Nuprl Lemma : rleq-implies-rleq
∀[a,b,c,d:ℝ]. (a ≤ b) supposing ((c ≤ d) and ((d - c) = (b - a)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rsub: x - y,
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
rsub: x - y,
guard: {T}
Lemmas referenced :
radd-preserves-rleq,
rminus_wf,
radd-rminus-both,
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
rleq_wf,
req_wf,
radd_wf,
int-to-real_wf,
rleq_functionality,
req_weakening,
radd_comm,
req_inversion,
rleq_transitivity,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination
Latex:
\mforall{}[a,b,c,d:\mBbbR{}]. (a \mleq{} b) supposing ((c \mleq{} d) and ((d - c) = (b - a)))
Date html generated:
2017_10_03-AM-08_25_39
Last ObjectModification:
2017_04_04-PM-02_19_18
Theory : reals
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