Nuprl Lemma : rv-compass-compass-lemma
∀a,b,c,d:ℝ^2.
(a ≠ c
⇒ (↓∃p,q:ℝ^2. (((d(a;b) = d(a;p)) ∧ (d(c;d) = d(c;q))) ∧ (d(c;p) ≤ d(c;d)) ∧ (d(a;q) ≤ d(a;b))))
⇒ (∃u,v:{p:ℝ^2| ab=ap ∧ cd=cp}
((↓∃p,q:ℝ^2. (((d(a;b) = d(a;p)) ∧ (d(c;d) = d(c;q))) ∧ (d(c;p) < d(c;d)) ∧ (d(a;q) < d(a;b))))
⇒ (r2-left(u;c;a) ∧ r2-left(v;a;c)))))
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec-dist: d(x;y),
real-vec: ℝ^n,
rleq: x ≤ y,
rless: x < y,
req: x = y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
real-vec-dist: d(x;y),
prop: ℙ,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real-vec-sep: a ≠ b,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
true: True,
guard: {T},
rge: x ≥ y,
cand: A c∧ B,
req_int_terms: t1 ≡ t2,
top: Top,
rv-congruent: ab=cd,
real-vec-add: X + Y,
real-vec-sub: X - Y,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
nat_plus: ℕ+,
less_than: a < b,
or: P ∨ Q,
r2-left: r2-left(p;q;r),
r2-det: |pqr|,
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
rv-circle-circle-lemma2',
real-vec-dist_wf,
real-vec-sub_wf,
squash_wf,
exists_wf,
real-vec_wf,
false_wf,
le_wf,
req_wf,
real_wf,
rleq_wf,
int-to-real_wf,
real-vec-sep_wf,
rnexp_wf,
radd_wf,
rsub_wf,
rmul_wf,
rless_functionality,
req_weakening,
real-vec-dist-symmetry,
rleq_functionality,
rnexp_functionality,
radd_functionality,
rsub_functionality,
rmul_functionality,
sq_stable__rleq,
rnexp-rleq,
real-vec-dist-nonneg,
real-vec-triangle-inequality,
true_wf,
radd_comm_eq,
iff_weakening_equal,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rabs_wf,
zero-rleq-rabs,
rabs-difference-bound-rleq,
radd-preserves-rleq,
radd_comm,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rnexp2-nonneg,
req_inversion,
rabs-rnexp,
rabs-of-nonneg,
rminus_wf,
radd-rminus-both,
itermMinus_wf,
req_transitivity,
rminus_functionality,
real_term_value_minus_lemma,
equal_wf,
itermMultiply_wf,
itermConstant_wf,
rnexp2,
real_term_value_mul_lemma,
req_functionality,
rleq-implies-rleq,
rmul-nonneg-case1,
radd-zero,
real-vec-add_wf,
rv-congruent_wf,
rless_wf,
r2-left_wf,
int_seg_wf,
real-vec-norm_wf,
real-vec-norm_functionality,
sq_stable__rless,
rnexp-rless,
less_than_wf,
radd-rminus-assoc,
radd-preserves-rless,
rless_transitivity2,
rabs-difference-bound-iff,
rless_functionality_wrt_implies,
rless-implies-rless,
rmul-is-positive,
rless_transitivity1,
rleq_weakening,
r2-det_wf,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
because_Cache,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaEquality,
productEquality,
applyEquality,
setElimination,
rename,
setEquality,
independent_isectElimination,
productElimination,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
universeEquality,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
functionEquality,
inlFormation,
promote_hyp
Latex:
\mforall{}a,b,c,d:\mBbbR{}\^{}2.
(a \mneq{} c
{}\mRightarrow{} (\mdownarrow{}\mexists{}p,q:\mBbbR{}\^{}2. (((d(a;b) = d(a;p)) \mwedge{} (d(c;d) = d(c;q))) \mwedge{} (d(c;p) \mleq{} d(c;d)) \mwedge{} (d(a;q) \mleq{} d(a;b))))
{}\mRightarrow{} (\mexists{}u,v:\{p:\mBbbR{}\^{}2| ab=ap \mwedge{} cd=cp\}
((\mdownarrow{}\mexists{}p,q:\mBbbR{}\^{}2
(((d(a;b) = d(a;p)) \mwedge{} (d(c;d) = d(c;q))) \mwedge{} (d(c;p) < d(c;d)) \mwedge{} (d(a;q) < d(a;b))))
{}\mRightarrow{} (r2-left(u;c;a) \mwedge{} r2-left(v;a;c)))))
Date html generated:
2017_10_03-AM-11_53_00
Last ObjectModification:
2017_08_13-PM-00_48_29
Theory : reals
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