Nuprl Lemma : rsub_functionality_wrt_rless
∀x,y,z,t:ℝ. ((x < z) ⇒ (x - y) < (z - t) supposing t ≤ y)
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
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,
rsub: x - y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
radd-preserves-rless,
rminus_wf,
rless_functionality,
radd_wf,
radd_comm,
rless_transitivity1,
rleq_wf,
rless_wf,
radd-preserves-rleq,
rminus-reverses-rleq,
rleq_functionality
Rules used in proof :
comment,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
voidElimination,
lemma_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_isectElimination,
because_Cache
Latex:
\mforall{}x,y,z,t:\mBbbR{}. ((x < z) {}\mRightarrow{} (x - y) < (z - t) supposing t \mleq{} y)
Date html generated:
2016_05_18-AM-07_09_47
Last ObjectModification:
2015_12_28-AM-00_38_55
Theory : reals
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