Nuprl Lemma : radd_functionality_wrt_rless2
∀x,y,z,t:ℝ. ((x < z) ⇒ (x + y) < (z + t) supposing y ≤ t)
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rless: x < y,
radd: a + b,
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q)
Lemmas referenced :
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
rless-iff-rpositive,
radd_wf,
rleq_wf,
rless_wf,
rpositive-radd2,
rminus_wf,
rpositive_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,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
voidElimination,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
independent_isectElimination,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}x,y,z,t:\mBbbR{}. ((x < z) {}\mRightarrow{} (x + y) < (z + t) supposing y \mleq{} t)
Date html generated:
2017_10_03-AM-08_25_20
Last ObjectModification:
2017_07_28-AM-07_23_51
Theory : reals
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