Nuprl Lemma : real-vec-norm-equal-iff
∀[n:ℕ]. ∀[x,y:ℝ^n]. uiff(||x|| = ||y||;x⋅x = y⋅y)
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec: ℝ^n,
req: x = y,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
real: ℝ
Lemmas referenced :
iff_weakening_uiff,
req_wf,
real-vec-norm_wf,
dot-product_wf,
rnexp_wf,
false_wf,
le_wf,
rleq_wf,
int-to-real_wf,
real-vec-norm-eq-iff,
req_witness,
uiff_wf,
real-vec_wf,
nat_wf,
real-vec-norm-nonneg,
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
req_functionality,
req_weakening,
real-vec-norm-squared
Rules used in proof :
cut,
addLevel,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
independent_pairFormation,
isect_memberFormation,
introduction,
independent_isectElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
productEquality,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
lambdaFormation,
independent_functionElimination,
cumulativity,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
dependent_functionElimination,
voidElimination,
applyEquality,
setElimination,
rename,
minusEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. uiff(||x|| = ||y||;x\mcdot{}x = y\mcdot{}y)
Date html generated:
2017_10_03-AM-10_49_37
Last ObjectModification:
2017_06_08-PM-06_52_00
Theory : reals
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