Nuprl Lemma : rless-implies-rless
∀a,b:ℝ. ∀[c,d:ℝ]. (a < b) supposing ((c < d) and ((d - c) = (b - a)))
Proof
Definitions occuring in Statement :
rless: x < y,
rsub: x - y,
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
sq_stable: SqStable(P),
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
prop: ℙ,
rsub: x - y,
guard: {T}
Lemmas referenced :
req_witness,
rsub_wf,
sq_stable__rless,
radd-preserves-rless,
rminus_wf,
radd-rminus-both,
rless_wf,
req_wf,
real_wf,
radd_wf,
int-to-real_wf,
rless_functionality,
req_weakening,
radd_comm,
req_inversion,
rless_transitivity1,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
rename,
dependent_functionElimination,
because_Cache,
productElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
natural_numberEquality,
independent_isectElimination
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}[c,d:\mBbbR{}]. (a < b) supposing ((c < d) and ((d - c) = (b - a)))
Date html generated:
2017_10_03-AM-08_25_32
Last ObjectModification:
2017_04_04-PM-02_18_41
Theory : reals
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