Nuprl Lemma : rabs-nonneg
∀[x:ℝ]. rnonneg(|x|)
Proof
Definitions occuring in Statement :
rnonneg: rnonneg(x),
rabs: |x|,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
rnonneg2: rnonneg2(x),
all: ∀x:A. B[x],
rabs: |x|,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
so_lambda: λ2x.t[x],
int_upper: {i...},
real: ℝ,
le: A ≤ B,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_apply: x[s],
rnonneg: rnonneg(x),
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
int_upper_properties,
nat_plus_subtype_nat,
mul_preserves_le,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
false_wf,
decidable__lt,
absval-non-neg,
real_wf,
less_than'_wf,
nat_plus_wf,
less_than_transitivity1,
absval_wf,
le_wf,
all_wf,
int_upper_wf,
less_than_wf,
rabs_wf,
rnonneg-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
lambdaFormation,
sqequalRule,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
setElimination,
rename,
lambdaEquality,
multiplyEquality,
minusEquality,
applyEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
independent_pairEquality,
voidElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
isect_memberEquality,
voidEquality,
intEquality,
int_eqEquality,
computeAll
Latex:
\mforall{}[x:\mBbbR{}]. rnonneg(|x|)
Date html generated:
2016_05_18-AM-07_02_41
Last ObjectModification:
2016_01_17-AM-01_49_48
Theory : reals
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