Nuprl Lemma : rn-ip_wf
∀[n:{2...}]. (ipℝ^n ∈ InnerProductSpace)
Proof
Definitions occuring in Statement :
rn-ip: ipℝ^n,
inner-product-space: InnerProductSpace,
int_upper: {i...},
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
int_upper: {i...},
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
rn-ip: ipℝ^n,
rv-n: vecℝ^n,
ss-point: Error :ss-point,
ss-eq: Error :ss-eq,
rn-ss: sepℝ^n,
mk-real-vector-space: mk-real-vector-space,
ss-sep: Error :ss-sep,
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
mk-ss: Error :mk-ss,
btrue: tt,
rv-mul: a*x,
rv-add: x + y,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
cand: A c∧ B,
rv-0: 0,
so_lambda: λ2x.t[x],
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q)
Lemmas referenced :
rv-n_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,
mk-inner-product-space_wf,
rec_select_update_lemma,
istype-void,
dot-product_wf,
real-vec_wf,
upper_subtype_nat,
istype-false,
real-vec-sep_wf,
dot-product-comm,
dot-product-linearity1,
dot-product-linearity2,
real_wf,
req_wf,
real-vec-add_wf,
radd_wf,
real-vec-mul_wf,
rmul_wf,
real-vec-sep-0-iff,
subtype_rel_self,
all_wf,
iff_wf,
int-to-real_wf,
int_seg_wf,
rless_wf,
real-vec-perp-exists,
int_upper_wf,
exists_wf,
istype-int_upper,
not-real-vec-sep-iff-eq,
dot-product_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
because_Cache,
inhabitedIsType,
applyEquality,
lambdaFormation_alt,
functionIsType,
productElimination,
productIsType,
instantiate,
functionEquality,
closedConclusion,
productEquality,
axiomEquality
Latex:
\mforall{}[n:\{2...\}]. (ip\mBbbR{}\^{}n \mmember{} InnerProductSpace)
Date html generated:
2020_05_20-PM-01_10_56
Last ObjectModification:
2019_12_10-AM-00_34_43
Theory : inner!product!spaces
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