Nuprl Lemma : r2-left-extend
∀a,b,x,y:ℝ^2. (r2-left(x;a;b) ⇒ rv-T(2;b;x;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: 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,
guard: {T},
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
rv-T: rv-T(n;a;b;c),
real-vec-be: real-vec-be(n;a;b;c),
exists: ∃x:A. B[x],
top: Top,
uiff: uiff(P;Q),
stable: Stable{P},
or: P ∨ Q,
rev_uimplies: rev_uimplies(P;Q),
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
r2-det: |pqr|,
req_int_terms: t1 ≡ t2,
iff: P ⇐⇒ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
subtype_rel: A ⊆r B,
real: ℝ
Lemmas referenced :
rless_transitivity1,
int-to-real_wf,
r2-det_wf,
rv-T_wf,
false_wf,
le_wf,
rless_wf,
real-vec_wf,
stable__rleq,
or_wf,
real-vec-sep_wf,
not_wf,
rleq_wf,
member_rccint_lemma,
not-real-vec-sep-iff-eq,
rleq_weakening_equal,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rleq_functionality,
r2-det_functionality,
req-vec_weakening,
req_weakening,
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-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,
rleq_weakening,
rmul-is-positive,
trivial-rsub-rleq,
rmul_preserves_rleq2,
rleq_weakening_rless,
less_than'_wf,
real_wf,
nat_plus_wf,
rminus_wf,
rmul-identity1,
rleq-implies-rleq,
itermMinus_wf,
real_term_value_minus_lemma,
radd-preserves-rless,
rless_functionality,
radd-zero,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesisEquality,
independent_functionElimination,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairFormation,
because_Cache,
functionEquality,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
applyEquality,
imageMemberEquality,
baseClosed,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberFormation,
independent_pairEquality,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}a,b,x,y:\mBbbR{}\^{}2. (r2-left(x;a;b) {}\mRightarrow{} rv-T(2;b;x;y) {}\mRightarrow{} r2-left(y;a;b))
Date html generated:
2017_10_03-AM-11_55_17
Last ObjectModification:
2017_06_09-PM-05_13_30
Theory : reals
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