Nuprl Lemma : radd-preserves-rleq
∀[x,y,z:ℝ]. uiff(x ≤ y;(z + x) ≤ (z + y))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
radd: a + b,
real: ℝ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
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
Lemmas referenced :
rleq_functionality_wrt_implies,
radd_wf,
radd_functionality_wrt_rleq,
rleq_weakening_equal,
rleq_weakening,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
int-to-real_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,
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
rleq_wf,
rminus_wf,
rmul_wf,
rleq_functionality,
req_transitivity,
itermMinus_wf,
itermMultiply_wf,
itermConstant_wf,
real_term_value_minus_lemma,
real_term_value_mul_lemma,
rmul-identity1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
natural_numberEquality,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_pairEquality,
applyEquality,
setElimination,
rename,
minusEquality,
axiomEquality
Latex:
\mforall{}[x,y,z:\mBbbR{}]. uiff(x \mleq{} y;(z + x) \mleq{} (z + y))
Date html generated:
2017_10_03-AM-08_25_26
Last ObjectModification:
2017_07_28-AM-07_23_54
Theory : reals
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