Nuprl Lemma : real-vec-right-angle-lemma
∀[n:ℕ]. ∀[x,z:ℝ^n]. uiff(x⋅z = r0;d(z;x) = d(z;r(-1)*x))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
dot-product: x⋅y,
real-vec-mul: a*X,
real-vec: ℝ^n,
req: x = y,
int-to-real: r(n),
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
real-vec_wf,
nat_wf,
req_witness,
rsub_wf,
dot-product_wf,
rmul_wf,
int-to-real_wf,
req_wf,
real-vec-mul_wf,
uiff_wf,
real-vec-sub_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
iff_weakening_uiff,
real-vec-dist-equal-iff,
req_functionality,
req_transitivity,
dot-product-linearity1-sub,
rsub_functionality,
req_weakening,
dot-product-linearity2,
rmul_functionality,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
dot-product-comm,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
radd-preserves-req,
radd_wf,
itermAdd_wf,
real_term_value_add_lemma,
rmul_preserves_req,
rless-int,
rless_wf,
rmul_comm,
rmul-zero-both,
req_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
because_Cache,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_pairFormation,
minusEquality,
natural_numberEquality,
independent_functionElimination,
cumulativity,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
sqequalRule,
addLevel,
productElimination,
independent_isectElimination,
dependent_functionElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
inrFormation,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,z:\mBbbR{}\^{}n]. uiff(x\mcdot{}z = r0;d(z;x) = d(z;r(-1)*x))
Date html generated:
2018_05_22-PM-02_26_27
Last ObjectModification:
2018_03_26-AM-09_56_25
Theory : reals
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