Nuprl Lemma : rsub_functionality_wrt_rleq
∀[x,y,z,t:ℝ]. ((x - y) ≤ (z - t)) supposing ((y ≥ t) and (x ≤ z))
Proof
Definitions occuring in Statement :
rge: x ≥ y,
rleq: x ≤ y,
rsub: 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,
rsub: x - y,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T}
Lemmas referenced :
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
rge_wf,
rleq_wf,
radd_wf,
rminus_wf,
uiff_transitivity,
radd-preserves-rleq,
rmul_wf,
int-to-real_wf,
rleq_functionality,
req_transitivity,
radd_functionality,
rminus-as-rmul,
req_weakening,
radd-assoc,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
radd-zero-both,
rminus-reverses-rleq,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
rleq_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
because_Cache,
lemma_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
independent_functionElimination,
independent_isectElimination,
addEquality
Latex:
\mforall{}[x,y,z,t:\mBbbR{}]. ((x - y) \mleq{} (z - t)) supposing ((y \mgeq{} t) and (x \mleq{} z))
Date html generated:
2016_05_18-AM-07_08_47
Last ObjectModification:
2015_12_28-AM-00_39_43
Theory : reals
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