Nuprl Lemma : real-vec-dist-dim1
∀[x,y:ℝ^1]. (d(x;y) = |(x 0) - y 0|)
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec: ℝ^n,
rabs: |x|,
rsub: x - y,
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,
real-vec-dist: d(x;y),
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
all: ∀x:A. B[x],
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,
real-vec-sub: X - Y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
real-vec-dist_wf,
istype-void,
istype-le,
rabs_wf,
rsub_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_wf,
real-vec-norm_wf,
real-vec-sub_wf,
subtype_rel_self,
int_seg_wf,
real_wf,
req_weakening,
req_functionality,
real-vec-norm-dim1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
lambdaFormation_alt,
voidElimination,
hypothesis,
hypothesisEquality,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
isect_memberEquality_alt,
universeIsType,
productIsType,
because_Cache,
isectIsTypeImplies,
functionEquality,
productElimination
Latex:
\mforall{}[x,y:\mBbbR{}\^{}1]. (d(x;y) = |(x 0) - y 0|)
Date html generated:
2019_10_30-AM-08_28_38
Last ObjectModification:
2019_06_25-PM-03_23_31
Theory : reals
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