Nuprl Lemma : real-vec-between-inner-trans
∀n:ℕ. ∀a,b,c,d:ℝ^n. (a-b-d ⇒ b-c-d ⇒ a-b-c)
Proof
Definitions occuring in Statement :
real-vec-between: a-b-c,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
real-vec-between: a-b-c,
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
guard: {T},
iff: P ⇐⇒ Q,
cand: A c∧ B,
top: Top,
rsub: x - y,
rneq: x ≠ y,
or: P ∨ Q,
rooint: (l, u),
i-member: r ∈ I,
real-vec-mul: a*X,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
nat: ℕ,
real-vec: ℝ^n,
rev_uimplies: rev_uimplies(P;Q),
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
real_term_value: real_term_value(f;t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
itermAdd: left (+) right,
itermMinus: "-"num,
itermMultiply: left (*) right,
itermVar: vvar
Lemmas referenced :
exists_wf,
real_wf,
i-member_wf,
rooint_wf,
int-to-real_wf,
req-vec_wf,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
real-vec_wf,
nat_wf,
req-vec_functionality,
req-vec_weakening,
real-vec-add_functionality,
real-vec-mul_functionality,
req_weakening,
rmul-one-both,
rmul_comm,
rmul-zero-both,
rless_functionality,
rleq_weakening_rless,
rless_transitivity2,
rmul_wf,
rmul_preserves_rless,
member_rooint_lemma,
radd-preserves-rless,
radd_wf,
rminus_wf,
rless_wf,
radd-zero-both,
radd_functionality,
radd-rminus-both,
radd_comm,
radd-ac,
rminus_functionality,
rdiv_wf,
rmul-rdiv-cancel2,
rmul_over_rminus,
rmul-distrib,
req_transitivity,
rminus-zero,
equal_wf,
int_seg_wf,
req_functionality,
rmul_functionality,
req_wf,
uiff_transitivity,
rmul-distrib1,
rmul-assoc,
rminus-rminus,
req_inversion,
radd-assoc,
radd-rminus-assoc,
rmul-ac,
rminus-radd,
rminus-as-rmul,
rmul_preserves_req,
squash_wf,
true_wf,
iff_weakening_equal,
rdiv_functionality,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
radd-int,
rmul-distrib2,
rmul-identity1,
radd-preserves-req
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
lambdaEquality,
productEquality,
natural_numberEquality,
hypothesisEquality,
because_Cache,
rename,
independent_isectElimination,
addLevel,
existsFunctionality,
independent_pairFormation,
andLevelFunctionality,
promote_hyp,
independent_functionElimination,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
levelHypothesis,
dependent_pairFormation,
inrFormation,
equalityTransitivity,
equalitySymmetry,
setElimination,
applyEquality,
minusEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
computeAll,
int_eqEquality,
intEquality,
addEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,d:\mBbbR{}\^{}n. (a-b-d {}\mRightarrow{} b-c-d {}\mRightarrow{} a-b-c)
Date html generated:
2017_10_03-AM-10_48_31
Last ObjectModification:
2017_07_28-AM-08_20_10
Theory : reals
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