Nuprl Lemma : real-fun-implies-sfun-general
∀[I:Interval]. ∀[f:I ⟶ℝ].
∀x,y:{x:ℝ| x ∈ I} . (f x ≠ f y ⇒ x ≠ y) supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ ((f x) = (f y)))
Proof
Definitions occuring in Statement :
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rneq: x ≠ y,
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
rfun: I ⟶ℝ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
not: ¬A,
guard: {T},
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
top: Top,
cand: A c∧ B,
subinterval: I ⊆ J ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
i-member_wf,
req_wf,
real_wf,
real-weak-Markov,
rneq_wf,
set_wf,
all_wf,
rfun_wf,
interval_wf,
rneq-cases,
not_wf,
rneq_irreflexivity,
rneq_functionality,
req_weakening,
rmin-rmax-subinterval,
sq_stable__i-member,
rmin-rleq-rmax,
rmin_wf,
rmax_wf,
equal_wf,
member_rccint_lemma,
rleq_wf,
rmin_ub,
rleq-rmax,
rmin-rleq,
rccint_wf,
rmax_lb,
req_functionality,
rmin_functionality,
rmax_functionality,
rmin-req,
rmax-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
because_Cache,
independent_functionElimination,
setEquality,
lambdaFormation,
independent_isectElimination,
functionEquality,
unionElimination,
inlFormation,
inrFormation,
voidElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidEquality,
independent_pairFormation,
productEquality
Latex:
\mforall{}[I:Interval]. \mforall{}[f:I {}\mrightarrow{}\mBbbR{}].
\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . (f x \mneq{} f y {}\mRightarrow{} x \mneq{} y)
supposing \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} ((f x) = (f y)))
Date html generated:
2017_10_03-AM-09_57_00
Last ObjectModification:
2017_08_31-AM-11_52_29
Theory : reals
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