Nuprl Lemma : rv-weak-triangle-inequality
∀n:ℕ. ∀a,b,x,p:ℝ^n. (ax=ab ⇒ a-x-p ⇒ p ≠ b)
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
rv-between: a-b-c,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rge: x ≥ y,
guard: {T},
rless: x < y,
sq_exists: ∃x:{A| B[x]}
Lemmas referenced :
real-vec-dist-between,
rv-between_wf,
rv-congruent_wf,
real-vec_wf,
nat_wf,
real-vec-triangle-inequality,
real-vec-dist_wf,
real_wf,
rleq_wf,
int-to-real_wf,
radd_wf,
rleq_functionality,
req_weakening,
radd_functionality,
radd-preserves-rleq,
rminus_wf,
rmul_wf,
uiff_transitivity,
req_transitivity,
rminus-as-rmul,
radd-assoc,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
radd-zero-both,
rless_functionality,
real-vec-dist-symmetry,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rv-between-symmetry,
rv-between-sep
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
natural_numberEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
minusEquality,
addEquality,
independent_pairFormation
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,x,p:\mBbbR{}\^{}n. (ax=ab {}\mRightarrow{} a-x-p {}\mRightarrow{} p \mneq{} b)
Date html generated:
2016_10_26-AM-11_04_03
Last ObjectModification:
2016_10_21-AM-11_42_59
Theory : reals
Home
Index