Nuprl Lemma : not-ip-triangle-implies
∀rv:InnerProductSpace. ∀a,b,c:Point. ((¬Δ(a;b;c)) ⇒ (¬((¬a_b_c) ∧ (¬b_c_a) ∧ (¬c_a_b))))
Proof
Definitions occuring in Statement :
ip-triangle: Δ(a;b;c),
ip-between: a_b_c,
inner-product-space: InnerProductSpace,
ss-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
or: P ∨ Q,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
ml-term-to-poly: ml-term-to-poly(t),
nil: [],
it: ⋅,
has-value: (a)↓,
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q),
rge: x ≥ y,
rgt: x > y,
cand: A c∧ B,
rsub: x - y,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
rv-sub: x - y,
rv-minus: -x,
itermConstant: "const",
rooint: (l, u),
i-member: r ∈ I,
ss-eq: x ≡ y
Lemmas referenced :
not_wf,
ip-between_wf,
ip-triangle_wf,
ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
false_wf,
or_wf,
ss-sep_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
not-ip-triangle,
ip-between-iff,
exists_wf,
real_wf,
i-member_wf,
rooint_wf,
int-to-real_wf,
ss-eq_wf,
rv-add_wf,
rv-mul_wf,
rsub_wf,
ss-sep-symmetry,
rabs-positive-iff,
radd-preserves-rless,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
rless-int,
rless_functionality,
real_polynomial_null,
evalall-sqequal,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
req-iff-rsub-is-0,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
rdiv_wf,
rminus_wf,
rless_wf,
radd-ac,
radd-rminus-both,
rmul-one-both,
rmul-distrib2,
rmul-identity1,
req_inversion,
rminus-as-rmul,
radd-int,
rmul_functionality,
radd_comm,
rmul-rdiv-cancel2,
rmul_over_rminus,
rmul-distrib,
req_transitivity,
rmul-zero-both,
rminus_functionality,
rminus-zero,
radd_functionality,
req_weakening,
radd-zero-both,
rmul_wf,
rmul_preserves_rless,
member_rooint_lemma,
rv-sub_wf,
ss-eq_functionality,
ss-eq_weakening,
rv-add_functionality,
rv-mul_functionality,
rmul_preserves_req,
equal_wf,
rinv_wf2,
itermMultiply_wf,
itermMinus_wf,
req_functionality,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
rmul-rinv,
req_wf,
squash_wf,
true_wf,
iff_weakening_equal,
radd-preserves-req,
rv-minus_wf,
rv-0_wf,
uiff_transitivity,
rv-mul-linear,
rv-add-assoc,
rv-mul-mul,
rv-add-swap,
rv-mul-add-alt,
rv-mul-add,
ss-eq_transitivity,
rinv-as-rdiv,
rminus-rdiv,
rv-mul0,
rv-add-0,
rsub_functionality,
rv-mul1,
trivial-rsub-rless,
real_term_polynomial,
rless-implies-rless,
radd-assoc,
rminus-rminus,
rmul-int,
rminus-radd,
rmul-minus,
rv-mul-1-add-alt,
rv-add-comm,
rv-0-add,
rleq_antisymmetry,
not-rless,
ss-sep_functionality,
ip-between_functionality,
ip-between-trivial2,
ip-between-trivial
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
productEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
functionEquality,
unionElimination,
dependent_functionElimination,
productElimination,
addLevel,
impliesFunctionality,
independent_pairFormation,
lambdaEquality,
natural_numberEquality,
andLevelFunctionality,
impliesLevelFunctionality,
imageMemberEquality,
baseClosed,
computeAll,
sqleReflexivity,
mlComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
dependent_pairFormation,
inrFormation,
levelHypothesis,
addEquality,
minusEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
universeEquality,
multiplyEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point. ((\mneg{}\mDelta{}(a;b;c)) {}\mRightarrow{} (\mneg{}((\mneg{}a\_b\_c) \mwedge{} (\mneg{}b\_c\_a) \mwedge{} (\mneg{}c\_a\_b))))
Date html generated:
2017_10_05-AM-00_00_45
Last ObjectModification:
2017_07_28-AM-08_54_44
Theory : inner!product!spaces
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