Nuprl Lemma : real-vec-sep_functionality
∀n:ℕ. ∀a1,a2,b1,b2:ℝ^n. (req-vec(n;a1;a2) ⇒ req-vec(n;b1;b2) ⇒ (a1 ≠ b1 ⇐⇒ a2 ≠ b2))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
real-vec-sep: a ≠ b,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
real-vec-sep_wf,
req-vec_wf,
real-vec_wf,
nat_wf,
int-to-real_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
rless_functionality,
req_weakening,
real-vec-dist_functionality,
req-vec_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
sqequalRule,
because_Cache,
dependent_functionElimination,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a1,a2,b1,b2:\mBbbR{}\^{}n. (req-vec(n;a1;a2) {}\mRightarrow{} req-vec(n;b1;b2) {}\mRightarrow{} (a1 \mneq{} b1 \mLeftarrow{}{}\mRightarrow{} a2 \mneq{} b2))
Date html generated:
2016_10_26-AM-10_29_28
Last ObjectModification:
2016_09_25-PM-00_54_27
Theory : reals
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