Nuprl Lemma : real-vec-sep-iff-rneq
∀n:ℕ. ∀a,c:ℝ^n. (a ≠ c ⇐⇒ ∃i:ℕn. a i ≠ c i)
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
rneq: x ≠ y,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
real-vec: ℝ^n,
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
exists_wf,
int_seg_wf,
rneq_wf,
rneq-iff-rabs,
rless_wf,
int-to-real_wf,
rabs_wf,
rsub_wf,
iff_wf,
real-vec-sep-iff,
real-vec-sep_wf,
real-vec_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
addLevel,
productElimination,
impliesFunctionality,
existsFunctionality,
dependent_functionElimination,
independent_functionElimination,
existsLevelFunctionality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,c:\mBbbR{}\^{}n. (a \mneq{} c \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}n. a i \mneq{} c i)
Date html generated:
2017_10_03-AM-11_00_58
Last ObjectModification:
2017_06_20-PM-02_06_49
Theory : reals
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