Nuprl Lemma : q-between-overlap
∀a,b,c,d,q:ℚ.
(∃x,y:ℚ. (x < q < y ∧ (∀q':ℚ. a < q' < b ∧ c < q' < d supposing x < q' < y))) supposing (c < q < d and a < q < b)
Proof
Definitions occuring in Statement :
q-between: a < b < c,
rationals: ℚ,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
q-between: a < b < c,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
qless_witness,
qmax_wf,
qmin_wf,
qmax_strict_lb,
qmin_strict_ub,
q-between_wf,
and_wf,
all_wf,
rationals_wf,
isect_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lemma_by_obid,
isectElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
rename,
dependent_pairFormation,
independent_isectElimination,
independent_pairFormation,
lambdaEquality
Latex:
\mforall{}a,b,c,d,q:\mBbbQ{}.
(\mexists{}x,y:\mBbbQ{}. (x < q < y \mwedge{} (\mforall{}q':\mBbbQ{}. a < q' < b \mwedge{} c < q' < d supposing x < q' < y))) supposing
(c < q < d and
a < q < b)
Date html generated:
2016_05_15-PM-11_05_34
Last ObjectModification:
2015_12_27-PM-07_46_35
Theory : rationals
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