Nuprl Lemma : real-vec-norm-dim1
∀[x:ℝ^1]. (||x|| = |x 0|)
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
real-vec: ℝ^n,
rabs: |x|,
req: x = y,
uall: ∀[x:A]. B[x],
apply: f a,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
dot-product: x⋅y,
subtract: n - m,
so_lambda: λ2x.t[x],
less_than: a < b,
squash: ↓T,
so_apply: x[s]
Lemmas referenced :
square-req-iff,
real-vec-norm_wf,
istype-void,
istype-le,
rabs_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
real-vec-norm-nonneg,
zero-rleq-rabs,
req_functionality,
rnexp_wf,
dot-product_wf,
real-vec-norm-squared,
rabs-rnexp2,
req_witness,
real-vec_wf,
rmul_wf,
rsum_single,
int_seg_properties,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
itermAdd_wf,
int_term_value_add_lemma,
int_seg_wf,
req_weakening,
rnexp2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
lambdaFormation_alt,
voidElimination,
hypothesis,
hypothesisEquality,
applyEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
universeIsType,
productIsType,
productElimination,
because_Cache,
setElimination,
rename,
imageElimination,
int_eqEquality,
addEquality
Latex:
\mforall{}[x:\mBbbR{}\^{}1]. (||x|| = |x 0|)
Date html generated:
2019_10_30-AM-08_08_26
Last ObjectModification:
2019_06_25-PM-03_20_09
Theory : reals
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