Nuprl Lemma : rv-orthogonal-iff
∀[rv:InnerProductSpace]. ∀f:Point(rv) ⟶ Point(rv). (Orthogonal(f) ⇐⇒ f 0 ≡ 0 ∧ Isometry(f))
Proof
Definitions occuring in Statement :
rv-isometry: Isometry(f),
rv-orthogonal: Orthogonal(f),
inner-product-space: InnerProductSpace,
rv-0: 0,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
ss-eq: Error :ss-eq,
not: ¬A,
false: False,
rv-isometry: Isometry(f),
rv-orthogonal: Orthogonal(f),
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
or: P ∨ Q,
rneq: x ≠ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
nat: ℕ,
decidable: Dec(P),
nat_plus: ℕ+,
rv-sub: x - y,
rv-minus: -x,
pi2: snd(t),
rtermConstant: "const",
rtermMultiply: left "*" right,
pi1: fst(t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rat_term_to_real: rat_term_to_real(f;t),
stable: Stable{P},
sq_exists: ∃x:A [B[x]],
rless: x < y,
real: ℝ,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
rational-approx: (x within 1/n),
so_apply: x[s],
so_lambda: λ2x.t[x],
rge: x ≥ y
Lemmas referenced :
rv-orthogonal_wf,
Error :ss-eq_wf,
rv-0_wf,
rv-isometry_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
req_witness,
rv-norm_wf,
rv-sub_wf,
rv-ip_wf,
rv-isometry-implies-functional,
rv-orthogonal-isometry,
int-to-real_wf,
rv-mul_wf,
Error :ss-eq_functionality,
Error :ss-eq_weakening,
rv-mul0,
Error :ss-eq_inversion,
rv-orthogonal-iff-norm-preserving,
real_wf,
rv-add_wf,
rless_wf,
rless-int,
rdiv_wf,
rv-midpoint-unique,
req_functionality,
rdiv_functionality,
req_weakening,
rv-mul_functionality,
rv-0-add,
uiff_transitivity,
rv-add_functionality,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
rmul_wf,
rv-mul-mul,
req_transitivity,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rv-mul1,
istype-nat,
subtract-1-ge-0,
istype-less_than,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
itermMinus_wf,
rminus_wf,
itermAdd_wf,
rsub_wf,
radd_wf,
subtract_wf,
rv-mul-add,
radd_functionality,
req_inversion,
rsub-int,
radd-int,
squash_wf,
true_wf,
rminus-int,
real_term_value_add_lemma,
real_term_value_minus_lemma,
Error :ss-eq_transitivity,
nat_plus_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
nat_plus_properties,
nat_plus_subtype_nat,
rv-minus_wf,
rv-sub-is-zero,
rv-add-minus2,
rmul-int,
rv-add-comm,
rtermVar_wf,
rtermConstant_wf,
rtermDivide_wf,
rtermMultiply_wf,
assert-rat-term-eq2,
decidable__le,
istype-le,
int_term_value_minus_lemma,
rv-minus_functionality,
stable_req,
minimal-double-negation-hyp-elim,
false_wf,
rneq_wf,
not_wf,
req_wf,
rmul-int-rdiv,
minus-one-mul,
mul-associates,
one-mul,
req-same,
minimal-not-not-excluded-middle,
rv-norm-is-zero,
not-rneq,
rless_irreflexivity,
rless_transitivity1,
rv-norm-nonneg,
small-reciprocal-real,
rv-norm-triangle-inequality2,
rleq_wf,
iff_weakening_uiff,
rleq_functionality,
rv-norm_functionality,
rv-sub_functionality,
rv-mul-sub,
rabs_wf,
rv-norm-mul,
r-archimedean-implies,
rational-approx-property,
int_term_value_mul_lemma,
nequal_wf,
int_subtype_base,
int_formula_prop_eq_lemma,
intformeq_wf,
int-rdiv_wf,
rmul_functionality,
rabs_functionality,
rsub_functionality,
int-rdiv-req,
less_than_wf,
set_subtype_base,
rneq-int,
zero-rleq-rabs,
rational-approx_wf,
rabs-difference-symmetry,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rmul_functionality_wrt_rleq2,
rleq_transitivity,
rleq_weakening,
rv-sub0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
sqequalRule,
productIsType,
applyEquality,
because_Cache,
functionIsType,
instantiate,
independent_isectElimination,
lambdaEquality_alt,
productElimination,
independent_pairEquality,
voidElimination,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
isectIsTypeImplies,
natural_numberEquality,
baseClosed,
imageMemberEquality,
inrFormation_alt,
closedConclusion,
approximateComputation,
int_eqEquality,
Error :memTop,
dependent_pairFormation_alt,
intWeakElimination,
minusEquality,
addEquality,
imageElimination,
unionElimination,
equalityIstype,
multiplyEquality,
dependent_set_memberEquality_alt,
unionEquality,
functionEquality,
unionIsType,
intEquality,
sqequalBase,
baseApply,
inlFormation_alt
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}f:Point(rv) {}\mrightarrow{} Point(rv). (Orthogonal(f) \mLeftarrow{}{}\mRightarrow{} f 0 \mequiv{} 0 \mwedge{} Isometry(f))
Date html generated:
2020_05_20-PM-01_12_35
Last ObjectModification:
2020_01_08-AM-11_05_25
Theory : inner!product!spaces
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