Nuprl Lemma : rv-weak-triangle-inequality2
∀n:ℕ. ∀a,b,x,p:ℝ^n. (ax=ab ⇒ a_x_p ⇒ bp ≥ xp)
Proof
Definitions occuring in Statement :
rv-be: a_b_c,
rv-ge: cd ≥ ab,
rv-congruent: ab=cd,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
prop: ℙ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rv-congruent: ab=cd,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
rv-ge-dist,
radd-zero-both,
rmul-zero-both,
radd-int,
rmul_functionality,
rmul-distrib2,
rmul-identity1,
req_inversion,
radd-assoc,
rminus-as-rmul,
req_transitivity,
uiff_transitivity,
radd_functionality,
req_weakening,
rleq_functionality,
rmul_wf,
rminus_wf,
radd-preserves-rleq,
radd_wf,
int-to-real_wf,
rleq_wf,
real_wf,
real-vec-dist_wf,
nat_wf,
real-vec_wf,
rv-congruent_wf,
rv-be_wf,
real-vec-triangle-inequality,
rv-be-dist
Rules used in proof :
dependent_functionElimination,
addEquality,
minusEquality,
independent_isectElimination,
productElimination,
sqequalRule,
natural_numberEquality,
setEquality,
rename,
setElimination,
lambdaEquality,
applyEquality,
because_Cache,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,x,p:\mBbbR{}\^{}n. (ax=ab {}\mRightarrow{} a\_x\_p {}\mRightarrow{} bp \mgeq{} xp)
Date html generated:
2016_10_28-AM-07_39_41
Last ObjectModification:
2016_10_27-PM-03_21_04
Theory : reals
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