Nuprl Lemma : extensional-interval-to-bool-constant
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:{x:ℝ| x ∈ [a, b]} ⟶ 𝔹.
∀x,y:{x:ℝ| x ∈ [a, b]} . f x = f y supposing ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ f x = f y)
Proof
Definitions occuring in Statement :
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
req: x = y,
real: ℝ,
bool: 𝔹,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
i-member: r ∈ I,
rccint: [l, u],
rev_uimplies: rev_uimplies(P;Q),
top: Top,
iff: P ⇐⇒ Q,
guard: {T}
Lemmas referenced :
rmax_wf,
rmin_wf,
rleq-rmax,
rmax_lb,
sq_stable__rleq,
rmin-rleq,
rleq_wf,
extensional-real-to-bool-constant,
subtype_rel_self,
real_wf,
i-member_wf,
rccint_wf,
req_functionality,
rmax_functionality,
rmin_functionality,
req_weakening,
req_wf,
bool_wf,
member_rccint_lemma,
istype-void,
rmin_ub,
rmax-req,
rmin-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
productElimination,
independent_pairFormation,
because_Cache,
independent_isectElimination,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
productIsType,
universeIsType,
dependent_functionElimination,
lambdaEquality_alt,
applyEquality,
setEquality,
inhabitedIsType,
setIsType,
axiomEquality,
functionIsTypeImplies,
functionIsType,
equalityIstype,
isect_memberEquality_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . \mforall{}f:\{x:\mBbbR{}| x \mmember{} [a, b]\} {}\mrightarrow{} \mBbbB{}.
\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . f x = f y supposing \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((x = y) {}\mRightarrow{} f x = f y)
Date html generated:
2019_10_30-AM-07_17_36
Last ObjectModification:
2019_04_10-PM-03_55_34
Theory : reals
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