Nuprl Lemma : rstar-rleq
∀[x,y:ℝ]. ((x)* ≤ (y)* ⇐⇒ x ≤ y)
Proof
Definitions occuring in Statement :
rstar: (x)*,
rleq*: x ≤ y,
rleq: x ≤ y,
real: ℝ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
rev_implies: P ⇐ Q,
rleq*: x ≤ y,
rrel*: R*(x,y),
exists: ∃x:A. B[x],
rstar: (x)*,
int_upper: {i...},
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
rleq*_wf,
rstar_wf,
rleq_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
false_wf,
int_upper_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
voidElimination,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
because_Cache,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}[x,y:\mBbbR{}]. ((x)* \mleq{} (y)* \mLeftarrow{}{}\mRightarrow{} x \mleq{} y)
Date html generated:
2018_05_22-PM-03_18_28
Last ObjectModification:
2017_10_10-PM-01_58_37
Theory : reals_2
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