Nuprl Lemma : rn-ip-between
∀n:{2...}. ∀a,b,c:ℝ^n. (a_b_c ⇐⇒ rv-T(n;a;b;c))
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
rn-ip: ipℝ^n,
rv-T: rv-T(n;a;b;c),
real-vec: ℝ^n,
int_upper: {i...},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
ss-point: Error :ss-point,
record-select: r.x,
rn-ip: ipℝ^n,
mk-inner-product-space: mk-inner-product-space,
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
rv-n: vecℝ^n,
mk-real-vector-space: mk-real-vector-space,
rn-ss: sepℝ^n,
mk-ss: Error :mk-ss,
btrue: tt,
real-vec: ℝ^n,
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_upper: {i...},
ss-eq: Error :ss-eq,
top: Top,
ss-sep: Error :ss-sep,
real-vec-sep: a ≠ b,
rless: x < y,
sq_exists: ∃x:A [B[x]],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
rv-T: rv-T(n;a;b;c),
nat_plus: ℕ+,
le: A ≤ B,
less_than': less_than'(a;b),
real-vec-be: real-vec-be(n;a;b;c),
cand: A c∧ B,
i-member: r ∈ I,
rccint: [l, u],
real-vec-add: X + Y,
rv-add: x + y,
radd: a + b,
accelerate: accelerate(k;f),
real-vec-mul: a*X,
rv-mul: a*x
Lemmas referenced :
int_seg_wf,
real_wf,
istype-int_upper,
member_rccint_lemma,
istype-void,
istype-top,
ip-between-iff2,
rn-ip_wf,
ip-between_wf,
subtype_rel_self,
rv-T_wf,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
real-vec-sep_wf,
rleq_wf,
int-to-real_wf,
nat_plus_properties,
rv-add_wf,
inner-product-space_subtype,
rv-mul_wf,
rsub_wf,
real-vec-be_wf,
iff_weakening_uiff,
not_wf,
req-vec_wf,
not-real-vec-sep-iff-eq,
upper_subtype_nat,
istype-false,
i-member_wf,
rccint_wf,
real-vec-add_wf,
real-vec-mul_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
functionEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
universeIsType,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
inhabitedIsType,
productElimination,
independent_functionElimination,
hyp_replacement,
applyEquality,
independent_pairFormation,
promote_hyp,
dependent_set_memberEquality_alt,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
functionIsType,
productIsType
Latex:
\mforall{}n:\{2...\}. \mforall{}a,b,c:\mBbbR{}\^{}n. (a\_b\_c \mLeftarrow{}{}\mRightarrow{} rv-T(n;a;b;c))
Date html generated:
2020_05_20-PM-01_13_44
Last ObjectModification:
2019_12_10-AM-00_34_39
Theory : inner!product!spaces
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