Nuprl Lemma : rfun_subtype_2
∀[a,b,c:ℝ]. ((a ≤ c) ⇒ (c ≤ b) ⇒ ([a, b] ⟶ℝ ⊆r [c, b] ⟶ℝ))
Proof
Definitions occuring in Statement :
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
real: ℝ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
rfun: I ⟶ℝ,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
rfun_wf,
rccint_wf,
rleq_wf,
real_wf,
i-member_wf,
member_rccint_lemma,
rleq_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
dependent_functionElimination,
axiomEquality,
because_Cache,
isect_memberEquality,
functionExtensionality,
setEquality,
setElimination,
rename,
voidElimination,
voidEquality,
applyEquality,
productElimination,
independent_isectElimination,
independent_pairFormation,
dependent_set_memberEquality,
productEquality
Latex:
\mforall{}[a,b,c:\mBbbR{}]. ((a \mleq{} c) {}\mRightarrow{} (c \mleq{} b) {}\mRightarrow{} ([a, b] {}\mrightarrow{}\mBbbR{} \msubseteq{}r [c, b] {}\mrightarrow{}\mBbbR{}))
Date html generated:
2016_10_26-AM-09_29_57
Last ObjectModification:
2016_08_20-AM-10_16_45
Theory : reals
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