Nuprl Lemma : rv-Tsep-alt
∀n:ℕ. ∀a,b,c,d:ℝ^n. ((¬¬(∃u,v,w:ℝ^n. (rv-T(n;u;v;w) ∧ ab=uv ∧ cd=uw))) ⇒ a ≠ b ⇒ c ≠ d)
Proof
Definitions occuring in Statement :
rv-T: rv-T(n;a;b;c),
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
real-vec-sep: a ≠ b,
member: t ∈ T,
guard: {T},
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
not: ¬A,
exists: ∃x:A. B[x],
false: False,
rv-T: rv-T(n;a;b;c),
or: P ∨ Q,
true: True,
rv-congruent: ab=cd,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rge: x ≥ y,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
rless_transitivity1,
int-to-real_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
real-vec-sep_wf,
not_wf,
exists_wf,
real-vec_wf,
rv-T_wf,
rv-congruent_wf,
nat_wf,
not-rless,
rless_wf,
false_wf,
or_wf,
true_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
real-vec-dist-be,
radd_wf,
req_functionality,
req_inversion,
radd_functionality,
req_weakening,
rless_functionality,
real-vec-dist-nonneg,
rless_functionality_wrt_implies,
radd_functionality_wrt_rleq,
rleq_weakening_equal,
nat_plus_properties,
nat_properties,
satisfiable-full-omega-tt,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
radd_comm,
radd-zero-both,
not-real-vec-sep-iff-eq,
rv-congruent_functionality,
req-vec_weakening,
real-vec-dist-same-zero,
rleq_weakening,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesisEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
sqequalRule,
because_Cache,
independent_functionElimination,
independent_isectElimination,
productEquality,
productElimination,
voidElimination,
functionEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
promote_hyp
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,d:\mBbbR{}\^{}n. ((\mneg{}\mneg{}(\mexists{}u,v,w:\mBbbR{}\^{}n. (rv-T(n;u;v;w) \mwedge{} ab=uv \mwedge{} cd=uw))) {}\mRightarrow{} a \mneq{} b {}\mRightarrow{} c \mneq{} d)
Date html generated:
2016_10_28-AM-07_29_46
Last ObjectModification:
2016_10_26-PM-02_08_56
Theory : reals
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