Nuprl Lemma : rv-line-circle-2
∀n:ℕ. ∀a,b:ℝ^n. ∀p:{p:ℝ^n| ab ≥ ap} . ∀q:{q:ℝ^n| aq ≥ ab} .
(a ≠ b ⇒ p ≠ q ⇒ (∃u:{u:ℝ^n| ab=au ∧ q_u_p} . (∃v:{ℝ^n| (ab=av ∧ q_p_v)})))
Proof
Definitions occuring in Statement :
rv-be: a_b_c,
rv-ge: cd ≥ ab,
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
guard: {T},
uiff: uiff(P;Q),
rv-congruent: ab=cd,
false: False,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
not: ¬A,
so_apply: x[s],
and: P ∧ Q,
so_lambda: λ2x.t[x],
sq_exists: ∃x:{A| B[x]},
exists: ∃x:A. B[x],
squash: ↓T,
uimplies: b supposing a,
sq_stable: SqStable(P),
prop: ℙ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x],
rv-ge: cd ≥ ab,
rv-be: a_b_c
Lemmas referenced :
radd-zero-both,
rmul-zero-both,
radd-int,
rmul_functionality,
rmul-distrib2,
rmul-identity1,
radd-assoc,
rminus-as-rmul,
req_transitivity,
rless_irreflexivity,
rless_transitivity1,
rmul_wf,
rminus_wf,
radd-preserves-rless,
rless_functionality,
real-vec-dist-nonneg,
req_weakening,
radd_functionality,
req_inversion,
req_functionality,
radd_wf,
rless_wf,
rv-T-dist,
rv-T-iff,
nat_wf,
exists_wf,
set_wf,
rv-between_wf,
real-vec-sep_wf,
not_wf,
rv-congruent_wf,
real-vec_wf,
sq_exists_wf,
not-rless,
int-to-real_wf,
rleq_wf,
real_wf,
real-vec-dist_wf,
sq_stable__rleq,
rv-line-circle-1
Rules used in proof :
promote_hyp,
addEquality,
minusEquality,
comment,
voidElimination,
independent_pairFormation,
productEquality,
dependent_pairFormation,
productElimination,
imageElimination,
baseClosed,
imageMemberEquality,
independent_isectElimination,
natural_numberEquality,
setEquality,
lambdaEquality,
applyEquality,
isectElimination,
independent_functionElimination,
because_Cache,
rename,
setElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
extract_by_obid,
introduction,
cut,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b:\mBbbR{}\^{}n. \mforall{}p:\{p:\mBbbR{}\^{}n| ab \mgeq{} ap\} . \mforall{}q:\{q:\mBbbR{}\^{}n| aq \mgeq{} ab\} .
(a \mneq{} b {}\mRightarrow{} p \mneq{} q {}\mRightarrow{} (\mexists{}u:\{u:\mBbbR{}\^{}n| ab=au \mwedge{} q\_u\_p\} . (\mexists{}v:\{\mBbbR{}\^{}n| (ab=av \mwedge{} q\_p\_v)\})))
Date html generated:
2016_10_28-AM-07_39_26
Last ObjectModification:
2016_10_27-PM-01_32_12
Theory : reals
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