Nuprl Lemma : rv-ge-dist
∀n:ℕ. ∀a,b,c,p:ℝ^n. (d(a;b) ≤ d(c;p) ⇐⇒ cp ≥ ab)
Proof
Definitions occuring in Statement :
rv-ge: cd ≥ ab,
real-vec-dist: d(x;y),
real-vec: ℝ^n,
rleq: x ≤ y,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
rv-ge: cd ≥ ab,
not: ¬A,
rv-be: a_b_c,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
true: True,
false: False,
exists: ∃x:A. B[x],
real-vec-sep: a ≠ b,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
real-vec-mul: a*X,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
nat: ℕ,
real-vec: ℝ^n,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
rdiv: (x/y),
real-vec-sub: X - Y,
rv-T: rv-T(n;a;b;c),
real-vec-be: real-vec-be(n;a;b;c),
cand: A c∧ B,
rv-congruent: ab=cd,
real-vec-dist: d(x;y),
satisfiable_int_formula: satisfiable_int_formula(fmla),
squash: ↓T,
less_than: a < b,
ge: i ≥ j ,
nat_plus: ℕ+,
sq_exists: ∃x:{A| B[x]},
rless: x < y
Lemmas referenced :
rleq_wf,
real-vec-dist_wf,
real_wf,
int-to-real_wf,
rv-ge_wf,
real-vec_wf,
nat_wf,
not_wf,
exists_wf,
real-vec-sep_wf,
rv-between_wf,
rv-congruent_wf,
false_wf,
or_wf,
true_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rv-be_wf,
real-vec-add_wf,
real-vec-mul_wf,
real-vec-sub_wf,
rdiv_wf,
rless_wf,
equal_wf,
rless_transitivity1,
int_seg_wf,
rmul_preserves_req,
radd_wf,
rmul_wf,
rsub_wf,
rminus_wf,
rinv_wf2,
req_functionality,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_transitivity,
itermAdd_wf,
itermConstant_wf,
itermMinus_wf,
real_term_value_add_lemma,
real_term_value_minus_lemma,
radd_functionality,
rminus_functionality,
rmul_functionality,
req_weakening,
rmul-rinv,
rmul-rinv3,
rv-T-iff,
i-member_wf,
rccint_wf,
req-vec_wf,
member_rccint_lemma,
rmul_preserves_rleq,
rleq_functionality,
radd-preserves-rleq,
real-vec-dist-nonneg,
not-real-vec-sep-iff-eq,
req-vec_weakening,
req-vec_functionality,
req-vec_inversion,
radd-preserves-req,
rv-be-dist,
real-vec-dist-symmetry,
real-vec-norm_wf,
rabs_wf,
real-vec-norm_functionality,
real-vec-norm-mul,
rabs-of-nonneg,
rv-congruent_functionality,
real-vec-dist-same-zero,
rless_functionality,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
intformless_wf,
satisfiable-full-omega-tt,
nat_properties,
nat_plus_properties,
not-rless,
req_inversion,
radd-zero-both,
rmul-zero-both,
radd-int,
rmul-distrib2,
rmul-identity1,
radd-assoc,
rminus-as-rmul,
rless_irreflexivity,
radd-preserves-rless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
natural_numberEquality,
sqequalRule,
because_Cache,
productEquality,
functionEquality,
independent_functionElimination,
unionElimination,
voidElimination,
dependent_pairFormation,
independent_isectElimination,
inrFormation,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
productElimination,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
applyLambdaEquality,
hyp_replacement,
existsLevelFunctionality,
andLevelFunctionality,
existsFunctionality,
addLevel,
impliesFunctionality,
imageElimination,
promote_hyp,
addEquality,
minusEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,p:\mBbbR{}\^{}n. (d(a;b) \mleq{} d(c;p) \mLeftarrow{}{}\mRightarrow{} cp \mgeq{} ab)
Date html generated:
2017_10_03-AM-11_33_37
Last ObjectModification:
2017_07_28-AM-08_27_59
Theory : reals
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