Nuprl Lemma : rv-ip-add-squared
∀[rv:InnerProductSpace]. ∀[x,y:Point(rv)]. (x + y^2 = ((x^2 + (r(2) * x ⋅ y)) + y^2))
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-add: x + y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
req_witness,
rv-ip_wf,
rv-add_wf,
inner-product-space_subtype,
radd_wf,
rmul_wf,
int-to-real_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
req_functionality,
req_transitivity,
rv-ip-add,
radd_functionality,
rv-ip-add2,
req_weakening,
rv-ip-symmetry,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
natural_numberEquality,
independent_functionElimination,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType,
instantiate,
independent_isectElimination,
productElimination,
dependent_functionElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
voidElimination
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x,y:Point(rv)]. (x + y\^{}2 = ((x\^{}2 + (r(2) * x \mcdot{} y)) + y\^{}2))
Date html generated:
2020_05_20-PM-01_11_20
Last ObjectModification:
2019_12_09-PM-11_48_35
Theory : inner!product!spaces
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