Nuprl Lemma : rleq*_weakening_rless
∀[x,y:ℝ*]. (x < y ⇒ x ≤ y)
Proof
Definitions occuring in Statement :
rleq*: x ≤ y,
rless*: x < y,
real*: ℝ*,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
rless*: x < y,
rrel*: R*(x,y),
exists: ∃x:A. B[x],
rleq*: x ≤ y,
member: t ∈ T,
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
real*: ℝ*,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rge: x ≥ y,
guard: {T},
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
real: ℝ,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
rleq_functionality_wrt_implies,
int_upper_subtype_nat,
rleq_weakening_rless,
rleq_weakening_equal,
rleq_weakening,
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
int_upper_wf,
all_wf,
rleq_wf,
rless*_wf,
real*_wf,
itermSubtract_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
int-to-real_wf,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
sqequalRule,
hypothesisEquality,
introduction,
cut,
extract_by_obid,
isectElimination,
applyEquality,
because_Cache,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
independent_pairEquality,
voidElimination,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}[x,y:\mBbbR{}*]. (x < y {}\mRightarrow{} x \mleq{} y)
Date html generated:
2018_05_22-PM-03_19_48
Last ObjectModification:
2017_10_06-PM-05_12_19
Theory : reals_2
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