Nuprl Lemma : rv-extend-2
∀n:ℕ. ∀a,b,c,d:ℝ^n. (a ≠ b ⇒ (∃x:{x:ℝ^n| bx=cd} . (c ≠ d ⇒ a-b-x)))
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
rv-between: a-b-c,
and: P ∧ Q,
real-vec-sep: a ≠ b,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rless: x < y,
sq_exists: ∃x:{A| B[x]},
rge: x ≥ y,
guard: {T}
Lemmas referenced :
rv-extend,
real-vec-sep_wf,
rv-between_wf,
real-vec_wf,
nat_wf,
real-vec-dist-between,
int-to-real_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
radd_wf,
rless_functionality,
req_weakening,
real-vec-dist-nonneg,
trivial-rless-radd,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
dependent_pairFormation,
isectElimination,
functionEquality,
setElimination,
rename,
independent_pairFormation,
natural_numberEquality,
applyEquality,
lambdaEquality,
setEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,d:\mBbbR{}\^{}n. (a \mneq{} b {}\mRightarrow{} (\mexists{}x:\{x:\mBbbR{}\^{}n| bx=cd\} . (c \mneq{} d {}\mRightarrow{} a-b-x)))
Date html generated:
2016_10_26-AM-10_40_02
Last ObjectModification:
2016_09_26-PM-09_21_04
Theory : reals
Home
Index