Nuprl Lemma : rless_functionality_wrt_implies
∀a,b,c,d:ℝ. ({(b < c) ⇒ (a < d)}) supposing ((c ≤ d) and (b ≥ a))
Proof
Definitions occuring in Statement :
rge: x ≥ y,
rleq: x ≤ y,
rless: x < y,
real: ℝ,
uimplies: b supposing a,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
guard: {T},
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
rge: x ≥ y,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ
Lemmas referenced :
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
rless_transitivity2,
rless_transitivity1,
rless_wf,
rleq_wf,
rge_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
voidElimination,
lemma_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}a,b,c,d:\mBbbR{}. (\{(b < c) {}\mRightarrow{} (a < d)\}) supposing ((c \mleq{} d) and (b \mgeq{} a))
Date html generated:
2016_05_18-AM-07_06_10
Last ObjectModification:
2015_12_28-AM-00_37_13
Theory : reals
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