Nuprl Lemma : Cauchy-Schwarz-proof2
∀[n:ℕ]. ∀[x,y:ℝ^n]. (|x⋅y| ≤ (||x|| * ||y||))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec: ℝ^n,
rleq: x ≤ y,
rabs: |x|,
rmul: a * b,
nat: ℕ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
stable: Stable{P},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
or: P ∨ Q,
rneq: x ≠ y,
guard: {T},
nat: ℕ,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
real-vec-sub: X - Y,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
rsub: x - y
Lemmas referenced :
stable__rleq,
rabs_wf,
dot-product_wf,
rmul_wf,
real-vec-norm_wf,
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
real-vec_wf,
nat_wf,
false_wf,
or_wf,
rneq_wf,
int-to-real_wf,
not_wf,
rleq_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
real-vec-norm-nonneg,
rless_transitivity1,
rless_irreflexivity,
square-rleq-implies,
rmul-nonneg-case1,
rnexp_wf,
le_wf,
rnexp2-nonneg,
rleq_functionality,
req_inversion,
rabs-rnexp,
req_weakening,
rabs-of-nonneg,
rnexp-rmul,
rmul_functionality,
real-vec-norm-squared,
rnexp-positive,
rless_functionality,
int_seg_wf,
req_wf,
radd_wf,
real-vec-mul_wf,
rdiv_wf,
rless_wf,
rminus_wf,
uiff_transitivity,
req_functionality,
radd_functionality,
radd_comm,
radd-rminus-assoc,
real-vec-add_wf,
real-vec-sub_wf,
dot-product-comm,
dot-product_functionality,
req-vec_inversion,
req_transitivity,
dot-product-linearity1,
real-vec-sub_functionality,
real-vec-mul_functionality,
req-vec_weakening,
rdiv_functionality,
dot-product-linearity1-sub,
rsub_functionality,
dot-product-linearity2,
rnexp_functionality,
real-vec-norm_functionality,
rmul-rdiv-cancel2,
rminus_functionality,
rmul_comm,
radd-assoc,
radd-ac,
rmul_preserves_req,
rmul-assoc,
rmul-ac,
rmul-rdiv-cancel,
rnexp2,
radd-preserves-rleq,
rminus-as-rmul,
rmul-identity1,
rmul-distrib2,
radd-int,
rmul-zero-both,
radd-zero-both,
rmul_preserves_rleq2,
not-rneq,
real-vec-norm-is-0,
rabs_functionality,
dot-product-zero,
rleq_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
minusEquality,
natural_numberEquality,
because_Cache,
isect_memberFormation,
dependent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
functionEquality,
independent_functionElimination,
lambdaFormation,
unionElimination,
independent_pairFormation,
dependent_set_memberEquality,
inrFormation,
addEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. (|x\mcdot{}y| \mleq{} (||x|| * ||y||))
Date html generated:
2016_10_26-AM-10_22_50
Last ObjectModification:
2016_10_02-PM-07_36_28
Theory : reals
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