Nuprl Lemma : r2-left-convex
∀a,b,x,y,z:ℝ^2. (r2-left(x;a;b) ⇒ r2-left(z;a;b) ⇒ rv-T(2;x;y;z) ⇒ 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: 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,
iff: P ⇐⇒ Q,
cand: A c∧ B,
stable: Stable{P},
uimplies: b supposing a,
or: P ∨ Q,
rev_implies: P ⇐ Q,
rge: x ≥ y,
guard: {T},
rv-T: rv-T(n;a;b;c),
real-vec-be: real-vec-be(n;a;b;c),
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
top: Top,
req_int_terms: t1 ≡ t2,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B
Lemmas referenced :
rv-T_wf,
false_wf,
le_wf,
rless_wf,
int-to-real_wf,
r2-det_wf,
real-vec_wf,
stable__rleq,
rmin_wf,
or_wf,
real-vec-sep_wf,
not_wf,
rleq_wf,
rmin_strict_ub,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rless_functionality_wrt_implies,
rleq_weakening_equal,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
rleq_functionality,
req_weakening,
r2-det_functionality,
req-vec_weakening,
radd_wf,
rmul_wf,
r2-det-convex1,
rmin-rleq,
member_rccint_lemma,
rmul_preserves_rleq2,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_wf,
req_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
radd-preserves-rleq,
radd-zero,
itermMultiply_wf,
real_term_value_mul_lemma,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
rleq_weakening,
req_transitivity,
rmul_functionality,
real-vec-sep-symmetry,
not-real-vec-sep-iff-eq,
rmin_functionality
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,
because_Cache,
functionEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
unionElimination,
voidElimination,
isect_memberEquality,
voidEquality,
promote_hyp,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}a,b,x,y,z:\mBbbR{}\^{}2. (r2-left(x;a;b) {}\mRightarrow{} r2-left(z;a;b) {}\mRightarrow{} rv-T(2;x;y;z) {}\mRightarrow{} r2-left(y;a;b))
Date html generated:
2017_10_03-AM-11_54_50
Last ObjectModification:
2017_07_28-AM-08_29_39
Theory : reals
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