous-implies,
differentiable-continuous,
sq_stable__icompact,
sq_stable__iproper,
function-is-continuous,
rdiv_wf,
int-to-real_wf,
subtype_rel_sets_simple,
real_wf,
i-member_wf,
sq_stable__i-member,
req_wf,
i-member-approx,
iproper_wf,
nat_plus_wf,
derivative_wf,
nonzero-on_wf,
rfun_wf,
interval_wf,
req_weakening,
rdiv_functionality,
Inorm-bound,
Inorm_wf,
rleq_wf,
rabs_wf,
imax_wf,
r-bound_wf,
imax_nat_plus,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
rmax_wf,
rmax_ub,
rleq_functionality,
req_inversion,
rmax-int,
r-bound-property,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
mul_nat_plus,
i-approx-approx,
rmin_wf,
rsub_wf,
rless_wf,
rmul_wf,
rminus_wf,
rless-int,
rmin_strict_ub,
rmin-rleq,
implies_weakening_uimplies,
rneq-int,
int_entire_a,
mul_nzero,
subtype_base_sq,
int_subtype_base,
nequal_wf,
rneq_wf,
rmul_reverses_rless,
rmul_preserves_rless,
squash_wf,
true_wf,
rabs-rminus,
subtype_rel_self,
iff_weakening_equal,
itermSubtract_wf,
itermMultiply_wf,
radd_wf,
rinv_wf2,
itermMinus_wf,
itermAdd_wf,
rless_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
uiff_transitivity,
req_functionality,
rmul_functionality,
rabs-rmul,
req_transitivity,
rabs_functionality,
rminus_functionality,
rsub_functionality,
rinv-of-rmul,
radd_functionality,
rinv-mul-as-rdiv,
rinv-as-rdiv,
rmul-rinv3,
rmul-rinv,
real_term_value_minus_lemma,
real_term_value_add_lemma,
rmul-nonneg-case1,
zero-rleq-rabs,
rmul_functionality_wrt_rleq2,
rleq-int,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rmul_preserves_rleq2,
rmul-identity1,
rmul-int,
uimplies_transitivity,
rleq_transitivity,
r-triangle-inequality,
rleq_weakening,
multiply_nat_plus,
int_term_value_mul_lemma,
radd_functionality_wrt_rleq,
rabs-difference-symmetry,
rabs-rmul-rleq,
rleq-int-fractions,
rmul-int-rdiv,
mul_bounds_1a,
nat_plus_subtype_nat,
rneq_functionality,
set_subtype_base,
less_than_wf,
rinv_functionality2,
rmul_preserves_rleq,
square-nonzero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesisEquality,
hypothesis,
isectElimination,
natural_numberEquality,
independent_functionElimination,
productElimination,
universeIsType,
applyEquality,
independent_isectElimination,
sqequalRule,
lambdaEquality_alt,
imageMemberEquality,
baseClosed,
imageElimination,
closedConclusion,
setIsType,
promote_hyp,
inhabitedIsType,
productIsType,
functionIsType,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
isectIsType,
applyLambdaEquality,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIsType1,
inlFormation_alt,
inrFormation_alt,
dependent_set_memberFormation_alt,
multiplyEquality,
instantiate,
cumulativity,
intEquality,
equalityIsType4,
universeEquality,
baseApply
Latex:
\mforall{}I:Interval. \mforall{}f,g:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} (g[x] = g[y])))
{}\mRightarrow{} f[x]\mneq{}r0 for x \mmember{} I
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.g[x] on I
{}\mRightarrow{} d((r1/f[x]))/dx = \mlambda{}x.(-(g[x])/f[x] * f[x]) on I)
Date html generated:
2019_10_30-AM-09_02_57
Last ObjectModification:
2018_11_12-AM-11_59_54
Theory : reals
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