Nuprl Lemma : continuous-rinv
∀I:Interval. ∀f:I ⟶ℝ. (f[x] continuous for x ∈ I ⇒ f[x]≠r0 for x ∈ I ⇒ (r1/f[x]) continuous for x ∈ I)
Proof
Definitions occuring in Statement :
nonzero-on: f[x]≠r0 for x ∈ I,
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
rdiv: (x/y),
int-to-real: r(n),
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
continuous: f[x] continuous for x ∈ I,
nonzero-on: f[x]≠r0 for x ∈ I,
member: t ∈ T,
sq_exists: ∃x:{A| B[x]},
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
and: P ∧ Q,
nat_plus: ℕ+,
squash: ↓T,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
guard: {T},
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
less_than': less_than'(a;b),
real: ℝ
Lemmas referenced :
nat_plus_wf,
set_wf,
icompact_wf,
i-approx_wf,
nonzero-on_wf,
i-member_wf,
real_wf,
continuous_wf,
rfun_wf,
interval_wf,
rless_wf,
int-to-real_wf,
all_wf,
less_than_wf,
rleq_wf,
rabs_wf,
i-member-approx,
less_than'_wf,
rsub_wf,
squash_wf,
sq_stable__and,
sq_stable__rless,
sq_stable__all,
sq_stable__rleq,
small-reciprocal-real,
rless_transitivity1,
rdiv_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,
rleq_weakening_rless,
rabs-positive-iff,
mul_nat_plus,
mul_bounds_1b,
rneq_wf,
equal_wf,
rmul_preserves_rleq,
rabs-neq-zero,
rmul_wf,
radd_wf,
rminus_wf,
rinv_wf2,
rleq_functionality,
req_transitivity,
rmul_functionality,
rabs_functionality,
rsub_functionality,
rmul-identity1,
req_weakening,
rinv-as-rdiv,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
radd_functionality,
rminus_functionality,
rinv-mul-as-rdiv,
uiff_transitivity,
req_inversion,
rabs-rmul,
rdiv_functionality,
rmul-rinv,
rmul-rinv3,
rabs-difference-symmetry,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rmul_preserves_rleq2,
rleq-int,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rmul-int,
false_wf,
rleq_transitivity,
rleq-implies-rleq,
rmul_functionality_wrt_rleq2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
cut,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
setEquality,
natural_numberEquality,
productElimination,
isect_memberEquality,
functionEquality,
because_Cache,
independent_functionElimination,
imageElimination,
minusEquality,
independent_pairEquality,
voidElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
independent_isectElimination,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
independent_pairFormation,
computeAll,
promote_hyp,
productEquality,
multiplyEquality,
isect_memberFormation,
inlFormation
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
(f[x] continuous for x \mmember{} I {}\mRightarrow{} f[x]\mneq{}r0 for x \mmember{} I {}\mRightarrow{} (r1/f[x]) continuous for x \mmember{} I)
Date html generated:
2017_10_03-AM-10_27_19
Last ObjectModification:
2017_07_28-AM-08_11_08
Theory : reals
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