Nuprl Lemma : radd_functionality_wrt_rleq
∀[x,y,z,t:ℝ]. ((x + y) ≤ (z + t)) supposing ((y ≤ t) and (x ≤ z))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
radd: a + b,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
rleq: x ≤ y,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rnonneg-radd,
rsub_wf,
less_than'_wf,
radd_wf,
real_wf,
nat_plus_wf,
rnonneg_wf,
rminus_wf,
rnonneg_functionality,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMinus_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,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
radd_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
lambdaEquality,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
independent_isectElimination,
computeAll,
int_eqEquality,
intEquality,
voidEquality
Latex:
\mforall{}[x,y,z,t:\mBbbR{}]. ((x + y) \mleq{} (z + t)) supposing ((y \mleq{} t) and (x \mleq{} z))
Date html generated:
2017_10_03-AM-08_25_09
Last ObjectModification:
2017_07_28-AM-07_23_43
Theory : reals
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