Nuprl Lemma : rv-Tsep
∀n:ℕ. ∀a,b:ℝ^n. ∀c:{c:ℝ^n| ¬(b ≠ c ∧ (¬a-b-c))} . (a ≠ b ⇒ a ≠ c)
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
real-vec-sep: a ≠ b,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
rge: x ≥ y,
guard: {T},
rleq: x ≤ y,
rnonneg: rnonneg(x),
stable: Stable{P},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
le: A ≤ B,
sq_stable: SqStable(P),
squash: ↓T,
rv-between: a-b-c,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B
Lemmas referenced :
real-vec-sep_wf,
set_wf,
real-vec_wf,
not_wf,
rv-between_wf,
nat_wf,
int-to-real_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
rless_functionality_wrt_implies,
rleq_weakening_equal,
decidable__le,
rsub_wf,
all_wf,
nat_plus_wf,
le_wf,
less_than'_wf,
sq_stable__rleq,
false_wf,
or_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
real-vec-dist-between,
radd_wf,
radd-preserves-rleq,
rminus_wf,
rmul_wf,
real-vec-dist-nonneg,
rleq_functionality,
req_weakening,
uiff_transitivity,
req_transitivity,
radd_functionality,
rminus-as-rmul,
req_inversion,
rmul-identity1,
rmul-distrib2,
radd-assoc,
rmul_functionality,
radd-int,
rmul-zero-both,
radd-zero-both,
not-real-vec-sep-iff-eq,
real-vec-dist_functionality,
req-vec_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
natural_numberEquality,
applyEquality,
setElimination,
rename,
setEquality,
because_Cache,
dependent_functionElimination,
independent_isectElimination,
independent_functionElimination,
isect_memberFormation,
minusEquality,
unionElimination,
voidElimination,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
functionEquality,
addEquality,
independent_pairFormation,
addLevel,
impliesFunctionality,
impliesLevelFunctionality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b:\mBbbR{}\^{}n. \mforall{}c:\{c:\mBbbR{}\^{}n| \mneg{}(b \mneq{} c \mwedge{} (\mneg{}a-b-c))\} . (a \mneq{} b {}\mRightarrow{} a \mneq{} c)
Date html generated:
2016_10_26-AM-10_38_06
Last ObjectModification:
2016_09_25-AM-01_06_33
Theory : reals
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