Nuprl Lemma : rv-nontrivial
∀n:{2...}. ∃a,b,c:ℝ^n. (a ≠ b ∧ b ≠ c ∧ c ≠ a ∧ (¬a-b-c) ∧ (¬b-c-a) ∧ (¬c-a-b))
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
int_upper: {i...},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
real-vec: ℝ^n,
uall: ∀[x:A]. B[x],
int_upper: {i...},
int_seg: {i..j-},
and: P ∧ Q,
cand: A c∧ B,
not: ¬A,
implies: P ⇒ Q,
rv-between: a-b-c,
real-vec-between: a-b-c,
top: Top,
req-vec: req-vec(n;x;y),
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
prop: ℙ,
real-vec-sep: a ≠ b,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
real-vec-mul: a*X,
real-vec-add: X + Y,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rsub: x - y,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
true: True,
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
rev_uimplies: rev_uimplies(P;Q),
real-vec-sub: X - Y,
dot-product: x⋅y,
bool: 𝔹,
unit: Unit,
it: ⋅,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
subtract: n - m,
real: ℝ,
sq_stable: SqStable(P),
nequal: a ≠ b ∈ T
Lemmas referenced :
int-to-real_wf,
int_seg_wf,
ifthenelse_wf,
eq_int_wf,
real_wf,
member_rooint_lemma,
false_wf,
nat_plus_properties,
int_upper_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
rv-between_wf,
int_upper_subtype_nat,
le_wf,
real-vec-sep_wf,
not_wf,
exists_wf,
real-vec_wf,
int_upper_wf,
req_wf,
radd_wf,
rmul_wf,
rminus_wf,
rless_transitivity2,
rleq_weakening_rless,
rless_transitivity1,
rleq_weakening,
rless_irreflexivity,
req_inversion,
uiff_transitivity,
req_functionality,
req_weakening,
radd_functionality,
req_transitivity,
rmul-distrib,
rmul_over_rminus,
rminus_functionality,
rmul-zero-both,
rminus-zero,
radd-zero-both,
rmul-one-both,
radd-ac,
radd_comm,
real-vec-dist_wf,
rless-int,
rless_functionality,
rsqrt_wf,
dot-product-nonneg,
real-vec-sub_wf,
dot-product_wf,
rleq_wf,
rleq-int,
rsqrt1,
rsqrt_functionality,
rsum_wf,
subtract_wf,
rsub_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
rsum-split-first,
rsum_functionality,
intformeq_wf,
int_formula_prop_eq_lemma,
rmul_functionality,
rsub-int,
rmul-int,
rsum-constant,
sq_stable__less_than,
radd-assoc,
rmul-identity1,
rmul-distrib2,
radd-int,
rsqrt-positive,
rless_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_pairFormation,
sqequalRule,
lambdaEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
setElimination,
rename,
hypothesisEquality,
because_Cache,
independent_pairFormation,
productElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
addLevel,
dependent_set_memberEquality,
unionElimination,
independent_isectElimination,
int_eqEquality,
intEquality,
computeAll,
levelHypothesis,
applyEquality,
productEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
setEquality,
addEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
multiplyEquality,
imageElimination
Latex:
\mforall{}n:\{2...\}. \mexists{}a,b,c:\mBbbR{}\^{}n. (a \mneq{} b \mwedge{} b \mneq{} c \mwedge{} c \mneq{} a \mwedge{} (\mneg{}a-b-c) \mwedge{} (\mneg{}b-c-a) \mwedge{} (\mneg{}c-a-b))
Date html generated:
2017_10_03-AM-11_15_03
Last ObjectModification:
2017_07_28-AM-08_24_54
Theory : reals
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