Nuprl Lemma : r2-left-between
∀a,b,x,y:ℝ^2. (r2-left(x;a;b) ⇒ rv-T(2;b;y;x) ⇒ b ≠ y ⇒ r2-left(y;a;b))
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
rv-T: rv-T(n;a;b;c),
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
r2-left: r2-left(p;q;r),
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
cand: A c∧ B,
rv-congruent: ab=cd,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rv-T: rv-T(n;a;b;c),
iff: P ⇐⇒ Q,
real-vec-be: real-vec-be(n;a;b;c),
top: Top,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
r2-det: |pqr|,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real-vec-sep: a ≠ b,
or: P ∨ Q,
rev_implies: P ⇐ Q,
rneq: x ≠ y,
guard: {T}
Lemmas referenced :
real-vec-sep_wf,
false_wf,
le_wf,
rv-T_wf,
rless_wf,
int-to-real_wf,
r2-det_wf,
real-vec_wf,
rv-Tsep-alt,
not_wf,
exists_wf,
rv-congruent_wf,
rv-congruent-sym,
req_weakening,
real-vec-dist_wf,
real-vec-sep-symmetry,
member_rccint_lemma,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
rmul_wf,
radd_wf,
lelt_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
req_functionality,
r2-det_functionality,
req-vec_weakening,
r2-det-add,
radd_functionality,
r2-det-mul,
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,
rmul_functionality,
rabs_wf,
rless_functionality,
real-vec-dist_functionality,
real-vec-dist-between-1,
rmul-is-positive,
rabs-positive-iff,
rless-implies-rless,
radd-preserves-rless,
radd-zero,
rless_transitivity2,
rless_transitivity1,
zero-rleq-rabs,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
because_Cache,
productEquality,
dependent_pairFormation,
applyEquality,
independent_isectElimination,
productElimination,
isect_memberEquality,
voidEquality,
imageMemberEquality,
baseClosed,
approximateComputation,
int_eqEquality,
intEquality,
promote_hyp,
unionElimination,
inlFormation
Latex:
\mforall{}a,b,x,y:\mBbbR{}\^{}2. (r2-left(x;a;b) {}\mRightarrow{} rv-T(2;b;y;x) {}\mRightarrow{} b \mneq{} y {}\mRightarrow{} r2-left(y;a;b))
Date html generated:
2017_10_03-AM-11_55_41
Last ObjectModification:
2017_06_14-PM-04_32_43
Theory : reals
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