Nuprl Lemma : continuous-rneq
∀I:Interval. ∀f:I ⟶ℝ. (f[x] continuous for x ∈ I ⇒ (∀a,b:{x:ℝ| x ∈ I} . (f[a] ≠ f[b] ⇒ a ≠ b)))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rneq: x ≠ y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
rfun: I ⟶ℝ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
exists: ∃x:A. B[x],
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
label: ...$L... t,
rneq: x ≠ y,
or: P ∨ Q,
guard: {T},
uimplies: b supposing a,
continuous: f[x] continuous for x ∈ I,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
rless: x < y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B
Lemmas referenced :
rneq-iff-rabs,
rless_irreflexivity,
rless_transitivity1,
rleq_weakening_rless,
not-rless,
sq_stable__rleq,
sq_stable__all,
sq_stable__rless,
sq_stable__and,
squash_wf,
nat_plus_wf,
less_than'_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
i-member-approx,
rleq_wf,
all_wf,
i-approx_wf,
icompact_wf,
i-approx-compact,
req_weakening,
radd-zero-both,
rless_functionality,
or_wf,
radd_wf,
interval_wf,
rfun_wf,
continuous_wf,
i-member_wf,
real_wf,
set_wf,
rneq_wf,
sq_stable__i-member,
i-approx-containing2,
rless_wf,
int-to-real_wf,
rabs-difference-lower-bound,
rsub_wf,
rabs_wf,
small-reciprocal-real
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality,
isectElimination,
applyEquality,
sqequalRule,
hypothesisEquality,
hypothesis,
natural_numberEquality,
productElimination,
independent_functionElimination,
setElimination,
rename,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
lambdaEquality,
setEquality,
unionElimination,
inrFormation,
inlFormation,
because_Cache,
addLevel,
orFunctionality,
independent_isectElimination,
isect_memberEquality,
functionEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
minusEquality,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}. (f[x] continuous for x \mmember{} I {}\mRightarrow{} (\mforall{}a,b:\{x:\mBbbR{}| x \mmember{} I\} . (f[a] \mneq{} f[b] {}\mRightarrow{} a \mneq{} b)))
Date html generated:
2016_05_18-AM-09_08_58
Last ObjectModification:
2016_01_17-AM-02_37_40
Theory : reals
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