Nuprl Lemma : rmin-nonneg
∀[x,y:ℝ]. rnonneg(rmin(x;y)) supposing rnonneg(x) ∧ rnonneg(y)
Proof
Definitions occuring in Statement :
rnonneg: rnonneg(x),
rmin: rmin(x;y),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
rmin: rmin(x;y),
squash: ↓T,
prop: ℙ,
le: A ≤ B,
real: ℝ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
le_wf,
squash_wf,
true_wf,
imin_unfold,
iff_weakening_equal,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
nat_plus_wf,
less_than'_wf,
rmin_wf,
real_wf,
rnonneg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lambdaFormation,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
sqequalRule,
applyEquality,
lambdaEquality,
imageElimination,
extract_by_obid,
isectElimination,
equalityTransitivity,
equalitySymmetry,
intEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
independent_functionElimination,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
independent_pairEquality,
minusEquality,
axiomEquality,
productEquality,
isect_memberEquality
Latex:
\mforall{}[x,y:\mBbbR{}]. rnonneg(rmin(x;y)) supposing rnonneg(x) \mwedge{} rnonneg(y)
Date html generated:
2017_10_03-AM-08_24_39
Last ObjectModification:
2017_07_28-AM-07_23_26
Theory : reals
Home
Index