Nuprl Lemma : real-vec-between_functionality
∀n:ℕ. ∀a1,a2,b1,b2,c1,c2:ℝ^n. (req-vec(n;a1;a2) ⇒ req-vec(n;b1;b2) ⇒ req-vec(n;c1;c2) ⇒ (a1-b1-c1 ⇐⇒ a2-b2-c2))
Proof
Definitions occuring in Statement :
real-vec-between: a-b-c,
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-between: a-b-c,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
uimplies: b supposing a,
uiff: uiff(P;Q)
Lemmas referenced :
i-member_wf,
rooint_wf,
int-to-real_wf,
req-vec_wf,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
real-vec-between_wf,
real-vec_wf,
nat_wf,
req-vec_functionality,
real-vec-add_functionality,
real-vec-mul_functionality,
req_weakening,
req-vec_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
promote_hyp,
hypothesis,
productEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
because_Cache,
independent_isectElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a1,a2,b1,b2,c1,c2:\mBbbR{}\^{}n.
(req-vec(n;a1;a2) {}\mRightarrow{} req-vec(n;b1;b2) {}\mRightarrow{} req-vec(n;c1;c2) {}\mRightarrow{} (a1-b1-c1 \mLeftarrow{}{}\mRightarrow{} a2-b2-c2))
Date html generated:
2016_10_26-AM-10_17_46
Last ObjectModification:
2016_09_25-PM-00_52_03
Theory : reals
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