Nuprl Lemma : trivial-rleq-radd
∀[a,d:ℝ]. (uiff(a ≤ (a + d);r0 ≤ d) ∧ uiff(a ≤ (d + a);r0 ≤ d))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
uimplies: b supposing a,
all: ∀x:A. B[x],
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rleq-implies-rleq,
int-to-real_wf,
radd_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rsub_wf,
less_than'_wf,
real_wf,
nat_plus_wf,
rleq_wf,
squash_wf,
true_wf,
radd_comm_eq,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_isectElimination,
dependent_functionElimination,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination
Latex:
\mforall{}[a,d:\mBbbR{}]. (uiff(a \mleq{} (a + d);r0 \mleq{} d) \mwedge{} uiff(a \mleq{} (d + a);r0 \mleq{} d))
Date html generated:
2017_10_03-AM-08_26_00
Last ObjectModification:
2017_07_28-AM-07_24_05
Theory : reals
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