Nuprl Lemma : ip-between-same
∀[rv:InnerProductSpace]. ∀[a,b:Point]. (a_b_a ⇒ b ≡ a)
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
inner-product-space: InnerProductSpace,
ss-eq: x ≡ y,
ss-point: Point,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uiff: uiff(P;Q),
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
false: False,
not: ¬A,
ss-eq: x ≡ y,
prop: ℙ,
ip-between: a_b_c,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
top: Top,
req_int_terms: t1 ≡ t2,
has-value: (a)↓,
it: ⋅,
nil: [],
ml-term-to-poly: Error :ml-term-to-poly,
label: ...$L... t,
rev_uimplies: rev_uimplies(P;Q),
true: True,
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
rneq: x ≠ y
Lemmas referenced :
req_weakening,
rnexp2,
rv-norm-squared,
req_inversion,
radd_functionality,
req_functionality,
int-to-real_wf,
le_wf,
false_wf,
rnexp_wf,
rv-ip_wf,
rv-sub_wf,
rv-norm_wf,
rmul_wf,
radd_wf,
ss-point_wf,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
real-vector-space_subtype1,
ss-sep_wf,
ip-between_wf,
req-iff-rsub-is-0,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
evalall-sqequal,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
itermAdd_wf,
itermSubtract_wf,
real_polynomial_null,
rmul-zero-both,
rmul_comm,
req_wf,
rleq_wf,
real_wf,
rless_wf,
rless-int,
rmul_preserves_req,
rv-0_wf,
ss-eq_wf,
iff_transitivity,
iff_weakening_uiff,
square-is-zero,
uiff_transitivity,
rv-norm-is-zero,
rv-sub-is-zero,
ss-eq_inversion
Rules used in proof :
productElimination,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
voidElimination,
isect_memberEquality,
independent_isectElimination,
instantiate,
applyEquality,
because_Cache,
dependent_functionElimination,
lambdaEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
sqequalHypSubstitution,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidEquality,
intEquality,
int_eqEquality,
mlComputation,
sqleReflexivity,
computeAll,
productEquality,
setEquality,
rename,
setElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
inrFormation
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b:Point]. (a\_b\_a {}\mRightarrow{} b \mequiv{} a)
Date html generated:
2018_05_22-PM-09_31_47
Last ObjectModification:
2018_05_18-PM-04_43_18
Theory : inner!product!spaces
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