Nuprl Lemma : real-vec-sep-cases
∀n:ℕ. ∀a,b:ℝ^n. (a ≠ b ⇒ (∀c:ℝ^n. (a ≠ c ∨ b ≠ c)))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
real-vec-sep: a ≠ b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q),
rdiv: (x/y),
false: False,
not: ¬A,
rge: x ≥ y,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
nat: ℕ,
nat_plus: ℕ+,
sq_exists: ∃x:{A| B[x]},
rless: x < y
Lemmas referenced :
real-vec_wf,
rless_wf,
int-to-real_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
nat_wf,
rless-cases,
rdiv_wf,
rless-int,
rmul_preserves_rless,
rmul_wf,
rinv_wf2,
rless_functionality,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
req-iff-rsub-is-0,
req_transitivity,
itermVar_wf,
real_term_value_var_lemma,
rmul-rinv3,
real-vec-triangle-inequality,
radd_wf,
rleq_functionality,
req_weakening,
radd_functionality,
real-vec-dist-symmetry,
radd-preserves-rless,
itermAdd_wf,
real_term_value_add_lemma,
rinv-mul-as-rdiv,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
radd_functionality_wrt_rless2,
radd-rdiv,
rmul_comm,
rmul-rdiv-cancel2,
radd-int,
rmul-distrib2,
rmul-identity1,
req_inversion,
rdiv_functionality,
rmul_functionality,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
intformless_wf,
satisfiable-full-omega-tt,
nat_properties,
nat_plus_properties
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
dependent_functionElimination,
because_Cache,
independent_isectElimination,
inrFormation,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
unionElimination,
inlFormation,
computeAll,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_pairFormation,
imageElimination,
addEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b:\mBbbR{}\^{}n. (a \mneq{} b {}\mRightarrow{} (\mforall{}c:\mBbbR{}\^{}n. (a \mneq{} c \mvee{} b \mneq{} c)))
Date html generated:
2017_10_03-AM-10_59_46
Last ObjectModification:
2017_07_28-AM-08_22_15
Theory : reals
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