Nuprl Lemma : r2-left-separated
∀a,b,c,d:ℝ^2. (r2-left(a;c;d) ⇒ r2-left(b;d;c) ⇒ a ≠ b)
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
real-vec-sep: a ≠ b,
r2-left: r2-left(p;q;r),
real-vec-dist: d(x;y),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
uiff: uiff(P;Q),
uimplies: b supposing a,
rev_implies: P ⇐ Q,
rge: x ≥ y,
rgt: x > y,
guard: {T},
real-vec: ℝ^n,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
subtype_rel: A ⊆r B,
nequal: a ≠ b ∈ T ,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
real-vec-sub: X - Y,
r2-det: |pqr|,
eq_int: (i =z j),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
iff: P ⇐⇒ Q
Lemmas referenced :
rless_wf,
int-to-real_wf,
r2-det_wf,
real-vec_wf,
false_wf,
le_wf,
radd_wf,
trivial-rless-radd,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
radd_functionality_wrt_rless1,
dot-product_wf,
real-vec-sub_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
rsub_wf,
lelt_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_seg_wf,
rmul_wf,
nat_plus_properties,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
req_functionality,
req_weakening,
r2-dot-product,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rless_functionality,
Cauchy-Schwarz,
rabs_wf,
real-vec-norm_wf,
rless_transitivity1,
rleq_functionality,
rabs-of-nonneg,
rmul-is-positive,
real-vec-norm-nonneg,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
dependent_set_memberEquality,
independent_pairFormation,
because_Cache,
dependent_functionElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
lambdaEquality,
setElimination,
rename,
unionElimination,
equalityElimination,
applyEquality,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
approximateComputation,
intEquality,
isect_memberEquality,
voidEquality,
int_eqEquality
Latex:
\mforall{}a,b,c,d:\mBbbR{}\^{}2. (r2-left(a;c;d) {}\mRightarrow{} r2-left(b;d;c) {}\mRightarrow{} a \mneq{} b)
Date html generated:
2017_10_03-AM-11_54_25
Last ObjectModification:
2017_06_09-PM-03_53_25
Theory : reals
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