Nuprl Lemma : rv-circle-circle
∀n:{2...}. ∀a,b,c,d:ℝ^n. ∀p:{p:ℝ^n| ab=ap} . ∀q:{q:ℝ^n| cd=cq} . ∀x:{x:ℝ^n| cp=cx ∧ (¬(c ≠ x ∧ x ≠ d ∧ (¬c-x-d)))} .
∀y:{y:ℝ^n| aq=ay ∧ (¬(a ≠ y ∧ y ≠ b ∧ (¬a-y-b)))} .
(a ≠ c ⇒ (∃u,v:{p:ℝ^n| ab=ap ∧ cd=cp} . ((x ≠ d ∧ y ≠ b) ⇒ u ≠ v)))
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec: ℝ^n,
int_upper: {i...},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
real-vec-sep: a ≠ b,
uimplies: b supposing a
Lemmas referenced :
rv-circle-circle-lemma3,
real-vec-sep_wf,
exists_wf,
real-vec_wf,
int_upper_subtype_nat,
false_wf,
le_wf,
rv-congruent_wf,
set_wf,
not_wf,
rv-between_wf,
int_upper_wf,
rv-non-strict-between-iff,
rv-Tsep,
real-vec-sep-symmetry,
rv-between-symmetry,
real-vec-dist-be,
real-vec-dist_wf,
real_wf,
rleq_wf,
int-to-real_wf,
radd_wf,
radd-preserves-rless,
rminus_wf,
rmul_wf,
rless_wf,
rless_functionality,
req_weakening,
radd-zero-both,
rmul-zero-both,
radd_functionality,
rmul_functionality,
radd-int,
req_transitivity,
rminus-as-rmul,
req_inversion,
rmul-identity1,
rmul-distrib2,
radd-assoc
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,
sqequalRule,
independent_pairFormation,
productEquality,
isectElimination,
applyEquality,
because_Cache,
setElimination,
rename,
functionEquality,
setEquality,
dependent_set_memberEquality,
natural_numberEquality,
lambdaEquality,
voidElimination,
minusEquality,
addEquality,
independent_isectElimination,
addLevel
Latex:
\mforall{}n:\{2...\}. \mforall{}a,b,c,d:\mBbbR{}\^{}n. \mforall{}p:\{p:\mBbbR{}\^{}n| ab=ap\} . \mforall{}q:\{q:\mBbbR{}\^{}n| cd=cq\} . \mforall{}x:\{x:\mBbbR{}\^{}n|
cp=cx
\mwedge{} (\mneg{}(c \mneq{} x \mwedge{} x \mneq{} d \mwedge{} (\mneg{}c-x-d)))\} \000C.
\mforall{}y:\{y:\mBbbR{}\^{}n| aq=ay \mwedge{} (\mneg{}(a \mneq{} y \mwedge{} y \mneq{} b \mwedge{} (\mneg{}a-y-b)))\} .
(a \mneq{} c {}\mRightarrow{} (\mexists{}u,v:\{p:\mBbbR{}\^{}n| ab=ap \mwedge{} cd=cp\} . ((x \mneq{} d \mwedge{} y \mneq{} b) {}\mRightarrow{} u \mneq{} v)))
Date html generated:
2016_10_26-AM-11_10_28
Last ObjectModification:
2016_10_04-PM-11_52_54
Theory : reals
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