Nuprl Lemma : continuous-mul
∀I:Interval. ∀f,g:I ⟶ℝ. (f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ I ⇒ f[x] * g[x] continuous for x ∈ I)
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
rmul: a * b,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
continuous: f[x] continuous for x ∈ I,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
exists: ∃x:A. B[x],
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
sq_exists: ∃x:{A| B[x]},
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,
sq_stable: SqStable(P),
cand: A c∧ B,
uiff: uiff(P;Q),
real: ℝ,
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_plus_wf,
set_wf,
icompact_wf,
i-approx_wf,
continuous_wf,
i-member_wf,
real_wf,
rfun_wf,
interval_wf,
rabs_wf,
continuous-max,
continuous-abs,
continuous_functionality_wrt_subinterval,
rmax_wf,
less_than_wf,
i-approx-is-subinterval,
rfun_subtype,
Inorm-bound,
r-bound_wf,
Inorm_wf,
less_than'_wf,
rsub_wf,
int-to-real_wf,
uall_wf,
rleq_wf,
r-bound-property,
rabs-bounds,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
mul_nat_plus,
rless_wf,
all_wf,
rdiv_wf,
rless-int,
multiply_nat_plus,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermMultiply_wf,
intformeq_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_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
squash_wf,
sq_stable__and,
sq_stable__rless,
sq_stable__all,
sq_stable__rleq,
rmin_wf,
rmul_wf,
rmin_strict_ub,
radd_wf,
r-triangle-inequality2,
zero-rleq-rabs,
rmul_functionality_wrt_rleq,
rmin-rleq,
rleq_functionality,
req_inversion,
rabs-rmul,
req_weakening,
radd_functionality,
rabs_functionality,
rmul-rsub-distrib,
rleq-rmax,
radd_functionality_wrt_rleq,
rmul_comm,
rleq-int-fractions2,
sq_stable__less_than,
sq_stable__icompact,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rleq_transitivity,
rneq-int,
int_entire_a,
equal-wf-base,
int_subtype_base,
equal-wf-T-base,
rmul_functionality_wrt_rleq2,
req_transitivity,
rmul-distrib2,
rmul_functionality,
radd-rdiv,
rdiv_functionality,
radd-int,
rmul_preserves_rleq,
mul_bounds_1b,
req_wf,
uiff_transitivity,
req_functionality,
rmul-assoc,
rmul-ac,
rmul-rdiv-cancel,
rmul-int,
rmul-int-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
promote_hyp,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
setEquality,
because_Cache,
independent_functionElimination,
natural_numberEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
isect_memberFormation,
productElimination,
independent_pairEquality,
voidElimination,
minusEquality,
axiomEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
isect_memberEquality,
functionEquality,
multiplyEquality,
inrFormation,
applyLambdaEquality,
unionElimination,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
imageElimination,
dependent_set_memberFormation,
productEquality,
inlFormation,
addEquality,
baseApply,
closedConclusion
Latex:
\mforall{}I:Interval. \mforall{}f,g:I {}\mrightarrow{}\mBbbR{}.
(f[x] continuous for x \mmember{} I {}\mRightarrow{} g[x] continuous for x \mmember{} I {}\mRightarrow{} f[x] * g[x] continuous for x \mmember{} I)
Date html generated:
2017_10_03-AM-10_26_27
Last ObjectModification:
2017_07_28-AM-08_10_31
Theory : reals
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