Nuprl Lemma : weak-continuity-principle-real-double
∀x:ℝ. ∀F,H:ℝ ⟶ 𝔹. ∀G:n:ℕ+ ⟶ {y:ℝ| x = y ∈ (ℕ+n ⟶ ℤ)} . (∃n:{ℕ+| (F x = F (G n) ∧ H x = H (G n))})
Proof
Definitions occuring in Statement :
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
bool: 𝔹,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
real: ℝ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
true: True,
sq_exists: ∃x:{A| B[x]},
cand: A c∧ B,
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
nat_plus_wf,
real_wf,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat_plus,
false_wf,
subtype_rel_self,
bool_wf,
WCPD_wf,
regularize-k-regular,
less_than_wf,
regularize_wf,
regular-int-seq_wf,
subtype_rel_sets,
set_wf,
squash_wf,
true_wf,
regularize-real,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
functionEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
setEquality,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
intEquality,
hypothesisEquality,
applyEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
functionExtensionality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
productEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_set_memberFormation,
productElimination,
equalityUniverse,
levelHypothesis,
imageElimination,
universeEquality
Latex:
\mforall{}x:\mBbbR{}. \mforall{}F,H:\mBbbR{} {}\mrightarrow{} \mBbbB{}. \mforall{}G:n:\mBbbN{}\msupplus{} {}\mrightarrow{} \{y:\mBbbR{}| x = y\} . (\mexists{}n:\{\mBbbN{}\msupplus{}| (F x = F (G n) \mwedge{} H x = H (G n))\})
Date html generated:
2017_10_03-AM-09_09_22
Last ObjectModification:
2017_09_12-PM-02_20_05
Theory : reals
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