Nuprl Lemma : r2-left-right-lemma
∀a,b,x,y:ℝ^2. (r2-left(x;a;b) ⇒ r2-left(y;b;a) ⇒ (∃t:ℝ. ((t ∈ [r0, r1]) ∧ (|t*x + r1 - t*yab| = r0))))
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
r2-det: |pqr|,
real-vec-mul: a*X,
real-vec-add: X + Y,
real-vec: ℝ^n,
rccint: [l, u],
i-member: r ∈ I,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: 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,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
uiff: uiff(P;Q),
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
top: Top,
req_int_terms: t1 ≡ t2,
r2-det: |pqr|,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
rneq: x ≠ y,
or: P ∨ Q,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y)
Lemmas referenced :
rless_wf,
int-to-real_wf,
r2-det_wf,
real-vec_wf,
false_wf,
le_wf,
radd_wf,
rmul_wf,
rsub_wf,
lelt_wf,
rminus_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
itermMinus_wf,
req-iff-rsub-is-0,
req_wf,
exists_wf,
real_wf,
rleq_wf,
real-vec-mul_wf,
i-member_wf,
rccint_wf,
real-vec-add_wf,
req_functionality,
r2-det-add,
req_weakening,
radd_functionality,
r2-det-mul,
member_rccint_lemma,
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,
real_term_value_minus_lemma,
equal_wf,
rless_functionality,
radd-preserves-rless,
radd-zero,
radd-rminus,
rless_transitivity2,
rleq_weakening_rless,
rdiv_wf,
rmul_preserves_rleq,
rmul_preserves_req,
rmul-zero-both,
rinv_wf2,
trivial-rleq-radd,
rleq_functionality,
req_transitivity,
rminus_functionality,
rmul-rinv3
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,
productElimination,
because_Cache,
applyEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
lambdaEquality,
productEquality,
addLevel,
existsFunctionality,
andLevelFunctionality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
approximateComputation,
int_eqEquality,
intEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_pairFormation,
inrFormation
Latex:
\mforall{}a,b,x,y:\mBbbR{}\^{}2.
(r2-left(x;a;b) {}\mRightarrow{} r2-left(y;b;a) {}\mRightarrow{} (\mexists{}t:\mBbbR{}. ((t \mmember{} [r0, r1]) \mwedge{} (|t*x + r1 - t*yab| = r0))))
Date html generated:
2017_10_03-AM-11_56_28
Last ObjectModification:
2017_06_09-PM-05_47_42
Theory : reals
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