Nuprl Lemma : real-vec-sep-msep-prod-metric
∀n:ℕ. ∀a,c:ℝ^n. (a ≠ c ⇐⇒ a # c)
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
rn-prod-metric: rn-prod-metric(n),
real-vec: ℝ^n,
msep: x # y,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
nat_plus: ℕ+,
sq_exists: ∃x:A [B[x]],
rless: x < y,
so_apply: x[s],
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
squash: ↓T,
less_than: a < b,
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
prop: ℙ,
real-vec: ℝ^n,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
nat: ℕ,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
top: Top,
member: t ∈ T,
uall: ∀[x:A]. B[x],
msep: x # y
Lemmas referenced :
nat_plus_properties,
subtract-add-cancel,
zero-rleq-rabs,
rsum-of-nonneg-positive-iff,
real-vec-sep_wf,
real-vec-sep-iff,
istype-nat,
real-vec_wf,
istype-less_than,
istype-le,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermSubtract_wf,
itermAdd_wf,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
subtract_wf,
rsum_wf,
rsub_wf,
rabs_wf,
int-to-real_wf,
rless_wf,
int_seg_wf,
istype-void,
mdist-rn-prod-metric
Rules used in proof :
promote_hyp,
inhabitedIsType,
addEquality,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
imageElimination,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
because_Cache,
applyEquality,
productElimination,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
universeIsType,
productIsType,
independent_pairFormation,
lambdaFormation_alt,
hypothesis,
voidElimination,
isect_memberEquality_alt,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,c:\mBbbR{}\^{}n. (a \mneq{} c \mLeftarrow{}{}\mRightarrow{} a \# c)
Date html generated:
2019_11_06-PM-00_33_04
Last ObjectModification:
2019_11_05-AM-11_28_24
Theory : reals
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