Nuprl Lemma : rv-T-dist
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-T(n;a;b;c) ⇒ (d(a;c) = (d(a;b) + d(b;c))))
Proof
Definitions occuring in Statement :
rv-T: rv-T(n;a;b;c),
real-vec-dist: d(x;y),
real-vec: ℝ^n,
req: x = y,
radd: a + b,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uiff: uiff(P;Q),
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
squash: ↓T,
less_than: a < b,
ge: i ≥ j ,
nat: ℕ,
nat_plus: ℕ+,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
real-vec-sep: a ≠ b,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
false: False,
true: True,
or: P ∨ Q,
not: ¬A,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
rv-T: rv-T(n;a;b;c)
Lemmas referenced :
radd-int,
real-vec-dist-same-zero,
radd_functionality,
req-vec_weakening,
real-vec-dist_functionality,
not-real-vec-sep-iff-eq,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformless_wf,
satisfiable-full-omega-tt,
nat_properties,
nat_plus_properties,
req_inversion,
req_weakening,
rneq_functionality,
real-vec-dist-be,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
true_wf,
or_wf,
false_wf,
nat_wf,
real-vec_wf,
not_wf,
real-vec-be_wf,
real-vec-sep_wf,
rneq_wf,
radd_wf,
int-to-real_wf,
rleq_wf,
real_wf,
real-vec-dist_wf,
not-rneq
Rules used in proof :
promote_hyp,
addEquality,
computeAll,
voidEquality,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
imageElimination,
dependent_functionElimination,
voidElimination,
unionElimination,
independent_functionElimination,
functionEquality,
productEquality,
independent_isectElimination,
because_Cache,
natural_numberEquality,
setEquality,
rename,
setElimination,
lambdaEquality,
applyEquality,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}\^{}n. (rv-T(n;a;b;c) {}\mRightarrow{} (d(a;c) = (d(a;b) + d(b;c))))
Date html generated:
2016_10_28-AM-07_29_27
Last ObjectModification:
2016_10_27-PM-00_45_42
Theory : reals
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