Nuprl Lemma : rv-sep-exists
∀n:{1...}. ∃a,b:ℝ^n. a ≠ b
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
int_upper: {i...},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
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-},
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
real-vec-sep: a ≠ b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
true: True,
uimplies: b supposing a,
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
real-vec-sub: X - Y,
dot-product: x⋅y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
eq_int: (i =z j),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
subtract: n - m
Lemmas referenced :
int-to-real_wf,
int_seg_wf,
ifthenelse_wf,
eq_int_wf,
real_wf,
real-vec-sep_wf,
int_upper_subtype_nat,
false_wf,
le_wf,
exists_wf,
real-vec_wf,
int_upper_wf,
real-vec-dist_wf,
rless-int,
rless_functionality,
req_weakening,
rsqrt_wf,
dot-product-nonneg,
real-vec-sub_wf,
dot-product_wf,
rleq_wf,
req_wf,
rleq-int,
rsqrt1,
req_functionality,
rsqrt_functionality,
rsum_wf,
subtract_wf,
rmul_wf,
rsub_wf,
radd_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,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rsum-split-first,
rsum_functionality,
intformeq_wf,
int_formula_prop_eq_lemma,
uiff_transitivity,
rmul_functionality,
rsub-int,
rmul-int,
radd_functionality,
rsum-constant,
rmul-zero-both,
radd_comm,
radd-zero-both
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,
applyEquality,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
setEquality,
productEquality,
addEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
multiplyEquality
Latex:
\mforall{}n:\{1...\}. \mexists{}a,b:\mBbbR{}\^{}n. a \mneq{} b
Date html generated:
2017_10_03-AM-11_15_23
Last ObjectModification:
2017_03_07-PM-00_02_21
Theory : reals
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